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BEGIN:VEVENT
SUMMARY:Myrto Manolaki (University College Dublin)
DTSTART;VALUE=DATE-TIME:20200512T130000Z
DTEND;VALUE=DATE-TIME:20200512T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/1
DESCRIPTION:Title: Me
rgelyan-type theorems in several complex variables\nby Myrto Manolaki
(University College Dublin) as part of CAvid: Complex Analysis video semin
ar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Walter Bergweiler (Christian-Albrechts Universität Kiel)
DTSTART;VALUE=DATE-TIME:20200519T130000Z
DTEND;VALUE=DATE-TIME:20200519T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/2
DESCRIPTION:Title: En
tire solutions of linear q-difference equations\nby Walter Bergweiler
(Christian-Albrechts Universität Kiel) as part of CAvid: Complex Analysis
video seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katsuya Ishizaki (Open University of Japan)
DTSTART;VALUE=DATE-TIME:20200526T130000Z
DTEND;VALUE=DATE-TIME:20200526T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/3
DESCRIPTION:Title: Me
romorphic solutions of Fermat type equations\nby Katsuya Ishizaki (Ope
n University of Japan) as part of CAvid: Complex Analysis video seminar\n\
nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aimo Hinkkanen (University of Illinois)
DTSTART;VALUE=DATE-TIME:20200602T130000Z
DTEND;VALUE=DATE-TIME:20200602T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/4
DESCRIPTION:Title: A
determinant problem for a third order ODE\nby Aimo Hinkkanen (Universi
ty of Illinois) as part of CAvid: Complex Analysis video seminar\n\nLectur
e held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuefei Wang (Chinese Academy of Sciences\, Beijing)
DTSTART;VALUE=DATE-TIME:20200609T130000Z
DTEND;VALUE=DATE-TIME:20200609T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/5
DESCRIPTION:Title: On
the dynamics of entire functions with symmetry\nby Yuefei Wang (Chine
se Academy of Sciences\, Beijing) as part of CAvid: Complex Analysis video
seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre Eremenko (Purdue University)
DTSTART;VALUE=DATE-TIME:20200616T130000Z
DTEND;VALUE=DATE-TIME:20200616T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/6
DESCRIPTION:Title: Mo
duli spaces for Lamé functions\nby Alexandre Eremenko (Purdue Univers
ity) as part of CAvid: Complex Analysis video seminar\n\nLecture held in N
/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Luca (University College London)
DTSTART;VALUE=DATE-TIME:20200623T130000Z
DTEND;VALUE=DATE-TIME:20200623T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/7
DESCRIPTION:Title: Mi
xed boundary value problems for slow viscous flows: new transform methods
and applications\nby Elena Luca (University College London) as part of
CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract
\nMotivated by microfluidics applications where it is required to manipula
te viscous fluids at small scales\, we present new transform methods for s
olving mixed boundary value problems for biharmonic fields arising therein
. The new methods provide a unified general approach to finding quasi-anal
ytical solutions to a variety of technologically important problems of slo
w viscous flows and lead to fast and accurate schemes for evaluation of th
e solutions. In this talk\, we focus on problems in simply and multiply co
nnected domains\, with boundaries consisting of straight-line or circular
edges.\n
LOCATION:https://researchseminars.org/talk/CAvid/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zinelâabidine Latreuch (University of Mostaganem\, Algeria)
DTSTART;VALUE=DATE-TIME:20200630T130000Z
DTEND;VALUE=DATE-TIME:20200630T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/8
DESCRIPTION:Title: On
meromorphic solutions of non-linear differential equations of Tumura-Clun
ie type\nby Zinelâabidine Latreuch (University of Mostaganem\, Algeri
a) as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A
.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grigor Barsegian (National Academy of Sciences of Armenia)
DTSTART;VALUE=DATE-TIME:20200707T130000Z
DTEND;VALUE=DATE-TIME:20200707T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/9
DESCRIPTION:Title: A
new property of arbitrary complex polynomials\nby Grigor Barsegian (Na
tional Academy of Sciences of Armenia) as part of CAvid: Complex Analysis
video seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Wang (Fudan University\, China)
DTSTART;VALUE=DATE-TIME:20200714T130000Z
DTEND;VALUE=DATE-TIME:20200714T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/10
DESCRIPTION:Title: J
ulia limiting directions of meromorphic functions\nby Jun Wang (Fudan
University\, China) as part of CAvid: Complex Analysis video seminar\n\nLe
cture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phil Rippon (Open University\, UK)
DTSTART;VALUE=DATE-TIME:20200908T130000Z
DTEND;VALUE=DATE-TIME:20200908T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/11
DESCRIPTION:Title: C
onstructing entire functions of small order - motivated by complex dynamic
s\nby Phil Rippon (Open University\, UK) as part of CAvid: Complex Ana
lysis video seminar\n\nLecture held in N/A.\n\nAbstract\nIn 1989\, Eremenk
o conjectured that for any transcendental entire function the escaping set
$I(f) = \\{z:f^n(z)\\to\\infty \\text{ as } n\\to\\infty\\}$ has no bound
ed components -- despite much work this conjecture is still open.\n\nFor r
eal entire functions $f$ of finite order with only real zeros\, we have sh
own that Eremenko's conjecture holds if there exists $r>0$ such that the i
terated minimum modulus $m^n(r)\\to\\infty$ as $n\\to\\infty$. Here $m(r)=
\\min_{|z|=r}|f(z)|$.\n\nWe discuss examples of families of entire functio
ns of small order for which this iterated minimum modulus condition holds\
, and construct examples of functions of small order for which it does not
hold\, including examples based on a new development of a method due to K
jellberg.\n\n(Joint work with Dan Nicks and Gwyneth Stallard.)\n\nPlease e
-mail Rod Halburd (r.halburd@ucl.ac.uk) for the Zoom link. Please let him
know if you would like to receive weekly announcements about CAvid (the C
omplex Analysis video seminar series).\n
LOCATION:https://researchseminars.org/talk/CAvid/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuen-Wai Ng (Hong Kong University)
DTSTART;VALUE=DATE-TIME:20200915T130000Z
DTEND;VALUE=DATE-TIME:20200915T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/12
DESCRIPTION:Title: T
he squeezing function on doubly-connected domains via the Loewner differen
tial equation\nby Tuen-Wai Ng (Hong Kong University) as part of CAvid:
Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nInspi
red by the work of Liu\, Sun and Yau (2004) on holomorphic homogeneous reg
ular (HHR) domains and Yeung (2009)’s work on domains with uniform squee
zing property (another name for HHR domains)\, Deng\, Guan and Zhang (2012
) introduced a new biholomorphic invariant\, namely\, the squeezing functi
on for bounded domains in the n-dimensional complex Euclidean space. Since
then it has been one of the most active area in several complex variables
in recent years.\n\nOn the other hand\, until now\, there is only one exp
licit example of non-constant squeezing functions\, namely the squeezing f
unction of the punctured ball. In this talk\, we will establish an explici
t formula for the squeezing functions of annuli and hence (up to biholomor
phisms) for any doubly connected planar domain. The main tools used to pro
ve this result are the Schottky-Klein prime function (following the work
of Crowdy) and a version of the Loewner differential equation on annuli du
e to Komatu. We will also show that these results can be used to obtain lo
wer bounds on the squeezing function for certain product domains in the n-
dimensional complex Euclidean space.\n\nThis is a joint work with Chiu Cha
k Tang and Jonathan Tsai.\n
LOCATION:https://researchseminars.org/talk/CAvid/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lukas Geyer (Montana State University)
DTSTART;VALUE=DATE-TIME:20200922T130000Z
DTEND;VALUE=DATE-TIME:20200922T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/13
DESCRIPTION:Title: G
ravitational lensing and critically fixed anti-rational maps\nby Lukas
Geyer (Montana State University) as part of CAvid: Complex Analysis video
seminar\n\nLecture held in N/A.\n\nAbstract\nStudying the dynamics of ant
i-rational maps\, i.e.\, complex conjugates of rational maps\, is a subjec
t closely related to holomorphic dynamics\, with intriguing connections to
problems in gravitational lensing. In particular\, the lens equation for
a single-plane gravitational lens made up of N point masses is known to be
a fixed point equation for an anti-rational map of degree N. These fixed
points are apparent images of a single (point) light source\, and it is kn
own from work of Rhie (2003) and Khavinson and Neumann (2006) that for N>1
there can be at most 5N-5 such images\, and that this bound is sharp.\n\n
Originally motivated by the goal of classifying maximal lensing configurat
ions\, i.e.\, configurations for which the bound 5N-5 is attained\, we rec
ently succeeded in giving a complete classification of anti-rational maps
for which all critical points are fixed\, through simple topological model
s associated with certain planar graphs. We will explain this classificati
on\, the main ideas in the proof\, and how this yields a partial classific
ation and new examples of maximal lensing configurations. Finally\, we wil
l discuss some open problems and questions.\n
LOCATION:https://researchseminars.org/talk/CAvid/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Norbert Steinmetz (Technische Universität Dortmund)
DTSTART;VALUE=DATE-TIME:20200929T130000Z
DTEND;VALUE=DATE-TIME:20200929T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/14
DESCRIPTION:Title: L
aplace contour integrals and linear differential equations\nby Norbert
Steinmetz (Technische Universität Dortmund) as part of CAvid: Complex An
alysis video seminar\n\nLecture held in N/A.\n\nAbstract\nAny linear diffe
rential equation with coefficients of degree one\n$$w^{(n)}+\\sum_{j=0}^{n
-1}(a_j+b_jz)w^{(j)}=0$$\nhas solutions that may be represented as\nLaplac
e contour integrals\n$$f(z)=\\frac1{2\\pi i}\\int_C\\phi(t)e^{-zt}\\\,dt.$
$\nWe will discuss the main properties of\nthese solutions and determine t
heir order of growth\, asymptotics\, Phragm\\'en-Lindel\\"of indicator\, d
istribution of zeros\,\nNevanlinna functions $T(r\,f)$ and $N(r\,1/f)$\, a
nd the existence of sub-normal and polynomial solutions.\n
LOCATION:https://researchseminars.org/talk/CAvid/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Nowak (Maria Curie-Skłodowska University\, Poland)
DTSTART;VALUE=DATE-TIME:20201006T130000Z
DTEND;VALUE=DATE-TIME:20201006T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/15
DESCRIPTION:Title: O
n kernels of Toeplitz operators\nby Maria Nowak (Maria Curie-Skłodows
ka University\, Poland) as part of CAvid: Complex Analysis video seminar\n
\nLecture held in N/A.\n\nAbstract\nLet $H^2$ denote the standard Hardy sp
ace on the unit disk $\\mathbb\nD$ and let $\\mathbb T=\\partial \\mathbb
D$. Every $f(z)=\\sum_{n=0}^{\\infty}a_nz^n\\in H^2$ has a nontangential l
imit $f(e^{i\\theta})$ a.e. on $\\mathbb {T}=\\partial\\mathbb {D}$ and th
is boundary function $f(e^{i\\theta})$ is in $L^2(\\mathbb {T})$.\nFurthe
rmore\, if $\\{c_n\\}$ are Fourier coefficients of $f(e^{i\\theta})$ then
$c_n=a_n$ for $n\\geq 0$ and $c_n=0$ for $n<0$.\n Actually\, the space $H^
2$ can be identified with a closed subspace of\n$L^2(\\mathbb {T})$ whos
e Fourier coefficients with negative indices vanish.\n\n\nFor $\\varphi\\i
n\nL^{\\infty}(\\mathbb T)$ the Toeplitz operator $T_{\\varphi}$ on $H^2$
is given by\n$T_{\\varphi}f=P_{+}(\\varphi f)$\, where $P_{+}$ is the orth
ogonal\nprojection of $L^2(\\mathbb T)$ onto $H^2$. It is a consequence o
f Hitt's Theorem that\n$\\ker T_{\\varphi}= fK_I$\, where $K_I= H^2\\omin
us IH^2$\nis the model space corresponding to the inner function $I$ such
that\n$I(0)=0$ and $f$ is an outer function of unit $H^2$ norm that\nacts
as an isometric multiplier from $K_I$ onto $f K_{I}$.\nHowever\, not all
spaces $fK_{I}$\, where $f$ and $K_I$ are as above\, can be kernels o
f Toeplitz operators.\nThe sufficient and necessary condition for the spac
e $fK_I$ to be the kernel of a Toeplitz operator was given by E. Hayashi (
1990).\nIn 1994 D. Sarason gave another proof of this condition based on d
e Branges-Rovnyak spaces theory.\nIf $M= fK_I$ is a kernel of a Toeplitz
operator\, then also we have $M=\\ker T_{\\frac{\\overline{If}}{f}}$\nIn t
he talk we consider the case when $fK_I\\varsubsetneq \\ker T_{\\frac{\\ov
erline{If}}{f}}$ and try to describe\nthe space $\\ker T_{\\frac{\\bar I\\
bar f}{f}}\\ominus fK_I$. We use Sarason's approach and the structure
of de Branges-Rovnyak spaces generated by nonextreme functions.\n\nThe ta
lk is based on joint work with P. Sobolewski\, A.\nSo{\\l}tysiak and M. Wo
{\\l}oszkiewicz-Cyll.\n
LOCATION:https://researchseminars.org/talk/CAvid/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shamil Makhmutov (Sultan Qaboos University\, Oman)
DTSTART;VALUE=DATE-TIME:20201013T130000Z
DTEND;VALUE=DATE-TIME:20201013T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/16
DESCRIPTION:Title: G
rowth estimates for meromorphic solutions of higher order algebraic differ
ential equations\nby Shamil Makhmutov (Sultan Qaboos University\, Oman
) as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.
\n\nAbstract\nPointwise growth estimates for the spherical derivative of s
olutions of the first order algebraic differential equations are obtained.
\nA generalization of this result to higher order equations is also given
. \nWe discuss the related question of when for a given class X of meromor
phic functions in the unit disc\, \ndefined by means of the spherical deri
vative and integer $n$\, $n>1$\, condition $f^n \\in X$ implies $f \\in X$
. \nAn affirmative answer to this is given in the case of UBC and some oth
er classes. \nHowever\, there are classes when the answer is negative.\n
LOCATION:https://researchseminars.org/talk/CAvid/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Min Ru (University of Houston)
DTSTART;VALUE=DATE-TIME:20201020T130000Z
DTEND;VALUE=DATE-TIME:20201020T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/17
DESCRIPTION:Title: R
ecent developments in Nevanlinna theory and Diophantine approximation\
nby Min Ru (University of Houston) as part of CAvid: Complex Analysis vide
o seminar\n\nLecture held in N/A.\n\nAbstract\nIn this talk\, I'll survey
the recent results in Nevanlinna theory and Diophantine approximation. I'
ll focus on the extension of H. Cartan's Second Main Theorem in Nevanlinna
theory.\n
LOCATION:https://researchseminars.org/talk/CAvid/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Catherine Bénéteau (University of South Florida)
DTSTART;VALUE=DATE-TIME:20201027T130000Z
DTEND;VALUE=DATE-TIME:20201027T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/18
DESCRIPTION:Title: A
survey of optimal polynomial approximants and connections to digital filt
ers\nby Catherine Bénéteau (University of South Florida) as part of
CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\
nIn this talk\, I will discuss the notion of optimal polynomial approximan
ts\, which are polynomials that approximate\, in some sense\, inverses of
functions in certain Hilbert spaces of analytic functions. In the last 10
years\, a number of papers have appeared examining the zeros of these poly
nomials\, rates of convergence\, efficient algorithms for their computatio
n\, and connections to orthogonal polynomials and reproducing kernels\, am
ong other topics. On the other hand\, in the 70s\, researchers in engineer
ing and applied mathematics introduced least squares inverses in the conte
xt of digital filters in signal processing. It turns out that in the Hardy
space $H^2$ the optimal polynomial approximants and the least squares inv
erses are identical. In this talk\, I will survey results related to the z
eros of optimal polynomial approximants and implications for the design of
ideal digital filters. This talk is based on a preprint of a survey paper
that is joint with Ray Centner.\n
LOCATION:https://researchseminars.org/talk/CAvid/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dierk Schleicher (Aix–Marseille Université)
DTSTART;VALUE=DATE-TIME:20201103T140000Z
DTEND;VALUE=DATE-TIME:20201103T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/19
DESCRIPTION:Title: F
inding polynomial roots using complex analysis\, dynamical systems\, compu
ter algebra\nby Dierk Schleicher (Aix–Marseille Université) as part
of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstr
act\nOne of the classical problems in all areas of mathematics is to find
roots of complex polynomials. It is well known that this can be done only
by methods of approximation. We discuss three classical methods: the Newto
n\, Weierstrass\, and Ehrlich-Aberth methods\; these are complex analytic
maps that\, under iteration\, are supposed to converge to one root\, resp.
all roots of the polynomial. Locally\, these methods converge fast\, but
the global dynamical properties are hard to describe.\n\nWe introduce thes
e complex analytic dynamical systems and describe recent progress towards
their global dynamical properties. In particular\, the Newton and Weierstr
ass methods are not globally convergent: for open sets of polynomials ther
e are open sets of initial points that fail to converge to roots. Moreover
\, for Weierstrass and Ehrlich-Aberth\, there are orbits that are always d
efined and converge\, but not to roots. For Newton\, there is meanwhile a
rich theory about its global dynamics\, but there are many open questions
for all these methods.\n\nMuch of this is joint work with members of my ER
C team\, in particular my PhD student Bernhard Reinke\, as well as with co
lleagues.\n
LOCATION:https://researchseminars.org/talk/CAvid/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nick Trefethen (University of Oxford)
DTSTART;VALUE=DATE-TIME:20201110T140000Z
DTEND;VALUE=DATE-TIME:20201110T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/20
DESCRIPTION:Title: A
pproximation on complex domains and Riemann surfaces\nby Nick Trefethe
n (University of Oxford) as part of CAvid: Complex Analysis video seminar\
n\nLecture held in N/A.\n\nAbstract\nLet f be a function analytic on a clo
sed Jordan region E apart\nfrom a finite number of branch point singularit
ies on the boundary.\nWe show how f can be approximated by rational functi
ons on E with\nroot-exponential convergence\, i.e.\, errors $O(\\exp(-C \\
sqrt n))$ with\n$C>0$. Such approximations lead to "lightning solvers" fo
r Laplace\nproblems in planar domains. Then we move to "reciprocal-log" o
r\n"log-lightning" approximations involving terms of the form\n$c/(\\log(z
-z_k) - s_k)$. Now one gets exponential-minus-log convergence\,\ni.e.\, $
O(\\exp(-C n/\\log n))$. Moreover\, the reciprocal-log functions\ncan be
analytically continued around the branch points to provide\napproximation
on further Riemann sheets. This work (with Yuji\nNakatsukasa) is very new
\, and there are many open questions.\n
LOCATION:https://researchseminars.org/talk/CAvid/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Núria Fagella (University of Barcelona)
DTSTART;VALUE=DATE-TIME:20201117T140000Z
DTEND;VALUE=DATE-TIME:20201117T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/21
DESCRIPTION:Title: W
andering in complex dynamics\nby Núria Fagella (University of Barcelo
na) as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/
A.\n\nAbstract\nIn a holomorphic dynamical system a wandering domain is a
component of the stable (or normal) set whose iterates never meet. This ty
pe of components only exist in the presence of essential singularities and
are the most unknown among all the possible kinds. In this talk I will ex
plain what is and is not known about wandering domains and some of the mos
t recent progress\, which relates wandering dynamics to sequences of holom
orphic functions on the unit disk (non-autonomous dynamics).\n
LOCATION:https://researchseminars.org/talk/CAvid/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gautam Bharali (Indian Institute of Science\, Bangalore)
DTSTART;VALUE=DATE-TIME:20201124T140000Z
DTEND;VALUE=DATE-TIME:20201124T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/22
DESCRIPTION:Title: T
he Wolff-Denjoy theorem beyond the unit disc\nby Gautam Bharali (India
n Institute of Science\, Bangalore) as part of CAvid: Complex Analysis vid
eo seminar\n\nLecture held in N/A.\n\nAbstract\nThe Wolff-Denjoy theorem h
as been the motivation for a host of results that resemble the classical t
heorem for holomorphic self-maps of the unit disc. In this talk\, we shall
look at yet another result in this class. This result applies to a rather
general class of bounded domains in one and higher dimensions\, which may
have rough boundaries and aren't necessarily contractible. While our tech
niques are motivated by the properties of holomorphic maps in several comp
lex variables\, the theory of such maps turns out to be incidental to thes
e techniques. In fact\, in this talk\, we shall spend some time examining
certain analogies between the Poincaré distance and the Hilbert distance
on convex domains. This is relevant as there exists a Wolff--Denjoy-type t
heorem\, by Beardon\, in the latter setting. It is these analogies that gi
ve rise to the fundamental concept that underlies our result(s): namely\,
a weak notion of negative curvature for spaces equipped with the Kobayashi
distance (of which the Poincaré distance is a special case). No knowledg
e of several complex variables will be assumed in this talk: indeed\, most
of the discussion will focus on basic complex analysis and on the propert
ies of metric spaces and contractive maps. A large part of this talk will
be based on joint work with Andrew Zimmer and Anwoy Maitra.\n
LOCATION:https://researchseminars.org/talk/CAvid/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Gauthier (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20201201T140000Z
DTEND;VALUE=DATE-TIME:20201201T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/23
DESCRIPTION:Title: A
symptotic first boundary value problem for holomorphic functions of sever
al complex variables\nby Paul Gauthier (Université de Montréal) as p
art of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAb
stract\n(Jointly with M. Shirazi)\n\nLet $M$ be a complex manifold endowed
with a distance $d$ and let $U\\subset M$ be an arbitrary Stein domain. L
et $\\mu$ be a regular Borel measure on $U\,$ such that non-empty open set
s of $U$ have positive $\\mu$ measure and $\\nu$ a regular Borel measure o
n $\\partial U.$ Let $\\psi$ be a \nBorel measurable function on $\\partia
l U\,$ \nwhose restriction to some closed subset $S\\subset\\partial U$ i
s continuous. \nThen\, \nthere exists a holomorphic function $f$ on $U\
,$ such that\, for $\\nu$-almost every $p\\in \\partial U$\, \nand for ev
ery $p\\in S\,$ $f(x)\\to \\psi(p)$\, as $x\\to p$ outside a set of $\\mu$
-density zero at $p$ \nrelative to $U.$\n
LOCATION:https://researchseminars.org/talk/CAvid/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kerstin Jordaan (University of South Africa)
DTSTART;VALUE=DATE-TIME:20201208T140000Z
DTEND;VALUE=DATE-TIME:20201208T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/24
DESCRIPTION:Title: A
characterisation of Askey-Wilson polynomials and the indeterminate moment
problem associated with a limiting case\nby Kerstin Jordaan (Universi
ty of South Africa) as part of CAvid: Complex Analysis video seminar\n\nLe
cture held in N/A.\n\nAbstract\n(Joint work with M. Kenfack Nangho)\n\nIn
this talk I will complete and prove a conjecture concerning a characterisi
ng relation for Askey-Wilson orthogonal polynomials and study a limiting c
ase of Askey-Wilson polynomials when one of the parameters goes to infinit
y. Solutions to the associated indeterminate moment problem are found and
an orthogonality relation is established.\n
LOCATION:https://researchseminars.org/talk/CAvid/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abhijit Banerjee (University of Kalyani\, India)
DTSTART;VALUE=DATE-TIME:20201215T140000Z
DTEND;VALUE=DATE-TIME:20201215T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/25
DESCRIPTION:Title: A
survey on different uniqueness and strong uniqueness polynomials and thei
r generating unique range sets\nby Abhijit Banerjee (University of Kal
yani\, India) as part of CAvid: Complex Analysis video seminar\n\nLecture
held in N/A.\n\nAbstract\nThe notion of unique range sets was introduced b
y Gross-Yang [Proc. Japan Acad.\, 58 (1982)\, 17-20]. Since the inception
of the definition\, it became an interesting topic for the researchers to
study. In course of time\, the research had been shifted to-wards the char
acterizations of the polynomial backbone of concerned sets. As a result\,
the uniqueness and strong uniqueness polynomial appeared in the literature
and made a lusting impression. \n\nIn 2000\, H. Fujimoto [H. Fujimoto\, O
n uniqueness of meromorphic functions sharing finite sets\, Amer. J. Math.
\, 122 (2000)\, 1175-1203.] first discovered a special property of a polyn
omial\, called it as “property (H)” which played a vital role in the r
esearch of uniqueness and strong uniqueness polynomial.\n\nWithin the real
m of Nevanlinna theory\, we wish to elaborately characterize the existing
uniqueness as well as strong uniqueness polynomials\, the relation between
them and their contribution in forming the unique range sets under relaxe
d sharing hypothesis. We also wish to discuss the scope for future researc
h and intend to present our humble contribution in this aspect.\n
LOCATION:https://researchseminars.org/talk/CAvid/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mónica Moreno Rocha (Centro de Investigación en Matemáticas\, M
exico)
DTSTART;VALUE=DATE-TIME:20210119T140000Z
DTEND;VALUE=DATE-TIME:20210119T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/26
DESCRIPTION:Title: H
erman rings of elliptic functions\nby Mónica Moreno Rocha (Centro de
Investigación en Matemáticas\, Mexico) as part of CAvid: Complex Analysi
s video seminar\n\nLecture held in N/A.\n\nAbstract\nConsider the family o
f iterates of a rational or transcendental meromorphic function $f$. A com
ponent of normality that is invariant under some $n$-iterate of $f$ is cal
led a Herman ring if over such a component\, $f^n$ is conformally conjugat
e to an irrational rotation acting on an annulus of finite conformal modul
us. In that case\, the positive iterates of the Herman ring form a cycle.
Showing the existence of cycles of Herman rings for meromorphic functions
is not an easy task\, and when they exist\, it is natural to ask oneself i
f an upper bound for the number of cycles is achievable.\n\nIn the late 19
80s Shishikura introduced the theory of quasiconformal surgery to construc
t examples of rational maps with cycles of Herman rings while also showing
that a rational map of degree d has at most d-2 cycles (thus\, rational m
aps of degree 2 cannot have Herman rings). In the case of elliptic functio
ns\, Hawkins & Koss showed in 2004 that the Weierstrass P function\, defin
ed over any given lattice\, cannot have cycles of Herman rings. This resul
t motivated the question of the existence of Herman rings for elliptic fun
ctions in terms of their order. In this talk\, I will present recent resul
ts obtained through the implementation of Shishikura’s surgery technique
s to the elliptic case. First\, we’ll see that Herman rings can be reali
zable by elliptic functions of order at least 3\, and in particular\, orde
r 2 elliptic functions cannot have cycles of Herman rings. Then\, I will p
resent an upper bound for the number of invariant Herman rings in terms of
the order of the elliptic function and show how to refine that bound usin
g the multiplicity of poles.\n
LOCATION:https://researchseminars.org/talk/CAvid/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Dyakonov (ICREA & Universitat de Barcelona\, Spain)
DTSTART;VALUE=DATE-TIME:20210202T140000Z
DTEND;VALUE=DATE-TIME:20210202T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/28
DESCRIPTION:Title: L
acunary polynomials in $L^1$: geometry of the unit sphere\nby Konstant
in Dyakonov (ICREA & Universitat de Barcelona\, Spain) as part of CAvid: C
omplex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nLet $\\
Lambda$ be a finite set of nonnegative integers\, and let $\\mathcal P(\\L
ambda)$ be the linear hull of the monomials $z^k$ with $k\\in\\Lambda$\, v
iewed as a subspace of $L^1$ on the unit circle. We characterize the extre
me and exposed points of the unit ball in $\\mathcal P(\\Lambda)$.\n
LOCATION:https://researchseminars.org/talk/CAvid/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linda Keen (CUNY\, USA)
DTSTART;VALUE=DATE-TIME:20210209T140000Z
DTEND;VALUE=DATE-TIME:20210209T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/29
DESCRIPTION:Title: P
arameter spaces of families of transcendental functions\nby Linda Keen
(CUNY\, USA) as part of CAvid: Complex Analysis video seminar\n\nLecture
held in N/A.\n\nAbstract\nThis lecture is based on joint work with Tao Che
n\, Nuria Fagella and Yunping Jiang. It is part of a more general program
to understand parameter spaces of transcendental maps.\n\nIf we perturb a
rational function by a topological conjugacy we obtain a rational function
\, so the dynamics depend on the coefficients\, which therefore form a nat
ural parameter space. It is not true that there is a natural way of parame
terizing general families of transcendental functions so that a perturbati
on of the function remains in the family. This makes it difficult to descr
ibe how the dynamics varies across these families. We will look at two exa
mples of reasonably general families of transcendental meromorphic functio
ns where one can overcome these difficulties. What this means is that we w
ill be able to describe the properties of the components defined by the bi
furcation locus. We will see at the end how these examples fit into the la
rger program.\n
LOCATION:https://researchseminars.org/talk/CAvid/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Demina (National Research University Higher School of Econom
ics\, Russia)
DTSTART;VALUE=DATE-TIME:20210216T140000Z
DTEND;VALUE=DATE-TIME:20210216T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/30
DESCRIPTION:Title: A
lgebraic invariants\, integrability\, and meromorphic solutions\nby Ma
ria Demina (National Research University Higher School of Economics\, Russ
ia) as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/
A.\n\nAbstract\nConsider an autonomous algebraic ordinary differential equ
ation of order higher than one. The aim of the talk is to address the foll
owing questions.\n\n1. Does there exist an autonomous algebraic first-orde
r ordinary differential equation compatible with the original equation?\n\
n2. If yes\, how to find all such equations?\n \n\nBivariate polynomials p
roducing autonomous algebraic first-order ordinary differential equations
compatible with the equation under consideration are called algebraic inva
riants. The main difficulty in deriving irreducible algebraic invariants l
ies in the fact that the degrees of related bivariate polynomials are not
known in advance.\n\nAlgebraic invariants are important from theoretical a
nd practical point of views. In the two-dimensional case algebraic invaria
nts are key objects in establishing Darboux and Liouvillian integrability
of the original ordinary differential equation. In addition\, algebraic in
variants can be used to perform the classification of W-meromorphic soluti
ons of ordinary differential equations. We shall pay some attention to the
se applications.\n
LOCATION:https://researchseminars.org/talk/CAvid/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Risto Korhonen (University of Eastern Finland)
DTSTART;VALUE=DATE-TIME:20210323T130000Z
DTEND;VALUE=DATE-TIME:20210323T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/31
DESCRIPTION:Title: D
elay differential equations and Nevanlinna theory\nby Risto Korhonen (
University of Eastern Finland) as part of CAvid: Complex Analysis video se
minar\n\nLecture held in N/A.\n\nAbstract\nThe idea that the existence of
sufficiently many finite-order meromorphic solutions could be used to sing
le out difference Painlevé equations was introduced by Ablowitz\, Halburd
and Herbst. In this talk necessary conditions are obtained for certain ty
pes of delay differential equations to admit a transcendental meromorphic
solution of hyper-order less than one. The equations obtained include dela
y Painlevé equations and equations solvable by elliptic functions. We con
clude with recent results on the existence of transcendental meromorphic s
olutions of first-order difference equations\, without growth conditions.\
n
LOCATION:https://researchseminars.org/talk/CAvid/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Chyzhykov (University of Warmia and Mazury\, Poland)
DTSTART;VALUE=DATE-TIME:20210223T140000Z
DTEND;VALUE=DATE-TIME:20210223T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/32
DESCRIPTION:Title: I
rregular solutions of complex linear differential equations in the unit di
sc\nby Igor Chyzhykov (University of Warmia and Mazury\, Poland) as pa
rt of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbs
tract\nIt is shown that the order and the lower order of growth are equal
for all non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coeffi
cient $A$ is analytic in the unit disc and $\\log^+ M(r\,A)/\\log(1-r)$ te
nds to a~finite limit as $r\\to 1^-$.\nA~family of examples is constructe
d\, where the order of solutions remain the same while the lower order may
vary on a~certain interval depending on the irregular growth of the coeff
icient.\nThese coefficients emerge as the logarithm of their modulus appro
ximates smooth radial subharmonic functions of prescribed irregular growth
on a~sufficiently large subset of the unit disc.\nA~result describing the
phenomenon behind these examples is also established. En route to\nresul
ts of general nature\, a~new sharp logarithmic derivative estimate involvi
ng the lower order of growth is discovered.\nIn addition to these estimate
s\,\narguments used are based\, in particular\, on the Wiman-Valiron theor
y adapted for the lower order.\n
LOCATION:https://researchseminars.org/talk/CAvid/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paweł Wójcicki (Warsaw University of Technology\, Poland)
DTSTART;VALUE=DATE-TIME:20210302T140000Z
DTEND;VALUE=DATE-TIME:20210302T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/33
DESCRIPTION:Title: O
n an Invariant distance induced by the Szego kernel and its applications\nby Paweł Wójcicki (Warsaw University of Technology\, Poland) as part
of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstr
act\nThe aim of my talk is to recall the notion of the so called Szego ker
nel and provide some new biholomorphic invariant by means of it. In fact\
, it is defined on a similar way as the co called Skwarczyński distance b
y means of the Bergman kernel. The relationship between completeness in
both cases will be examined. It turns out that the new biholomorphic inva
riant gives rise to some other new invariant\, by means of which we can es
timate the so called Bergman metric by means of the so called Szego metric
.\n\nThis is a joint work with Professor Steven Krantz (WUST\, MO\, USA)\n
LOCATION:https://researchseminars.org/talk/CAvid/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chengfa Wu (Shenzhen University\, China)
DTSTART;VALUE=DATE-TIME:20210316T130000Z
DTEND;VALUE=DATE-TIME:20210316T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/34
DESCRIPTION:Title: E
lliptic functions and their applications in complex differential equations
\nby Chengfa Wu (Shenzhen University\, China) as part of CAvid: Comple
x Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nThis talk fo
cuses on the applications of elliptic functions in complex differential eq
uations. First\, we discuss classifications of meromorphic solutions of ce
rtain autonomous complex differential equations. In particular\, we will f
ocus on the Loewy factorizable algebraic ODEs. Then we move to the study o
f the autonomous Schwarzian differential equations (SDEs). Ishizaki showed
that there are six canonical types of autonomous SDEs that have transcend
ental meromorphic solutions. We will construct all transcendental meromorp
hic solutions of five canonical types explicitly. In particular\, the solu
tions of four types are shown to be elliptic functions. Also\, all transce
ndental meromorphic solutions that possess a Picard exceptional value are
characterized for the remaining canonical type. This talk is based on join
t works with Ng and Liao respectively.\n
LOCATION:https://researchseminars.org/talk/CAvid/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sushil Gorai (Indian Institute of Science Education and Research K
olkata)
DTSTART;VALUE=DATE-TIME:20210330T130000Z
DTEND;VALUE=DATE-TIME:20210330T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/35
DESCRIPTION:Title: P
olynomial convexity and real surfaces with singularity\nby Sushil Gora
i (Indian Institute of Science Education and Research Kolkata) as part of
CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\
nIn this talk I will first discuss briefly about polynomial convexity and
its application in polynomial approximations. Then\, I will discuss the qu
estions of polynomial convexity and approximation on compact subsets of a
couple of classes of real surfaces in $\\mathbb{C}^2$ with singularity\, n
amely\, the union of three totally real subspaces and surfaces with isolat
ed CR-singularity.\n
LOCATION:https://researchseminars.org/talk/CAvid/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Caterina Stoppato (Università di Firenze\, Italy)
DTSTART;VALUE=DATE-TIME:20210309T140000Z
DTEND;VALUE=DATE-TIME:20210309T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/36
DESCRIPTION:Title: R
egularity in one hypercomplex variable\nby Caterina Stoppato (Universi
tà di Firenze\, Italy) as part of CAvid: Complex Analysis video seminar\n
\nLecture held in N/A.\n\nAbstract\nSince the 1930s\, several function the
ories have been introduced over the algebra of quaternions and other alter
native algebras. The aim of such constructions is to recover in higher dim
ensions the refined tools available in the theory of holomorphic functions
of one complex variable. The peculiar properties of the higher-dimensiona
l algebras considered are reflected in the different theories introduced.\
n\nA relatively recent breakthrough was the introduction of the class of s
lice regular functions of one quaternionic variable by Gentili and Struppa
in 2006. This study\, generalized to alternative $*$-algebras by Ghiloni
and Perotti in 2011\, has rapidly developed into a full-fledged theory.\n\
nThe talk will overview the general problem of function theory in one hype
rcomplex variable\, the main features of the theory of slice regular funct
ions and its applications to open problems from other areas of mathematics
.\n
LOCATION:https://researchseminars.org/talk/CAvid/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yueyang Zhang (University of Science and Technology Beijing)
DTSTART;VALUE=DATE-TIME:20210420T130000Z
DTEND;VALUE=DATE-TIME:20210420T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/37
DESCRIPTION:Title: O
n entire function $e^{p(z)}\\int_0^{z}\\beta(t)e^{-p(t)}dt$ with applicati
ons to Tumura--Clunie equations and complex dynamics\nby Yueyang Zhang
(University of Science and Technology Beijing) as part of CAvid: Complex
Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nLet $p(z)$ be
a non-constant polynomial and $\\beta(z)$ be a small entire function of $e
^{p(z)}$ in the sense of Nevanlinna. By using the classical Phragm\\'{e}n-
-Lindel\\"{o}f theorem\, we analyze the growth behavior of the entire func
tion $H(z):=e^{p(z)}\\int_0^{z}\\beta(t)e^{-p(t)}dt$ on the complex plane
$\\mathbb{C}$. We then apply these results to Tumura--Clunie type differen
tial equation $f(z)^n+P(z\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$\, where $
b_1(z)$ and $b_2(z)$ are non-zero polynomials\, $p_1(z)$ and $p_2(z)$ are
two polynomials of the same degree~$k\\geq 1$ and $P(z\,f)$ is a different
ial polynomial in $f$ of degree $\\leq n-1$ with meromorphic functions of
order less than~$k$ as coefficients\, and precisely characterize entire so
lutions of this equation. This gives an answer to a problem in the literat
ure and allows to find all zero-free solutions of the second-order differe
ntial equation $f''-(b_1e^{p_1}+b_2e^{p_2}+b_3)f=0$\, where $b_3$ is a pol
ynomial. We also use the Phragm\\'{e}n--Lindel\\"{o}f theorem to prove a t
heorem on certain first-order non-homogeneous linear differential equation
related to complex dynamics.\n
LOCATION:https://researchseminars.org/talk/CAvid/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chiara de Fabritiis (Università Politecnica delle Marche\, Italy)
DTSTART;VALUE=DATE-TIME:20210427T130000Z
DTEND;VALUE=DATE-TIME:20210427T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/38
DESCRIPTION:Title: *
-products\, *-exponential\, *-logarithm: some peculiarities of slice regul
ar functions on the quaternions\nby Chiara de Fabritiis (Università P
olitecnica delle Marche\, Italy) as part of CAvid: Complex Analysis video
seminar\n\nLecture held in N/A.\n\nAbstract\nSlice regular functions on qu
aternions were introduced in 2006 by Gentili and Struppa in order to gener
alize the notion of holomorphic functions on complex numbers (for an effec
tive introduction you can refer to C. Stoppato's seminar (https://mediace
ntral.ucl.ac.uk/Play/59248/). The theory had a quick development in severa
l directions by many authors\, in this talk I will focus on three unexpect
ed behaviours of these functions. The first aspect we deal with is the *-p
roduct\, which is the analogous of pointwise product for holomorphic funct
ions\; in particular we give an interpretation of this operation via two o
perators which resemble the scalar product and the vector product on R^3.
The second point we investigate is a suitable extension of the notion of e
xponential of a slice regular function\, namely the *-exponential exp_*(f)
(originally introduced by Colombo\, Sabadini and Struppa)\; we will descr
ibe some of its features\, especially with regard to the non-commutativity
of the *-product and to its connections with *-sine and *-cosine. Lastly\
, we study the possible existence and uniqueness of a *-logarithm of a nev
er vanishing slice regular function\, both on slice and on product domains
of the quaternions. We give some existence and non-existence results for
*-logarithm of never-vanishing slice regular functions (according to the s
plitting in real and vectorial part) and an accurate description of the po
ssible uniqueness of the *-logarithm.\nThis is a joint work with Amedeo Al
tavilla (Università di Bari).\n
LOCATION:https://researchseminars.org/talk/CAvid/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oliver Roth (University of Würzburg)
DTSTART;VALUE=DATE-TIME:20210504T130000Z
DTEND;VALUE=DATE-TIME:20210504T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/39
DESCRIPTION:Title: A
new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps\nby Oliver Roth (University of Würzburg) as part of CAvid: Complex Ana
lysis video seminar\n\nLecture held in N/A.\n\nAbstract\nWe establish seve
ral invariant boundary versions of the (infinitesimal) Schwarz-Pick lemma
for conformal pseudometrics on the unit disk and for holomorphic selfmaps
of strongly convex domains in CN in the spirit of the boundary Schwarz lem
ma of Burns-Krantz. Firstly\, we focus on the case of the unit disk and pr
ove a general boundary rigidity theorem for conformal pseudometrics with v
ariable curvature. In its simplest cases this result already includes new
types of boundary versions of the lemmas of Schwarz-Pick\, Ahlfors-Schwarz
and Nehari-Schwarz. The proof is based on a new Harnack-type inequality a
s well as a boundary Hopf lemma for conformal pseudometrics which extend e
arlier interior rigidity results of Golusin\, Heins\, Beardon\, Minda and
others. Secondly\, we prove similar rigidity theorems for sequences of con
formal pseudometrics\, which even in the interior case appear to be new. F
or instance\, a first sequential version of the strong form of Ahlfors' le
mma is obtained. As an auxiliary tool we establish a Hurwitz-type result a
bout preservation of zeros of sequences of conformal pseudometrics. Thirdl
y\, we apply the one-dimensional sequential boundary rigidity results toge
ther with a variety of techniques from several complex variables to prove
a boundary version of the Schwarz-Pick lemma for holomorphic maps of stron
gly convex domains in $\\C^N$ for $N>1$.\n\n(This is joint work with Filip
po Bracci and Daniela Kraus)\n
LOCATION:https://researchseminars.org/talk/CAvid/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Margaret Stawiska-Friedland\\ (American Mathematical Society/Mathe
matical Reviews\, USA)
DTSTART;VALUE=DATE-TIME:20210511T130000Z
DTEND;VALUE=DATE-TIME:20210511T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/40
DESCRIPTION:Title: A
potential-theoretic characterization of polynomials in holomorphic dynami
cs in one variable}\nby Margaret Stawiska-Friedland\\ (American Mathe
matical Society/Mathematical Reviews\, USA) as part of CAvid: Complex Anal
ysis video seminar\n\nLecture held in N/A.\n\nAbstract\nIn 1960s Hans Brol
in initiated systematic application of potential-theoretic methods in the
dynamics of holomorphic maps. Among other things\, he proved the now-famou
s equidistribution theorem: for a complex polynomial $f$ of degree greater
than $1$ the preimages\, under successive iterates of $f$\, of a Dirac me
asure at an arbitrary point of the complex plane (except at most two so-ca
lled exceptional points) converge weakly to the equilibrium measure (with
pole at infinity) for the Julia set $J_f$ of $f$. To a general rational ma
p $f$ of degree $d \\geq 2$ on the Riemann sphere $\\mathbb{CP}^1$ one can
associate another measure $\\mu$\, called the balanced measure. It is sup
ported on the Julia set for $f$ and satisfies $f*\\mu=d \\cdot \\mu$. Sin
ce it also can be obtained as the limit of preimages of quite general pro
babilistic measures on $\\mathbb{CP}^1$ (thanks to the results of M. Lyubi
ch and independently Freire-Lopes-R. Ma\\~ne from 1980s)\, a question aris
es whether it always equals the equilibrium measure for $J_f$ (when the la
tter notion makes sense). Several mathematicians noticed that equality of
these two measures (under suitable assumptions on $f$) implies that $f$ i
s a polynomial. However\, all ``proofs'' of this implication from befor
e 1990s contained gaps. The proof by S. Lalley from 1992 was fully succes
sful\, but it was based on the theory of Brownian motions. In this talk\,
I will present a general version of this implication with a proof using
mainly classical and weighted potential theory: Let $f: \\mathbb{CP}^1 \
\to \\mathbb{CP}^1$ be a rational function of degree $d \\geq 2$ whose Jul
ia set does not contain the point $\\infty$. The following are equivalent:
(i) $f \\circ f$ is a polynomial\; (ii) the balanced measure for $f$ and
the equilibrium measure for the Julia set $J_f$ with pole at infinity ar
e equal. This is joint work with Y\\^usuke Okuyama from Kyoto Institute o
f Technology.\n
LOCATION:https://researchseminars.org/talk/CAvid/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Khavinson (University of South Florida)
DTSTART;VALUE=DATE-TIME:20210518T130000Z
DTEND;VALUE=DATE-TIME:20210518T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/41
DESCRIPTION:Title: A
lgebra and PDE : Some Less Traveled Paths Connecting Them\nby Dmitry
Khavinson (University of South Florida) as part of CAvid: Complex Analysis
video seminar\n\nLecture held in N/A.\n\nAbstract\nHere are samples of qu
estions I plan to discuss.\n\n- Let $F(u\,v)$ be a rational function of tw
o variables that has no linear factors and a meromorphic function $u(x\,y)
$ solves the PDE $F(\\nabla u)=0$ near the origin\, say. Then $u$ is a lin
ear function\, i.e.\, $u=ax+by+c$. Why? Is it true in three variables?\n\
n- Does there exist a harmonic polynomial in $\\mathbb{R}^n$ divisible by
a non-negative polynomial?\n\n- Let $P(D)[u^k]=0$\, where $P(D)$ is a part
ial differential operator with constant\, polynomial \, or even entire coe
fficients and k runs over an arithmetic progression of positive integers\,
e. g.\, $k=2n+3$\, $n=1\,2\,\\ldots$. Then the Hessian\, Hess $u$\, vani
shes identically\, so the mapping grad $u:\\\, \\mathbb{C}^n\\mapsto\\math
bb{C^n}$ is degenerate\, i.e.\, the range is an algebraic variety. Is it t
rue? \n\n- When we are solving the Dirichlet problem in a domain with an a
lgebraic boundary\, and the Dirichlet data is a polynomial\, a rational or
an algebraic function\, is the solution algebraic as well?\n
LOCATION:https://researchseminars.org/talk/CAvid/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Chuaqui Farrú (Pontificia Universidad Católica de Chile)
DTSTART;VALUE=DATE-TIME:20210601T130000Z
DTEND;VALUE=DATE-TIME:20210601T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/42
DESCRIPTION:Title: A
hlfors’ Schwarzians for curves\nby Martin Chuaqui Farrú (Pontificia
Universidad Católica de Chile) as part of CAvid: Complex Analysis video
seminar\n\nLecture held in N/A.\n\nAbstract\nWe discuss Ahlfors' Schwarzia
n derivatives for curves in euclidean space introduced some three decades
ago. The definitions consider separate generalizations of the real and ima
ginary part of the classical operator in the complex plane that have impor
tant invariance properties with respect to the Möbius group in euclidean
n-space. We will describe some of the applications of the real Schwarzian
to the study of simple curves in n-space\, to knots in 3-space\, as well
as to the injectivity of the conformal parametrization of minimal surfaces
in 3-space. The role of the imaginary Schwarzian will be presented in euc
lidean 3-space\, highlighting its connection with the osculating sphere\,
a new transformation law under the Möbius group\, and theorems on the exi
stence and uniqueness of parametrized curves with prescribed real and imag
inary Schwarzians.\n
LOCATION:https://researchseminars.org/talk/CAvid/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vasiliki Evdoridou (Open University\, UK)
DTSTART;VALUE=DATE-TIME:20210608T130000Z
DTEND;VALUE=DATE-TIME:20210608T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/43
DESCRIPTION:Title: W
andering on the boundary\nby Vasiliki Evdoridou (Open University\, UK)
as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\
n\nAbstract\nIn the theory of iteration of transcendental entire functions
\, wandering domains\, i.e. connected components of the Fatou set that are
not eventually periodic\, have been extensively studied in recent years.
For example\, a nine-way classification of the internal dynamics in simply
connected wandering domains has been given. In this talk we focus on the
dynamical behaviour on the boundaries of simply connected wandering domain
s. In particular\, we consider the possibility that most boundary orbits c
onverge together in a certain sense\, and give sufficient conditions for s
uch a convergence to hold. Our results are motivated by and extend classic
al results on the boundary dynamics of inner functions.\n\nThis is work in
progress joint with A.M. Benini\, N. Fagella\, P. Rippon and G. Stallard.
\n
LOCATION:https://researchseminars.org/talk/CAvid/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Galina Filipuk (University of Warsaw)
DTSTART;VALUE=DATE-TIME:20210615T130000Z
DTEND;VALUE=DATE-TIME:20210615T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/44
DESCRIPTION:Title: A
spects of nonlinear differential equations\nby Galina Filipuk (Univers
ity of Warsaw) as part of CAvid: Complex Analysis video seminar\n\nLecture
held in N/A.\n\nAbstract\nNonlinear differential equations may have compl
icated singularities in the\ncomplex plane. Painleve equations are nonline
ar second order differential\nequations solutions of which have no movable
critical points. They have a\nlot of nice properties.\n\nIn this talk I s
hall mainly review connection between solutions of the \nPainlev\\'e equa
tions and recurrence coefficients of semi-classical\northogonal polynomi
als.\n
LOCATION:https://researchseminars.org/talk/CAvid/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jujie Wu (Sun Yat-Sen University)
DTSTART;VALUE=DATE-TIME:20210622T130000Z
DTEND;VALUE=DATE-TIME:20210622T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/45
DESCRIPTION:Title: W
eighted L^2 polynomial approximation in C\nby Jujie Wu (Sun Yat-Sen Un
iversity) as part of CAvid: Complex Analysis video seminar\n\nLecture held
in N/A.\n\nAbstract\nWe study the density of polynomials in $H^2(\\Omega\
, \\varphi)$\, the space of square integrable holomorphic functions in a b
ounded domain $\\Omega$ in $\\C$\, where $\\varphi$ is a subharmonic funct
ion. In particular\, we prove that the density holds in Caratheodory doma
ins for any subharmonic function $\\varphi$ in a neighborhood of the closu
re of $\\Omega$. In non-Caratheodory domains\, we prove that the density d
epends on the weight function\, giving examples. We also give a weighted $
L^2$ version of Weierstrass theorem and give the example. Some $L^2$ appro
ximation in higher dimension also will be state here\, which part are in p
rogress now.\n\nThis is joint with Severine Biard and John Erik Fornaess.\
n
LOCATION:https://researchseminars.org/talk/CAvid/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Walter Van Assche (KU Leuven)
DTSTART;VALUE=DATE-TIME:20210706T130000Z
DTEND;VALUE=DATE-TIME:20210706T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/46
DESCRIPTION:Title: H
ermite-Padé approximation to two function with branch points\nby Walt
er Van Assche (KU Leuven) as part of CAvid: Complex Analysis video seminar
\n\nLecture held in N/A.\n\nAbstract\nHermite-Padé approximation to two f
unctions is rational approximation to both functions with a common denomin
ator and close contact at one point (we will use infinity). The common den
ominator is a polynomial with orthogonality conditions for two measures. I
f the two functions have branch points in the complex plane\, then the asy
mptotic behaviour of the zeros (the poles of the Hermite-Padé approximant
s) is determined by algebraic functions satisfying a cubic relation.\nWe w
ill sketch how to get the full asymptotics of the common denominator using
the Riemann-Hilbert problem for matrix valued functions for some particul
ar choices of branch points\, which appear in applications in number theor
y.\n
LOCATION:https://researchseminars.org/talk/CAvid/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Bishop (Stony Brook University\, USA)
DTSTART;VALUE=DATE-TIME:20210914T130000Z
DTEND;VALUE=DATE-TIME:20210914T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/47
DESCRIPTION:Title: F
ast conformal mapping via computational and hyperbolic geometry\nby Ch
ris Bishop (Stony Brook University\, USA) as part of CAvid: Complex Analys
is video seminar\n\nLecture held in N/A.\n\nAbstract\nThe conformal map fr
om the unit disk to the interior of a polygon P is given by the Schwarz-Ch
ristoffel formula\, but this is stated in terms of parameters that are har
d to compute from P. After some background and motivation\, I explain how
the medial axis of a domain\, a concept from computational geometry\, can
be used to give a fast approximation to these parameters\, with bounds on
the accuracy that are independent of P. The precise statement involves qua
siconformal mappings\, and proving these bounds uses a result about hyperb
olic convex sets originating in Thurston's work on 3-manifolds. If time p
ermits\, I will mention some applications to optimal meshing and triangul
ation of planar polygons.\n
LOCATION:https://researchseminars.org/talk/CAvid/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kirill Lazebnik (University of Toronto\, Canada)
DTSTART;VALUE=DATE-TIME:20210921T130000Z
DTEND;VALUE=DATE-TIME:20210921T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/48
DESCRIPTION:Title: T
ranscendental Julia Sets of Minimal Hausdorff Dimension\nby Kirill Laz
ebnik (University of Toronto\, Canada) as part of CAvid: Complex Analysis
video seminar\n\nLecture held in N/A.\n\nAbstract\nWe discuss an approach
to the construction of entire functions with Julia sets having minimal Hau
sdorff dimension. This talk will not assume a background in complex dynami
cs. This talk is based on joint work with Jack Burkart.\n
LOCATION:https://researchseminars.org/talk/CAvid/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yilin Wang (MIT\, USA)
DTSTART;VALUE=DATE-TIME:20210928T130000Z
DTEND;VALUE=DATE-TIME:20210928T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/49
DESCRIPTION:Title: L
oewner-Kufarev energy and foliations by Weil-Petersson quasicircles\nb
y Yilin Wang (MIT\, USA) as part of CAvid: Complex Analysis video seminar\
n\nLecture held in N/A.\n\nAbstract\nWe use Loewner-Kufarev equation to de
scribe evolutions of univalent functions and introduce an energy on the dr
iving measure\, called Loewner-Kufarev energy. We show that when this ener
gy is finite\, the boundaries of the evolving image domains are Weil-Peter
sson quasicircles which form a foliation of the Riemann sphere. Weil-Peter
sson quasicircles are studied in Teichmuller theory\, geometric function t
heory\, and string theory by both mathematicians and physicists. More than
20 equivalent definitions of this class of Jordan curves are discovered s
o far. In particular\, it is characterized as the class of curves having f
inite Loewner energy which was also introduced recently. Furthermore\, we
show that the Loewner-Kufarev energy is dual to the Loewner energy and exh
ibits remarkable symmetries. Both energies and their duality result are in
spired by ideas from the probabilistic theory of Schramm-Loewner evolution
s. This is a joint work with Fredrik Viklund (KTH).\n\nReferences: \n\nThe
Loewner-Kufarev energy and foliations by Weil-Petersson quasicircles\nFre
drik Viklund\, Yilin Wang (2020)\nhttps://arxiv.org/abs/2012.05771\n
LOCATION:https://researchseminars.org/talk/CAvid/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Istvan Prause (University of Eastern Finland)
DTSTART;VALUE=DATE-TIME:20211005T130000Z
DTEND;VALUE=DATE-TIME:20211005T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/50
DESCRIPTION:Title: P
robabilistic limit shapes and harmonic functions\nby Istvan Prause (Un
iversity of Eastern Finland) as part of CAvid: Complex Analysis video semi
nar\n\nLecture held in N/A.\n\nAbstract\nLimit shapes are surfaces in $\\m
athbb{R^3}$ which arise in the scaling limit of discrete random surfaces a
ssociated to various probability models such as domino tilings\, random Yo
ung tableaux or the 5-vertex model. The limit surface is a minimiser of a
gradient variational problem with a surface tension which encodes the loca
l entropy of the model. I'll show that in an intrinsic complex variable th
ese limit shapes can all be parametrised by harmonic functions across a va
riety of models. Some new features beyond determinantal settings will be d
iscussed. The talk is based on joint works with Rick Kenyon.\n
LOCATION:https://researchseminars.org/talk/CAvid/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jörg Winkelmann (Ruhr-Universität Bochum\, Germany)
DTSTART;VALUE=DATE-TIME:20211012T130000Z
DTEND;VALUE=DATE-TIME:20211012T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/51
DESCRIPTION:Title: O
n the existence of dense entire holomorphic curves in rationally connected
manifolds\nby Jörg Winkelmann (Ruhr-Universität Bochum\, Germany) a
s part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\
nAbstract\nJoint work with Frederic Campana. We prove that for every ratio
nally connected\nprojective manifold X there exists a holomorphic map from
the complex line to X with\ndense image and deduce some related results.\
n
LOCATION:https://researchseminars.org/talk/CAvid/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhi-Tao Wen (Shantou University\, China)
DTSTART;VALUE=DATE-TIME:20211019T130000Z
DTEND;VALUE=DATE-TIME:20211019T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/52
DESCRIPTION:Title: D
ifference radical in terms of shifting zero and applications to the Stothe
rs-Mason theorem\nby Zhi-Tao Wen (Shantou University\, China) as part
of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstra
ct\nIn this talk\, we show the shifting zeros with its heights and an anal
ogue to difference radical. We focus on the Stothers-Mason theorem by usin
g falling factorials. As applications\, we discuss the difference version
of the Fermat type functional equations. Some examples are given. It is a
joint work with Katsuya Ishizaki.\n
LOCATION:https://researchseminars.org/talk/CAvid/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeinab Mansour (Cairo University\, Egypt)
DTSTART;VALUE=DATE-TIME:20211026T130000Z
DTEND;VALUE=DATE-TIME:20211026T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/53
DESCRIPTION:Title: L
idstone expansions of entire functions\nby Zeinab Mansour (Cairo Unive
rsity\, Egypt) as part of CAvid: Complex Analysis video seminar\n\nLecture
held in N/A.\n\nAbstract\nLidstone expansions express an entire function
$f(z)$ in terms of the values of the derivatives of even orders at $0\,1$.
The polynomials in the expansion are called Lidstone polynomials. They ar
e Bernoulli polynomials\; many authors introduced necessary and (or) suffi
cient conditions for the absolute convergence of the series in the expansi
on. The classical exponential function plays an essential role in derivin
g the Lidstone series. In the $q$ theory\, we have three $q$-difference op
erators\, the Jackson $q$-difference operator\, the symmetric $q$-differen
ce operator\, and the Askey-Wilson $q$-difference operator. Each operator
is associated with a $q$-analog of the exponential function. In this talk\
, we introduce $q$-extensions to the Lidstone expansion associated with th
ese operators. New three $q$-analogs of Bernoulli polynomials with nice pr
operties are coming out. \n\nJoint work with Professor Mourad Ismail\, Uni
versity of Central Florida\, USA.\n
LOCATION:https://researchseminars.org/talk/CAvid/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leticia Pardo-Simón (University of Manchester\, UK)
DTSTART;VALUE=DATE-TIME:20211102T130000Z
DTEND;VALUE=DATE-TIME:20211102T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/54
DESCRIPTION:Title: T
he maximum modulus set of an entire function\nby Leticia Pardo-Simón
(University of Manchester\, UK) as part of CAvid: Complex Analysis video s
eminar\n\nLecture held in N/A.\n\nAbstract\nThe set of points where an ent
ire function achieves its maximum modulus is known as the maximum modulus
set\, and usually consists of a collection of disjoint analytic curves. In
this talk\, we discuss recent progress on the description of the features
that this set might exhibit. Namely\, on the existence of discontinuities
\, singleton components\, and on its structure near the origin. This is ba
sed on joint work with D. Sixsmith and V. Evdoridou.\n
LOCATION:https://researchseminars.org/talk/CAvid/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Mityushev (Cracow Technological University\, Poland)
DTSTART;VALUE=DATE-TIME:20211109T140000Z
DTEND;VALUE=DATE-TIME:20211109T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/55
DESCRIPTION:Title: R
iemann-Hilbert problem for a multiply connected domain and its application
s to the effective properties of 2D random composites\nby Vladimir Mit
yushev (Cracow Technological University\, Poland) as part of CAvid: Comple
x Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nIn this talk
we answer the following question. "Why did James Bond prefer shaken\, not
stirred martini with ice?" The posed question is resolved by reduction to
the scalar Riemann-Hilbert problem Re (a f) = g for a multiply connected
domain and its complete solution. Relations to the ℝ-linear problem and
the effective properties of 2D random composites are discussed.\n
LOCATION:https://researchseminars.org/talk/CAvid/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter A. Clarkson (University of Kent\, UK)
DTSTART;VALUE=DATE-TIME:20211116T140000Z
DTEND;VALUE=DATE-TIME:20211116T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/56
DESCRIPTION:Title: S
pecial polynomials associated with the Painlevá equations\nby Peter A
. Clarkson (University of Kent\, UK) as part of CAvid: Complex Analysis vi
deo seminar\n\nLecture held in N/A.\n\nAbstract\nThe six Painlevé equatio
ns\, whose solutions are called the Painlevé transcendents\, were derived
by Painlevé and his colleagues in the late 19th and early 20th centuries
in a classification of second order ordinary differential equations whose
solutions have no movable critical points. In the 18th and 19th centuries
\, the classical special functions such as Bessel\, Airy\, Legendre and hy
pergeometric functions\, were recognized and developed in response to the
problems of the day in electromagnetism\, acoustics\, hydrodynamics\, elas
ticity and many other areas. \n\nAround the middle of the 20th century\, a
s science and engineering continued to expand in new directions\, a new cl
ass of functions\, the Painlevé functions\, started to appear in applicat
ions. The list of problems now known to be described by the Painlevé equa
tions is large\, varied and expanding rapidly. The list includes\, at one
end\, the scattering of neutrons off heavy nuclei\, and at the other\, the
distribution of the zeros of the Riemann-zeta function on the critical li
ne Re(z) =1/2. Amongst many others\, there is random matrix theory\, the a
symptotic theory of orthogonal polynomials\, self-similar solutions of int
egrable equations\, combinatorial problems such as the longest increasing
subsequence problem\, tiling problems\, multivariate statistics in the imp
ortant asymptotic regime where the number of variables and the number of s
amples are comparable and large\, and also random growth problems.\n\nThe
Painlevé equations possess a plethora of interesting properties including
a Hamiltonian structure and associated isomonodromy problems\, which expr
ess the Painlevé equations as the compatibility condition of two linear s
ystems. Solutions of the Painlevé equations have some interesting asympto
tics which are useful in applications. They possess hierarchies of rationa
l solutions and one-parameter families of solutions expressible in terms o
f the classical special functions\, for special values of the parameters.
Further the Painlevé equations admit symmetries under affine Weyl groups
which are related to the associated Bäcklund transformations. \n\nIn this
talk I shall discuss special polynomials associated with rational solutio
ns of Painlevé equations. Although the general solutions of the six Painl
evé equations are transcendental\, all except the first Painlevé equatio
n possess rational solutions for certain values of the parameters. These s
olutions are expressed in terms of special polynomials. The roots of these
special polynomials are highly symmetric in the complex plane and specula
ted to be of interest to number theorists. The polynomials arise in applic
ations such as random matrix theory\, vortex dynamics\, in the description
of rogue wave patterns\, in supersymmetric quantum mechanics\, as coeffic
ients of recurrence relations for semi-classical orthogonal polynomials an
d are examples of exceptional orthogonal polynomials.\n
LOCATION:https://researchseminars.org/talk/CAvid/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thierry Meyrath (University of Luxembourg\, Luxembourg)
DTSTART;VALUE=DATE-TIME:20211123T140000Z
DTEND;VALUE=DATE-TIME:20211123T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/57
DESCRIPTION:Title: O
n covering properties of non-normal families of meromorphic functions\
nby Thierry Meyrath (University of Luxembourg\, Luxembourg) as part of CAv
id: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nWe
study the behaviour of families of meromorphic functions in the neighbour
hood of points of non-normality and prove certain covering properties that
complement Montel's Theorem. Moreover\, we obtain characterizations of no
n-normality in terms of such properties. This talk is based on joint work
with Jürgen Müller.\n
LOCATION:https://researchseminars.org/talk/CAvid/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olubunmi A. Fadipe-Joseph (University of Ilorin\, Nigeria)
DTSTART;VALUE=DATE-TIME:20211130T140000Z
DTEND;VALUE=DATE-TIME:20211130T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/58
DESCRIPTION:Title: T
he sigmoid function in geometric function theory\nby Olubunmi A. Fadip
e-Joseph (University of Ilorin\, Nigeria) as part of CAvid: Complex Analys
is video seminar\n\nLecture held in N/A.\n\nAbstract\nGeometric Function T
heory (GFT) is a branch of complex analysis which studies geometric proper
ties of analytic functions. Moreover\, in spite of the famous coefficient
problems\, Bieberbach conjecture that was solved by Louis de Branges in 19
84 suggested various approaches and directions for study in geometric func
tion theory. Therefore\, one of the major interests in GFT is finding the
coefficient bounds of univalent and multivalent functions. The bounds det
ermine the growth\, distortion properties among others of the analytic fun
ctions. Special functions are of great interest in mathematics\, mathemati
cal\nphysics\, engineering and other fields of science. They are rich in t
erms of practical applications in solving a wide range of\nproblems. Recen
tly\, we investigate special functions in geometric function theory. In pa
rticular\, the connection between sigmoid function and geometric function
theory was established.\n
LOCATION:https://researchseminars.org/talk/CAvid/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Loredana Lanzani (Syracuse University\, USA)
DTSTART;VALUE=DATE-TIME:20211207T140000Z
DTEND;VALUE=DATE-TIME:20211207T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/59
DESCRIPTION:Title: T
he Cauchy-Szegö projection and its commutator for domains in $\\mathbb C^
n$ with minimal smoothness\nby Loredana Lanzani (Syracuse University\,
USA) as part of CAvid: Complex Analysis video seminar\n\nLecture held in
N/A.\n\nAbstract\nLet $D\\subset\\C^n$ be a bounded\, strongly pseudoconve
x domain whose boundary $bD$ satisfies the minimal regularity condition of
class $C^2$. A 2017 result of Lanzani and Stein states that \nthe Cauchy
-Szegö projection $S_\\omega$ defined with respect to any Leray Levi-like
measure $\\omega$ is bounded in $L^p(bD\, \\omega)$ for any $1 < p < \\in
fty$.\n(For this class of domains\, induced Lebesgue measure is Leray Lev
i-like.)\n Here we show that $S_\\omega$\n is in fact bounded in $L^p(bD
\, \\Omega_p)$ for any $1 < p< \\infty$ and for any $\\Omega_p$ in the opt
imal\n class\n of $A_p$ measures\, that is $\\Omega_p = \\psi_p\\sigma$
where $\\sigma$ is induced Lebesgue measure and $\\psi_p$ is any Muckenho
upt $A_p$-weight.\n As an application\, we\n characterize boundedness an
d compactness in $L^p(bD\, \\Omega_p)$ for any $1 < p < \\infty$ and for
any $A_p$ measure $\\Omega_p$\, of the commutator $[b\, S_p]$ for any Lera
y Levi-like measure $\\omega$. \n We next introduce the notion of holomor
phic Hardy spaces for $A_p$ measures\,\n $1 < p < \\infty$\, \n and \n
we characterize\n boundedness and compactness in $L^2(bD\, \\Omega_2)$ o
f the commutator \n $\\displaystyle{[b\,S_{\\Omega_2}]}$ of the Cauchy-Sze
gö projection defined with respect to any \n $A_2$ measure $\\Omega_2$.\n
Earlier closely related results \n rely upon an asymptotic expansion\, a
nd subsequent pointwise estimates\, of the Cauchy-Szegö kernel\, but thes
e are unavailable in the settings of minimal regularity {of $bD$} and/or $
A_p$ measures\; it turns out that the real harmonic analysis method of ext
rapolation is an appropriate replacement for the missing tools.\n\n \nThi
s is joint work with Xuan Thinh Duong (Macquarie University)\, Ji Li (Macq
uarie University) and Brett Wick (Washington University in St. Louis).\n
LOCATION:https://researchseminars.org/talk/CAvid/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adri Olde Daalhuis (University of Edinburgh\, UK)
DTSTART;VALUE=DATE-TIME:20211214T140000Z
DTEND;VALUE=DATE-TIME:20211214T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/60
DESCRIPTION:Title: A
symptotics and complex analysis\nby Adri Olde Daalhuis (University of
Edinburgh\, UK) as part of CAvid: Complex Analysis video seminar\n\nLectur
e held in N/A.\n\nAbstract\nI will discuss the tools from complex analysis
that are needed in the study of (uniform) asymptotic expansions\nof speci
al functions. For many of these divergent expansions it is possible to con
struct very efficient\nintegral representations for the coefficients and r
emainders\, and these are needed in implementations\nand sharp error-bound
s. I might also discuss exponential asymptotics of the perturbed first Pai
nlevé equation.\n
LOCATION:https://researchseminars.org/talk/CAvid/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Miller (University of Michigan)
DTSTART;VALUE=DATE-TIME:20220111T140000Z
DTEND;VALUE=DATE-TIME:20220111T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/61
DESCRIPTION:Title: R
ational solutions of the Painlevé-IV equation with large parameters\n
by Peter Miller (University of Michigan) as part of CAvid: Complex Analysi
s video seminar\n\nLecture held in N/A.\n\nAbstract\nThe Painlevé-IV equa
tion has two families of rational solutions\, which can be represented in
terms of special polynomials called generalized Hermite polynomials and ge
neralized Okamoto polynomials\, respectively. The generalized Hermite pol
ynomials have a convenient representation in terms of Hankel determinants
for a suitable weight and hence can be identified with norming constants f
or certain pseudo-orthogonal polynomials. This connection provides a path
to the analysis of the generalized Hermite rationals when the parameters
are large\; however it is not known whether the generalized Okamoto polyno
mials have a similar representation. In this talk\, we explain how the is
omonodromic approach places both families of rational solutions in terms o
f special cases of the Riemann-Hilbert inverse monodromy problem for Painl
evé-IV. This allows techniques from steepest descent theory to be used t
o analyze both families of rational solutions within a common analytical f
ramework. This is joint work with Robert Buckingham.\n
LOCATION:https://researchseminars.org/talk/CAvid/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Miriam Benini (Università di Parma)
DTSTART;VALUE=DATE-TIME:20220118T140000Z
DTEND;VALUE=DATE-TIME:20220118T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/62
DESCRIPTION:Title: B
ifurcations arise when there is a drastic change in the solutions of some
equation depending on a parameter\, as the parameter\nby Anna Miriam B
enini (Università di Parma) as part of CAvid: Complex Analysis video semi
nar\n\nLecture held in N/A.\n\nAbstract\nBifurcations arise when there is
a drastic change in the solutions of some equation depending on a paramete
r\, as the parameter varies.\nIn this talk we study bifurcations in holomo
rphic families of meromorphic maps with finitely many singular values. Th
e equation(s) that we will study are the equations defining periodic point
s of period n. Such equations are crucial in complex dynamics because the
Julia set (the set on which the dynamics is chaotic) is the closure of rep
elling periodic points. The celebrated results by Mane-Sad-Sullivan for fa
milies of rational maps (and independently by Lyubich\, and by Levin for p
olynomials) show that in a set of parameters where no bifurcations of pe
riodic points occur\, the Julia set stays almost the same and so does the
dynamics\; precisely speaking\, all maps are topologically conjugate in s
uch set. Moreover\, they establish a precise correlation between bifurca
tions of periodic points and a change of behaviour in the orbits of singul
ar values.\nThe key new feature that appears for families of meromorphic
maps is that periodic points can disappear at infinity at specific param
eters\, creating a new type of bifurcations. Our work connects this new ty
pe of bifurcations with change of behaviour in singular orbits\, to establ
ish Mane-Sad-Sullivan's Theorem for meromorphic maps.\nThis is joint wor
k with Matthieu Astorg and Nùria Fagella.\n
LOCATION:https://researchseminars.org/talk/CAvid/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alastair Fletcher (Northern Illinois University)
DTSTART;VALUE=DATE-TIME:20220201T140000Z
DTEND;VALUE=DATE-TIME:20220201T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/63
DESCRIPTION:Title: C
antor sets and Julia sets\nby Alastair Fletcher (Northern Illinois Uni
versity) as part of CAvid: Complex Analysis video seminar\n\nLecture held
in N/A.\n\nAbstract\nOne does not have to study much complex dynamics befo
re coming across examples of Julia sets which are Cantor sets. It is then
a natural question to ask which Cantor sets can be Julia sets? The rigidit
y of holomorphic maps precludes certain natural examples\, and so we will
ask this question in the context of uniformly quasiregular mappings with a
focus on dimensions two and three. This talk is based on joint work with
Dan Stoertz and Vyron Vellis.\n
LOCATION:https://researchseminars.org/talk/CAvid/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joan Lind (University of Tennessee)
DTSTART;VALUE=DATE-TIME:20220208T140000Z
DTEND;VALUE=DATE-TIME:20220208T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/64
DESCRIPTION:Title: T
he Loewner equation with complex-valued driving functions\nby Joan Lin
d (University of Tennessee) as part of CAvid: Complex Analysis video semin
ar\n\nLecture held in N/A.\n\nAbstract\nThe chordal Loewner equation provi
des a correspondence between real-valued functions\, called driving functi
ons\, and certain growing 2-dimensional sets\, called hulls. In this talk
\, we will consider the generalization to complex-valued driving functions
\, which was first studied by Huy Tran. We will discuss some key differen
ces between the hulls in the complex-valued setting and those in the real-
valued setting\, including the question of the phase transition from simpl
e-curve hulls. This is joint work with Jeffrey Utley.\n
LOCATION:https://researchseminars.org/talk/CAvid/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kathy Driver (University of Cape Town)
DTSTART;VALUE=DATE-TIME:20220215T140000Z
DTEND;VALUE=DATE-TIME:20220215T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/65
DESCRIPTION:Title: I
nterlacing of zeros of Laguerre polynomials\nby Kathy Driver (Universi
ty of Cape Town) as part of CAvid: Complex Analysis video seminar\n\nLectu
re held in N/A.\n\nAbstract\nThe sequence of Laguerre polynomials $\\{L_{n
}^{(\\alpha)}(x)\\} _{n=0}^\\infty$ is orthogonal on $(0\, \\infty)$ with
respect to the weight function $e^{-x} x^{\\alpha}\,\\alpha > -1$ and the
real distinct positive zeros of $L_{n-1}^{(\\alpha)}(x)$ and $L_{n}^{(\\al
pha)}(x)$ are interlacing for $\\alpha >-1\, n \\geq 2.$ D-Muldoon (2015
-2019) proved that for $\\alpha >-1\,$ the zeros of $L_{n-1}^{(\\alpha+t)
}(x)$ and $L_{n}^{(\\alpha)}(x)$ are interlacing for $0 \\leq t \\leq 2\;
$ the zeros of the equal degree Laguerre polynomials $L_{n}^{(\\alpha)}(
x)$ and $L_{n}^{(\\alpha+t)}(x)$ interlace for $0 < t \\leq 2$\, and the
interval $0 \\leq t \\leq 2$ is sharp for interlacing to hold for every $
n \\in \\mathbb{N}$. Further\, the zeros of $L_{n-k}^{(\\alpha+t)}(x)$ and
$L_{n}^{(\\alpha)}(x)$ are interlacing (in the Stieltjes sense) for $0 \
\leq t \\leq 2k$\, $1 < k < n$ and the interval $0 \\leq t \\leq 2k$ is sh
arp. \nAt OPSFA 2019\, Alan Sokal: What happens to interlacing of roots if
you increase parameter and increase degree of one polynomial relative to
the other? Simplest case: Are the zeros of $L_{n}^{(\\alpha)}(x)$ and $L
_{n+1}^{(\\alpha+1)}(x)$ interlacing for $\\alpha > -1$ and each $n \\in \
\mathbb{N}$? We discuss this and related cases.\n
LOCATION:https://researchseminars.org/talk/CAvid/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Si Duc Quang (Hanoi National University of Education)
DTSTART;VALUE=DATE-TIME:20220222T140000Z
DTEND;VALUE=DATE-TIME:20220222T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/66
DESCRIPTION:Title: S
econd main theorems for meromorphic mappings into projective varieties and
arbitrary families of hypersurfaces\nby Si Duc Quang (Hanoi National
University of Education) as part of CAvid: Complex Analysis video seminar\
n\nLecture held in N/A.\n\nAbstract\nIn this talk\, we will give a short i
ntroduction to Nevanlinna theory for meromorphic mappings into projective
varieties. Our main aim is to present a second main theorem for meromorphi
c mappings with arbitrary families of hypersurfaces in projective varietie
s. This result is a generalization of the second main theorem for the mapp
ings with families of hypersurfaces in subgeneral position.\n
LOCATION:https://researchseminars.org/talk/CAvid/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Nicks (University of Nottingham)
DTSTART;VALUE=DATE-TIME:20220308T140000Z
DTEND;VALUE=DATE-TIME:20220308T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/67
DESCRIPTION:Title: I
terating the minimum modulus\nby Dan Nicks (University of Nottingham)
as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n
\nAbstract\nFor an entire function $f$ there may or may not exist an $r >
0$ such that the iterated minimum modulus $m^n(r)$ tends to infinity. Here
$m(r) = m(r\,f) = \\min\\{ |f(z)| : |z|=r \\}$. Focussing mainly on the c
lass of real transcendental entire functions of finite order with only rea
l zeroes\, we discuss some results about the existence of an $r > 0$ such
that $m^n(r) \\to \\infty$. This is motivated by the result that\, for fun
ctions in this class\, the existence of such an r implies connectedness of
the escaping set $\\{ z : f^n(z) \\to \\infty \\}$.\n\nThis is joint work
with Phil Rippon and Gwyneth Stallard.\n
LOCATION:https://researchseminars.org/talk/CAvid/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lasse Rempe (University of Liverpool)
DTSTART;VALUE=DATE-TIME:20220315T140000Z
DTEND;VALUE=DATE-TIME:20220315T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/68
DESCRIPTION:Title: A
counterexample to Eremenko's Conjecture\nby Lasse Rempe (University o
f Liverpool) as part of CAvid: Complex Analysis video seminar\n\nLecture h
eld in N/A.\n\nAbstract\nI shall speak within my lecture\n\nabout an inter
esting conjecture\n\nof Eremenko from a fine \n\npaper of 1989.\n\n>>>\n\n
He asked if each escaping point\n\ncan to infinity be joined\n\nusing a co
nnected shape\n\nall points of which themselves escape.\n\n>>>\n\nAlthough
quite simple it appears\,\n\nthis question has for many years\n\ncaused m
e and others some despair\,\n\nsleepless nights and greying hair.\n\n>>>\n
\nThrough our intense investigation\n\nof transcendental iteration\,\n\nmu
ch progress was indeed obtained\,\n\nbut the conjecture\, it remained - \n
\n>>>\n\ntill now! By work with Waterman\n\nand Martí-Pete\, now I can\n\
ndescribe to you\, within my lecture\,\n\na counterexample to Eremenko's C
onjecture.\n
LOCATION:https://researchseminars.org/talk/CAvid/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Duc Quang Si (Hanoi National University of Education)
DTSTART;VALUE=DATE-TIME:20220322T130000Z
DTEND;VALUE=DATE-TIME:20220322T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/69
DESCRIPTION:by Duc Quang Si (Hanoi National University of Education) as pa
rt of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\nAbstr
act: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajitha Ranasinghe (University of Peradeniya\, Sri Lanka)
DTSTART;VALUE=DATE-TIME:20220329T130000Z
DTEND;VALUE=DATE-TIME:20220329T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/70
DESCRIPTION:by Rajitha Ranasinghe (University of Peradeniya\, Sri Lanka) a
s part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\nA
bstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adi Glücksam (Northwestern University)
DTSTART;VALUE=DATE-TIME:20220405T130000Z
DTEND;VALUE=DATE-TIME:20220405T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/71
DESCRIPTION:Title: T
he Combinatorial Method for Stopping Time Arguments\nby Adi Glücksam
(Northwestern University) as part of CAvid: Complex Analysis video seminar
\n\nLecture held in N/A.\n\nAbstract\nIn this talk I will present a refine
ment of the combinatorial technique used by Jones and Makarov in '95. This
method can be used for stopping time arguments in different settings. I w
ill describe the method\, and present two applications that were already k
nown\, and one new application. Moreover\, I will give an example showing
this method is optimel. Lastly\, I will discuss future directions and open
problems.\n\nThe talk is based on work in progress.\n
LOCATION:https://researchseminars.org/talk/CAvid/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Marti-Pete (University of Liverpool)
DTSTART;VALUE=DATE-TIME:20220426T130000Z
DTEND;VALUE=DATE-TIME:20220426T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/72
DESCRIPTION:Title: W
andering domains in transcendental dynamics: topology and dynamics\nby
David Marti-Pete (University of Liverpool) as part of CAvid: Complex Anal
ysis video seminar\n\nLecture held in N/A.\n\nAbstract\nFor a transcendent
al entire or meromorphic function\, the Fatou set is the largest open set
on which its iterates are defined and form a normal family. A wandering do
main is a connected component of the Fatou set which is not eventually per
iodic. The first example of a transcendental entire function with a wander
ing domain was constructed by Baker in the 1970s. \n\nWandering domains\,
which do not exist for rational maps\, play an important role in transcend
ental dynamics and in the last decade there has been a resurgence in their
interest. For example\, Bishop proved that the Julia sets of transcendent
al entire functions can have Hausdorff dimension 1 by constructing a funct
ion with wandering domains. \n\nWandering domains are very diverse in term
s of both their topology (simply connected or multiply connected) and thei
r dynamics (escaping\, oscillating or\, perhaps\, even have bounded orbit)
. Recently\, Boc Thaler proved the surprising result that every bounded re
gular domain such that its closure has a connected complement is the wande
ring domain of some transcendental entire function. Inspired by this resul
t\, together with Rempe and Waterman\, we were able to obtain wandering do
mains that form Lakes of Wada. \n\nIn this talk\, I will describe the main
topological and dynamical properties of wandering domains (and their boun
daries) and give an overview of the current open questions.\n
LOCATION:https://researchseminars.org/talk/CAvid/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Girela (Universidad de Málaga)
DTSTART;VALUE=DATE-TIME:20220503T130000Z
DTEND;VALUE=DATE-TIME:20220503T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/73
DESCRIPTION:Title: O
n BMOA and the Bloch space\, normal functions\, and pointwise multipliers<
/a>\nby Daniel Girela (Universidad de Málaga) as part of CAvid: Complex A
nalysis video seminar\n\nLecture held in N/A.\n\nAbstract\nLet $\\mathbb D
$ be the unit disc in $\\mathbb C$ and let ${\\rm Hol} (\\mathbb D)$ denot
e the space of all holomorphic functions in $\\mathbb D$. In this talk we
shall be concerned with a number of subspaces of ${\\rm Hol}(\\mathbb D)$\
, especially with the space $H^\\infty $ of all bounded analytic functions
in $\\mathbb D$\, the space $BMOA$ which consists of those $f\\in H^1$ wh
ose boundary values have bounded mean oscillation on $\\partial \\mathbb D
$\, and the Bloch space $\\mathcal B$ which consists of those $f\\in{\\rm
Hol} (\\mathbb D)$ for which $$\\sup_{z\\in \\mathbb D}(1-\\vert z\\vert ^
2)\\vert f^\\prime (z)\\vert <\\infty .$$ It is well known that $H^\\infty
\\subset BMOA\\subset \\mathcal B$\, and that these inclusions are strict
. \\par A function $f$\, analytic in $\\mathbb D$\, is a normal function (
in the sense of Lehto-Virtanen) if $$\\sup_{z\\in \\mathbb D}(1-\\vert z\\
vert ^2)\\frac{\\vert f^\\prime (z)\\vert }{1+\\vert f(z)\\vert ^2}<\\inft
y .$$ We shall let $\\mathcal N$ denote the class of all normal analytic f
unctions in $\\mathbb D$. We have that $\\mathcal B\\subset \\mathcal N$ a
nd the inclusion is strict. In fact\, the class $\\mathcal N$ is much bigg
er that the Bloch space. \\par Clearly\, $H^\\infty $ is an algebra\, that
is\, the product of two $H^\\infty $-functions lies in $H^\\infty $. Howe
ver\, if $f\\in H^\\infty $ and $g$ is a $BMOA$ function or a Bloch functi
on\, then the product $g\\cdot f$ may not be a normal function: there exis
t pairs of functions $(f\, g)$ with $f\\in H^\\infty $ and $g\\in \\mathca
l B$ such that the product $f\\cdot g$ is not a normal function (or at lea
st it is not a Bloch function). In this talk we shall present distinct exa
mples of such pairs of functions starting with the first ones which were g
iven in the 1960's and finishing with other which have been recently obtai
ned. We shall reformulate these results in the language of pointwise multi
pliers.\n
LOCATION:https://researchseminars.org/talk/CAvid/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irina Markina (University of Bergen)
DTSTART;VALUE=DATE-TIME:20220510T130000Z
DTEND;VALUE=DATE-TIME:20220510T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/74
DESCRIPTION:by Irina Markina (University of Bergen) as part of CAvid: Comp
lex Analysis video seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Baddoo (MIT)
DTSTART;VALUE=DATE-TIME:20220517T130000Z
DTEND;VALUE=DATE-TIME:20220517T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/75
DESCRIPTION:Title: U
nderstanding interacting aerofoils with complex analysis\nby Peter Bad
doo (MIT) as part of CAvid: Complex Analysis video seminar\n\nLecture held
in N/A.\n\nAbstract\nWhen two or more aerofoils move together\, their int
eractions can significantly affect the characteristics of the surrounding
fluid. We develop a rigorous mathematical theory for these interactions us
ing conformal maps\, multiply connected function theory\, and modified Sch
warz problems. Via the transcendental Schottky–Klein prime function\, ou
r theory is valid for any connectivity (any number of aerofoils). Accordin
gly\, our approach is very general and permits many aerofoil motions (pitc
hing\, heaving\, undulatory) and configurations (tandem\, in-line\, ground
effect). We focus on the (doubly connected) case where there are two inte
racting swimmers and find that our theory yields excellent agreement with
experimental data. We also develop an asymptotic solution that captures th
e salient features of the prime function solution.\n
LOCATION:https://researchseminars.org/talk/CAvid/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roger Barnard (Texas Tech University)
DTSTART;VALUE=DATE-TIME:20220524T130000Z
DTEND;VALUE=DATE-TIME:20220524T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/76
DESCRIPTION:Title: O
n sharp bounds for ratios of k-balanced hypergeometric functions\nby R
oger Barnard (Texas Tech University) as part of CAvid: Complex Analysis vi
deo seminar\n\nLecture held in N/A.\n\nAbstract\n(Joint work with Kendall
C. Richards\, Southwestern University and Elyssa N. Sliheet\, Southwestern
Methodist University)\n\nIn this talk we begin with a brief history of ho
w the authors’ research\, originally in Geometric Function Theory\, deve
loped into applications of Special Function Theory to a variety of fields\
, giving examples. Then we discuss one of our latest results in Special Fu
nction Theory i.e. determining the sharp bounds for ratios of k-balanced h
ypergeometric functions.\n
LOCATION:https://researchseminars.org/talk/CAvid/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clare Dunning (University of Canterbury at Kent)
DTSTART;VALUE=DATE-TIME:20220531T130000Z
DTEND;VALUE=DATE-TIME:20220531T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/77
DESCRIPTION:Title: P
olynomials and partitions\nby Clare Dunning (University of Canterbury
at Kent) as part of CAvid: Complex Analysis video seminar\n\nLecture held
in N/A.\n\nAbstract\nWronskians of orthogonal polynomials appear in a rang
e of applications including in random matrix theory\, vortex dynamics and
supersymmetric quantum mechanics. They are also associated with the ration
al solution of Painlevé equations. We discuss how the partitions that lab
el the set of orthogonal polynomials in a particular Wronskian play a role
beyond simple notation. Curiously\, various aspects of the Wronksian poly
nomials can be expressed in terms of partition data.\n
LOCATION:https://researchseminars.org/talk/CAvid/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Heather Wilber (University of Texas at Austin)
DTSTART;VALUE=DATE-TIME:20220607T130000Z
DTEND;VALUE=DATE-TIME:20220607T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/78
DESCRIPTION:Title: L
ow rank numerical methods via rational function approximation\nby Heat
her Wilber (University of Texas at Austin) as part of CAvid: Complex Analy
sis video seminar\n\nLecture held in N/A.\n\nAbstract\nIn this talk\, we a
pply classical ideas in approximation theory to design low rank numerical
methods for a range of applications in scientific computing\, including th
e solving of certain linear systems\, matrix equations\, and partial diffe
rential equations. The primary workhorse in our approach and analysis is t
he alternating direction implicit (ADI) method\, and we explore how this s
pecial splitting algorithm is linked to a wealth of concepts from applied
mathematics\, including Laplace’s equation and conformal maps for doubly
-connected regions\, matrix and operator function evaluation\, digital fil
ter design\, and the low rank properties of matrices with special displace
ment structures.\n
LOCATION:https://researchseminars.org/talk/CAvid/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elias Wegert (Technische Universität Bergakademie Freiberg)
DTSTART;VALUE=DATE-TIME:20220614T130000Z
DTEND;VALUE=DATE-TIME:20220614T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/79
DESCRIPTION:Title: N
umerical range\, Blaschke products and Poncelet polygons\nby Elias Weg
ert (Technische Universität Bergakademie Freiberg) as part of CAvid: Comp
lex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\n(Joint wor
k with Ilya Spitkovsky\, New York University Abu Dhabi)\n\nIn 2016\, Gau\,
Wang and Wu conjectured that a partial isometry\nA acting on a $n$-dimens
ional complex Hilbert space cannot have \na circular numerical range with
a non-zero center.\nIn this talk we present a proof for operators with ran
k $A=n-1$ \nand any n. It is based on the unitary similarity of A to a com
pressed\nshift operator generated by a finite Blaschke product $B(z)$.\nWe
then use the description of the numerical range by Poncelet\npolygons ass
ociated with $zB(z)$\, a special representation of \nBlaschke products rel
ated to boundary interpolation\, and an \nexplicit formula for the barycen
ters of the vertices of Poncelet \npolygons involving elliptic functions.\
n
LOCATION:https://researchseminars.org/talk/CAvid/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jane Hawkins (University of North Carolina)
DTSTART;VALUE=DATE-TIME:20220621T130000Z
DTEND;VALUE=DATE-TIME:20220621T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/80
DESCRIPTION:Title: D
oubly periodic Julia and Fatou sets for iterated meromorphic functions: d
ynamics on unbounded components\nby Jane Hawkins (University of North
Carolina) as part of CAvid: Complex Analysis video seminar\n\nLecture held
in N/A.\n\nAbstract\nElliptic functions give rise under iteration to Juli
a and Fatou sets that are invariant under the action of translation by ele
ments of the period lattice. However doubly periodic Julia and Fatou sets
can arise for non-elliptic meromorphic functions as well. Unbounded Fatou
components in both settings exhibit dynamics different from those of rati
onal maps and are called toral bands since they can be viewed on a torus (
a fundamental region in the plane with identifications). We discuss how t
he dynamics depend on the function and the lattice\, both its shape and it
s size\, and what parameter choices produce unbounded components. We will
also touch on the stability and connectivity of these components.\n
LOCATION:https://researchseminars.org/talk/CAvid/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fiana Jacobzon (Braude College of Engineering\, Karmiel\, Israel)
DTSTART;VALUE=DATE-TIME:20220628T130000Z
DTEND;VALUE=DATE-TIME:20220628T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/82
DESCRIPTION:Title: A
n "inverse Fekete-Szegö problem" and filtration of generators\nby Fia
na Jacobzon (Braude College of Engineering\, Karmiel\, Israel) as part of
CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\
nIn this talk we introduce and discuss a question that can be interpreted
as an "inverse Fekete-Szegö problem". It turns out that this problem lin
ks to the so-called filtration of infinitesimal generators. Several filtra
tion classes have recently been studied\, including their applications to
semigroups of holomorphic mappings in the unit disk.\nTo address the circl
e of questions that arise in this context we introduce new filtration clas
ses using the non-linear differential operator\n\\[\\alpha\\cdot \\frac{f(
z)}{z}+\\beta\\cdot \\frac{zf'(z)}{f(z)}+(1-\\alpha-\\beta)\\cdot \\left[1
+\\frac{zf''(z)}{f'(z)}\\right]\,\\]\nand establish certain properties of
these classes. \nSharp upper bounds of the modulus of the Fekete--Szegö f
unctional over some filtration classes are found. \nWe also present open p
roblems for further study.\n\n\nJoint work with Mark Elin (Braude College
of Engineering\, Karmiel\, Israel) and \nNikola Tuneski (Ss. Cyril and Met
hodius University\, Skopje\, Republic of North Macedonia)\n
LOCATION:https://researchseminars.org/talk/CAvid/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Waterman (Stony Brook University)
DTSTART;VALUE=DATE-TIME:20220920T130000Z
DTEND;VALUE=DATE-TIME:20220920T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/83
DESCRIPTION:Title: M
averick points on the boundary of wandering domains\nby James Waterman
(Stony Brook University) as part of CAvid: Complex Analysis video seminar
\n\nLecture held in N/A.\n\nAbstract\nWandering domains\, or wandering Fat
ou components\, are a central object of study in the iteration of transcen
dental entire functions. We will introduce several basic properties of wan
dering domains. Moreover\, focusing on behavior on the boundary of these d
omains\, we will discuss the existence of boundary points of a wandering d
omain with accumulation behavior distinct from that of the wandering domai
n itself. We call such points maverick points. This is joint work with Dav
id Martí-Pete and Lasse Rempe.\n
LOCATION:https://researchseminars.org/talk/CAvid/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Darren Crowdy (Imperial College London)
DTSTART;VALUE=DATE-TIME:20221011T130000Z
DTEND;VALUE=DATE-TIME:20221011T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/84
DESCRIPTION:Title: W
ater waves with vorticity and the Schwarz function\nby Darren Crowdy (
Imperial College London) as part of CAvid: Complex Analysis video seminar\
n\nLecture held in N/A.\n\nAbstract\nThe theory of water waves is centurie
s old\, but it remains a vibrant area of research. Most theoretical work o
n water waves takes the flow to be irrotational\, but there has been growi
ng interest\, especially recently\, in the effect of vorticity on the stru
cture of the waves. The assumption of irrotationality has the theoretical
advantage that complex analysis techniques can be used to analyze the prob
lem in the two-dimensional setting. This talk will present a novel theoret
ical formulation of the problem of steadily-travelling water waves in the
presence of vorticity (where the assumption of irrotationality is dropped)
but in the absence of gravity or capillarity. The approach is based on th
e notion of a Schwarz function of a curve. It unifies our understanding of
several recent results in the water wave literature and provides a wealth
of new exact mathematical solutions to this challenging free boundary pro
blem.\n
LOCATION:https://researchseminars.org/talk/CAvid/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maurice Kenfack Nangho (University of Dschang\, Cameroon)
DTSTART;VALUE=DATE-TIME:20221018T130000Z
DTEND;VALUE=DATE-TIME:20221018T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/85
DESCRIPTION:Title: A
characterization of Askey-Wilson polynomials: proof of a conjecture by Mo
urad Ismail\nby Maurice Kenfack Nangho (University of Dschang\, Camero
on) as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/
A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tamara Grava (University of Bristol and SISSA)
DTSTART;VALUE=DATE-TIME:20220927T130000Z
DTEND;VALUE=DATE-TIME:20220927T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/86
DESCRIPTION:Title: T
he Stieltjes-Fekete problem and degenerate orthogonal polynomials\nby
Tamara Grava (University of Bristol and SISSA) as part of CAvid: Complex A
nalysis video seminar\n\nLecture held in N/A.\n\nAbstract\nA result of Sti
eltjes famously relates the zeroes of the classical orthogonal polynomials
with the configurations of points on the line that minimize a suitable lo
garithmic energy\, or equivalently the solutions\nof a suitable weighted F
ekete problem. The optimal configuration satisfies an algebraic set of equ
ations with the logarithmic derivative of the weight function as ``externa
l field": we call this set of algebraic\nequations the Stieltjes-Fekete pr
oblem. In this work we consider the\nStieltjes-Fekete problem with an arbi
trary rational external field. We\nshow that its solutions are in one-to-o
ne correspondence with the zeroes of certain non-hermitean orthogonal poly
nomials that satisfy an excess of orthogonality conditions and are thus te
rmed ``degenerate". This generalizes the above mentioned result of Stieltj
es.\n
LOCATION:https://researchseminars.org/talk/CAvid/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yang Chen (University of Macau)
DTSTART;VALUE=DATE-TIME:20221025T130000Z
DTEND;VALUE=DATE-TIME:20221025T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/87
DESCRIPTION:Title: L
aguerre Unitary Ensembles with Multiple Discontinuities\, PDE\, and the Co
upled Painlevé V System\nby Yang Chen (University of Macau) as part o
f CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstrac
t\nWe study the Hankl generated by the Laguerre weight with jump\ndisconti
nuities at $t_k$\, $k=1\,2\,\\ldots\,m$. By employing the ladder operator
approach \nwe establish (multi-time) Riccati equations\, to show that $\\s
igma_n(t_1\, ...\,t_m)$\,\nthe log derivative of the $n\\times n$ Hankel d
eterminant\, satisfies a generalization of the $\\sigma$ of a Painlev\\'e
V equation. Through investigating the Riemann-Hibert problem (or Homogenou
s Hilbert Problem ) for the orthogonal polynomials\ngenerated by the LUEM
D and via Lax pair\, we express $\\sigma_n$ in terms of \nsolutions of a c
oupled Painlev\\'e V system. We also build relations between the auxiliary
quantities introduced in the above two methods\, which provide\nconnectio
ns between the Riccati equations and the Lax Pair. \n\nIn addition\, when
each $t_k$ tends to the hard edge of the spectrum and $n$ goes to infinity
\, the scaled $\\sigma_n$ is shown to satisfy a generalized Painlev\\'e II
I system.\n\nYang Chen (University of Macau\, Macau)\, Shulin Lyu (Qilu Un
iversity of Technology\, Shandong Academy of Science)\, Shuai-Xia Xu (Inst
itut Franco-Chinois de l'Energie Nculearie\, Sun Yat-sen University\, Gua
ngzhou\, China\n
LOCATION:https://researchseminars.org/talk/CAvid/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jasmin Raissy (Université de Bordeaux)
DTSTART;VALUE=DATE-TIME:20221122T140000Z
DTEND;VALUE=DATE-TIME:20221122T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/88
DESCRIPTION:by Jasmin Raissy (Université de Bordeaux) as part of CAvid: C
omplex Analysis video seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianhua Zheng (Tsinghua University)
DTSTART;VALUE=DATE-TIME:20221101T140000Z
DTEND;VALUE=DATE-TIME:20221101T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/89
DESCRIPTION:Title: C
lassification of Baker domains of meromorphic functions\nby Jianhua Zh
eng (Tsinghua University) as part of CAvid: Complex Analysis video seminar
\n\nLecture held in N/A.\n\nAbstract\nFirst we introduce the definition of
Baker domain of a meromorphic\n function\, an absorbing domain of a Baker
domain and a classification of Baker domains with\nconnectivity of its ab
sorbing domain. Secondly we introduce a more carful classification of Bake
r domains according to characteristic of the M\\"obius transformation whic
h the function sem-conjugates on the Baker domain in question. Thirdly\, w
e say criteria of Baker domain types. The talk mainly comes from a unpubli
shed paper I finished one year ago.\n
LOCATION:https://researchseminars.org/talk/CAvid/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Argyris Christodoulou (Aristotle University of Thessaloniki)
DTSTART;VALUE=DATE-TIME:20221115T140000Z
DTEND;VALUE=DATE-TIME:20221115T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/90
DESCRIPTION:Title: G
eneralising the Denjoy-Wolff theorem\nby Argyris Christodoulou (Aristo
tle University of Thessaloniki) as part of CAvid: Complex Analysis video s
eminar\n\nLecture held in N/A.\n\nAbstract\nThe starting point for this ta
lk is the classical Denjoy-Wolff theorem\, which completely describes the
behaviour of the iterates of a holomorphic self-map of the unit disc. Sinc
e its inception there have been many attempts at generalising this result
to include compositions of more than one map\, but as of yet there is no d
efinitive result of this type. We approach this subject by asking the foll
owing question: Is the result of the Denjoy-Wolff theorem stable when we p
erturb the iterated function? In particular\, we study the dynamical behav
iour of compositions arising from a sequence of self-maps of a Riemann sur
face\, when the sequence itself converges to a holomorphic map. Based on j
oint work with Marco Abate and Ian Short.\n
LOCATION:https://researchseminars.org/talk/CAvid/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marta Kosek (Jagiellonian University\, Poland)
DTSTART;VALUE=DATE-TIME:20230110T140000Z
DTEND;VALUE=DATE-TIME:20230110T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/91
DESCRIPTION:Title: O
n some limits in the theory of Julia sets\nby Marta Kosek (Jagiellonia
n University\, Poland) as part of CAvid: Complex Analysis video seminar\n\
nLecture held in N/A.\n\nAbstract\nWe will speak about polynomial Julia se
ts in the complex plane\, even\nthough most subjects can be investigated a
lso in higher dimensions. We\nconsider some approximation problems. One of
them is approximation of some\nregular sets by polynomial Julia sets. It
can be seen that a good tool for\nthis approximation is Klimek’s metric
defined with use of Green's\nfunctions of complex sets\, which is more app
ropriate than the classical\nHausdorff metric. Another problem concerns cr
eating computer pictures of\nsome composite Julia sets. Finally\, we deal
with some sequences defined\nwith use of (compositions of) Chebyshev polyn
omials and obtain their uniform limit.\n
LOCATION:https://researchseminars.org/talk/CAvid/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mohamed Nasser (Wichita State University\, USA)
DTSTART;VALUE=DATE-TIME:20230117T140000Z
DTEND;VALUE=DATE-TIME:20230117T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/92
DESCRIPTION:Title: A
boundary integral method for the Riemann–Hilbert problem on multiply co
nnected domains\nby Mohamed Nasser (Wichita State University\, USA) as
part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\n
Abstract\nLet $G$ be a multiply connected domain in the extended complex p
lane and let $A$ be a complex function on the boundary $\\partial G$ with
$A\\ne0$. \nFor a given real function $\\gamma$ on $\\partial G$\, the Rie
mann--Hilbert (RH) boundary value problem requires determining a function
$f$ analytic in $G$ (vanishing at infinity for unbounded $G$)\, continuous
in the closure $\\overline{G}$\, and satisfying the boundary condition Re
$[Af]=\\gamma$ on $\\partial G.$\n\nA boundary integral method for solving
the above RH problem will be presented in this talk. The method is based
on an integral equation known as {the boundary integral equation with the
generalized Neumann kernel}. Applications of the method will be also prese
nted.\n
LOCATION:https://researchseminars.org/talk/CAvid/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Vishnyakova (V. N. Karazin Kharkiv National University\, Ukra
ine and Holon Institute of Technology\, Israel and Holon Institute of
Technology\, Israel)
DTSTART;VALUE=DATE-TIME:20230124T140000Z
DTEND;VALUE=DATE-TIME:20230124T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/93
DESCRIPTION:Title: N
ecessary and sufficient conditions for entire functions to belong to the L
aguerre-Polya class\nby Anna Vishnyakova (V. N. Karazin Kharkiv Nation
al University\, Ukraine and Holon Institute of Technology\, Israel an
d Holon Institute of Technology\, Israel) as part of CAvid: Complex Analys
is video seminar\n\nLecture held in N/A.\n\nAbstract\nThe famous Laguerre-
Polya class consists of\nentire functions which are uniform on the\ncompac
ts limits of real polynomials having all\nreal zeros. The Laguerre-Polya c
lass is of interest\nto many areas of mathematics such as complex analysis
\,\nstatistical physics\, combinatorics\, asymptotic analysis\,\nthe theor
y of mock modular forms and others. We present\nnew necessary and new suff
icient conditions for an entire\nfunction to belong to the Laguerre-Polya
class in terms\nof Taylor coefficients of the function. The partial theta-
function\n$g_a(z) =\\sum_{k=0}^{\\infty} \\frac {z^k}{a^{k^2}}\, a>1\,$\np
lays an important role in our investigations. It is known\nthat there exis
ts a constant $ q_\\infty\\approx 3{.}23363666\,$\nsuch that the partial t
heta-function belongs to the Laguerre-Polya\nclass if and only if $a^2 \\g
eq q_\\infty.$ The following statement\nis an example of our results. Let
$f(z)=\\sum_{k=0}^\\infty a_k z^k $\nbe an entire function with positive
coefficients. Suppose that the\nsequence $\\frac{a_n^2}{a_{n-1} a_{n+1}}$
is decreasing in $n$\,\nand the limit of this sequence is greater than or
equal to\n$\\ q_\\infty.$ Then the function $f$ belongs to the\nLaguerre-P
olya class.\n
LOCATION:https://researchseminars.org/talk/CAvid/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Short (Open University\, UK)
DTSTART;VALUE=DATE-TIME:20230207T140000Z
DTEND;VALUE=DATE-TIME:20230207T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/94
DESCRIPTION:Title: I
terated function systems in holomorphic dynamics\nby Ian Short (Open U
niversity\, UK) as part of CAvid: Complex Analysis video seminar\n\nLectur
e held in N/A.\n\nAbstract\nMotivated by classical results in continued fr
action theory\, we explore iterated function systems of holomorphic self-m
aps of the disc and other Riemann surfaces. The primary tools in this ende
avour are the hyperbolic metric and Pick's theorem that holomorphic maps a
re contractions of the hyperbolic metric. We will review selected results
from this field over the last few decades and finish with work in preparat
ion advancing these results by use of the hyperbolic derivative.\n
LOCATION:https://researchseminars.org/talk/CAvid/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonatan Lenells (KTH\, Sweden)
DTSTART;VALUE=DATE-TIME:20230131T140000Z
DTEND;VALUE=DATE-TIME:20230131T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/95
DESCRIPTION:by Jonatan Lenells (KTH\, Sweden) as part of CAvid: Complex An
alysis video seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adolfo Guillot (National Autonomous University of Mexico)
DTSTART;VALUE=DATE-TIME:20230214T140000Z
DTEND;VALUE=DATE-TIME:20230214T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/96
DESCRIPTION:Title: M
eromorphic vector fields on algebraic surfaces having univalent solutions<
/a>\nby Adolfo Guillot (National Autonomous University of Mexico) as part
of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstra
ct\nWe consider algebraic\, first-order\, autonomous ordinary\ndifferentia
l equations in two complex variables (meromorphic vector\nfields on compac
t algebraic surfaces\, for instance\, those coming from\nrational vector f
ields on affine surfaces)\, and discuss the very\nstrong constraints impos
ed by the existence of one transcendental\nunivalent solution: either ther
e is some variable that integrates\nindependently (the vector field preser
ves a fibration on the surface)\,\nor the surface is an abelian one and th
e vector field is linear.\n
LOCATION:https://researchseminars.org/talk/CAvid/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Martinez-Finkelshtein (Baylor University\, USA and Universi
ty of Almería\, Spain)
DTSTART;VALUE=DATE-TIME:20230221T140000Z
DTEND;VALUE=DATE-TIME:20230221T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/97
DESCRIPTION:by Andrei Martinez-Finkelshtein (Baylor University\, USA and U
niversity of Almería\, Spain) as part of CAvid: Complex Analysis video se
minar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ragnar Sigurdsson (University of Iceland)
DTSTART;VALUE=DATE-TIME:20230321T130000Z
DTEND;VALUE=DATE-TIME:20230321T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/98
DESCRIPTION:by Ragnar Sigurdsson (University of Iceland) as part of CAvid:
Complex Analysis video seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Sokal (University College London)
DTSTART;VALUE=DATE-TIME:20230228T140000Z
DTEND;VALUE=DATE-TIME:20230228T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/99
DESCRIPTION:Title: M
otion of zeros of polynomial solutions of the one-dimensional heat equatio
n: A first-order Calogero-Moser system\nby Alan Sokal (University Coll
ege London) as part of CAvid: Complex Analysis video seminar\n\nLecture he
ld in N/A.\n\nAbstract\nI study the motion of zeros of polynomial solution
s $\\phi(x\, t)=\\prod_{i=1}^n[x-x_{i}(t)]$\nof the one-dimensional heat e
quation \n$\\displaystyle\\frac{\\partial \\phi}{\\partial t}=\\kappa\\fra
c{\\partial^2\\phi}{\\partial x^2}$\; they satisfy the first-order\nCaloge
ro–Moser system \n\\[\n\\frac{{\\rm d}x_i}{{\\rm d}t}=\\sum_{j\\ne i}\\f
rac{-2\\kappa}{x_i-x_j}.\n\\]\nI am interested in the behavior at complex
time $t$ (usually with real initial conditions). My goals are to\n\n(a) De
termine the complex times t at which collisions can or cannot occur\; and\
n\n(b) Control the location of $x_1(t)\,\\ldots\, x_n(t)$ in the complex p
lane. I have no nontrivial theorems\, but many interesting conjectures.\n
LOCATION:https://researchseminars.org/talk/CAvid/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sayani Bera (Indian Association for the Cultivation of Science\, K
olkata)
DTSTART;VALUE=DATE-TIME:20230307T140000Z
DTEND;VALUE=DATE-TIME:20230307T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/100
DESCRIPTION:Title:
Attracting basins of non-autonomous families\nby Sayani Bera (Indian A
ssociation for the Cultivation of Science\, Kolkata) as part of CAvid: Com
plex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nThe goal
of this talk is to explore the basins of non-autonomous or iterative famil
ies of automorphisms of $\\mathbb{C}^m$ \, m ≥ 2\, admitting a common at
tracting fixed point\, and their connection to the classical ‘stable man
ifold theorem’.\nFurther\, we affirmatively answer a conjecture (formula
ted by Fornæss and Stensønes in 2004) on non-autonomous basins\, by gene
ralising appropriate techniques from the (iterative) dynamics of Hénon/re
gular maps in $\\mathbb{C}^m$\,m ≥ 2. This\, also confirms a stronger ve
rsion of the stable manifold theorem\, originally raised as a question by
Bedford in 2000.\nThis is a joint work with K. Verma.\n
LOCATION:https://researchseminars.org/talk/CAvid/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michel Zinsmeister (Université d'Orléans\, France)
DTSTART;VALUE=DATE-TIME:20230314T130000Z
DTEND;VALUE=DATE-TIME:20230314T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/101
DESCRIPTION:Title:
Coullet-Tresser Cascade of Bifurcations in the logistic Family and Hausdor
ff Dimension of real quadratic Julia Sets\nby Michel Zinsmeister (Univ
ersité d'Orléans\, France) as part of CAvid: Complex Analysis video semi
nar\n\nLecture held in N/A.\n\nAbstract\nIn a paper with L. Jacksztas (Adv
Math 2020) we have proven that if $c_0$ is a parabolic parameter (i.e. wi
th a parabolic cycle) in $(c_{Feig}\,1/4)$ ($c_{Feig}$ being the limit poi
nt of the cascade of bifurcations) then the function $d(c)=$ Hausdorff dim
ension of the Julia set $J_c$ of $z^2+c$ has an infinite derivative at $c_
0$ if $d(c_0)\\leq 4/3$\, while it is $C^1$ across $c_0$ if $d(c_0)>4/3$.\
n\nRecently A. Dudko\, I. Gorobovickis and W. Tucker have proven that $d(c
)>4/3$ on $[-1.53\,-1.23]$ (arXiv:2204.07880). The combination of these tw
o results implies that $d$ is $C^1$ on $(c_{Feig}\,-3/4)$ while $d'(-3/4)=
-\\infty$ (a former result of L. Jacksztas).\\\\\nAfter some description (
including a history) of the Coullet-Tresser Feigenbaum phenomenon\, I will
outline the proof of J-Z theorem and briefly describe D-G-T's result. \n\
n(Joint work with L. Jacksztas)\n
LOCATION:https://researchseminars.org/talk/CAvid/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thu Hien Nguyen (Leipzig University\, Germany & V. N. Karazin Khar
kiv University\, Ukraine)
DTSTART;VALUE=DATE-TIME:20230425T130000Z
DTEND;VALUE=DATE-TIME:20230425T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/102
DESCRIPTION:Title:
Some results on entire functions from the Laguerre-Pólya class: proof ide
as and techniques\nby Thu Hien Nguyen (Leipzig University\, Germany &
V. N. Karazin Kharkiv University\, Ukraine) as part of CAvid: Complex Anal
ysis video seminar\n\nLecture held in N/A.\n\nAbstract\nThe Laguerre-P\\'o
lya class is a class of entire functions that are locally the uniform limi
t of a sequence of real polynomials that have only real zeros. We present
some simple necessary and sufficient conditions for entire functions to b
elong to the Laguerre–Pólya class in terms of their Taylor coefficients
. For an entire function $f(z) = \\sum_{k=0}^{\\infty} a_k z^k$\, we defi
ne the second quotients of Taylor coefficients as $q_n(f) := \\frac{a_{n-1
}^2}{a_{n-2} a_{n}}$\, $n\\geq 2$\, and find conditions on $q_n(f)$ for
$f$ to belong to the Laguerre--P\\'olya class\, or to have only real zero
s. In this talk\, we will focus on the entire functions with increasing s
econd quotients of Taylor coefficients\, and discuss proof ideas and techn
iques we used. \n \n This is joint work with Anna Vishnyakova.\n
LOCATION:https://researchseminars.org/talk/CAvid/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Solynin (Texas Tech University\, USA)
DTSTART;VALUE=DATE-TIME:20230502T130000Z
DTEND;VALUE=DATE-TIME:20230502T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/103
DESCRIPTION:Title:
Quadratic differentials in complex analysis and beyond\nby Alexander S
olynin (Texas Tech University\, USA) as part of CAvid: Complex Analysis vi
deo seminar\n\nLecture held in N/A.\n\nAbstract\nI will discuss the role o
f quadratic differentials in the extremal\nproblems in Complex Analysis an
d beyond. We start with main\ndefinitions\, then discuss \nJenkins' theory
of extremal partitioning\, and then I will\nmention main results of the d
ifferentiation theory for the\nJenkins' weighted sum of moduli suggested b
y this speaker in\n1985-2000.\n\nTurning to applications\, I show first ho
w quadratic differentials\ncan be used to study fingerprints of (complex)
polynomial\nlemniscates. The main result here includes\, as special cases\
,\nEbenfelt-Khavinson-Shapiro characterization of fingerprints of\npolynom
ial lemniscates as well as Younsi characterization of\nrational lemniscate
s. Then I will show that every real algebraic\ncurve can be treated as a t
rajectory of a quadratic differential\ndefined on a certain Riemann surfac
e.\n\n\nAfter that\, we will discuss how quadratic differentials on\n$\\ov
erline{\\mathbf{C}}$ with the minimal possible number of poles\n(that is $
4$) can be used to solve the problem on the canonical\nembeddings of pairs
of arcs\, studied recently by M. Bonk and\nA. Eremenko\, and in several o
ther extremal problems on ring\ndomains.\n
LOCATION:https://researchseminars.org/talk/CAvid/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Navneet Lal Sharma (Gati Shakti Vishwavidyalaya\, India)
DTSTART;VALUE=DATE-TIME:20230516T130000Z
DTEND;VALUE=DATE-TIME:20230516T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/104
DESCRIPTION:by Navneet Lal Sharma (Gati Shakti Vishwavidyalaya\, India) as
part of CAvid: Complex Analysis video seminar\n\nLecture held in N/A.\nAb
stract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cinzia Bisi (Ferrara University\, Italy)
DTSTART;VALUE=DATE-TIME:20230530T130000Z
DTEND;VALUE=DATE-TIME:20230530T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/105
DESCRIPTION:Title:
Invariants and automorphisms of slice regular functions\nby Cinzia Bis
i (Ferrara University\, Italy) as part of CAvid: Complex Analysis video se
minar\n\nLecture held in N/A.\n\nAbstract\nLet $A$ be one of the following
Clifford Algebras : C\, H = R2 and R3. For the algebra A\, the automorphi
sm group Aut(A) and its invariants are well known. The talk will describe
the invariants of the automorphism group of the algebra of slice regular f
unctions over $A$ = H = R2 and over $A$ = R3. This is a joint project with
J. Winklelmann.\n
LOCATION:https://researchseminars.org/talk/CAvid/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Kostov (Université d'Azur\, CNRS\, LJAD)
DTSTART;VALUE=DATE-TIME:20230509T130000Z
DTEND;VALUE=DATE-TIME:20230509T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/106
DESCRIPTION:Title:
Analytic properties of the partial theta function\nby Vladimir Kostov
(Université d'Azur\, CNRS\, LJAD) as part of CAvid: Complex Analysis vide
o seminar\n\nLecture held in N/A.\n\nAbstract\nWe consider the partial the
ta function $\\theta (q\,x):=\\sum\n_{j=0}^{\\infty}q^{j(j+1)/2}x^j$\, whe
re $x$ is a variable and $q$ a\nparameter\n($|q|<1$). We deal with the two
possible situations\, when $q$ is real or\ncomplex. In the talk we focus
on the\nanalytic properties of $\\theta$\, such as asymptotic expansions f
or its\nzeros\, its spectrum (i.e. the set of values of the parameter $q$\
nfor which $\\theta (q\,.)$ has multiple zeros)\, behaviour of its zeros\,
\nespecially of its complex conjugate pairs\, when\nthe parameter $q$ vari
es\, separation in modulus of the zeros etc.\n
LOCATION:https://researchseminars.org/talk/CAvid/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Navneet Lal Sharma (Gati Shakti Vishwavidyalaya)
DTSTART;VALUE=DATE-TIME:20230523T130000Z
DTEND;VALUE=DATE-TIME:20230523T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/107
DESCRIPTION:Title:
Estimates logarithmic coefficients for certain classes of univalent functi
ons\nby Navneet Lal Sharma (Gati Shakti Vishwavidyalaya) as part of CA
vid: Complex Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nL
et $\\mathcal{S}$ be the family of analytic and univalent functions $f$ in
the unit disk $\\mathbb{D}$\nwith the normalization $f(0)=f'(0)-1=0$.\nTh
e logarithmic coefficients $\\gamma_n$ of $f\\in \\mathcal{S}$ are defined
by the formula\n$$\\log\\left(\\frac{f(z)}{z}\\right)\\\,=\\\,2\\sum_{n=1
}^{\\infty}\\gamma_n(f)z^n.\n$$\nIn this talk\, we will discuss bounds for
the logarithmic coefficients for certain geometric subfamilies of univale
nt functions as starlike\, convex\, close-to-convex and Janowski starlike
functions. Also\, we consider the families $\\mathcal{F}(c)$ and \n$\\math
cal{G}(\\delta)$ of functions $f\\in \\mathcal{S}$ defined by\n$$ {\\rm R
e} \\left ( 1+\\frac{zf''(z)}{f'(z)}\\right )>1-\\frac{c}{2}\\\, \\mbox{
and } \\\,\n{\\rm Re} \\left ( 1+\\frac{zf''(z)}{f'(z)}\\right )<1+\\frac
{\\delta}{2}\,\\quad z\\in \\mathbb{D} $$\nfor some $c\\in(0\,3]$ and $\\d
elta\\in (0\,1]$\, respectively. We obtain the sharp upper bound for $|\\g
amma_n|$ when $n=1\,2\,3$ and $f$ belongs to the classes \n$\\mathcal{F}(c
)$ and $\\mathcal{G}(\\delta)$\, respectively. We conclude with the follow
ing two conjectures:\n\n* If $f\\in\\mathcal{F}(-1/2)$\, then $ \\display
style |\\gamma_n|\\le \\frac{1}{n}\\left(1-\\frac{1}{2^{n+1}}\\right)$\n f
or $n\\ge 4$\, and\n$$ \\sum_{n=1}^{\\infty}|\\gamma_{n}|^{2} \\leq \\fra
c{\\pi^2}{6}+\\frac{1}{4} ~{\\rm Li\\\,}_{2}\\left(\\frac{1}{4}\\right)\n
-{\\rm Li\\\,}_{2}\\left(\\frac{1}{2}\\right)\, $$\nwhere ${\\rm Li}_2(x
)$ denotes the dilogarithm function. \n\n* If $f\\in \\mathcal{G}(\\delta)
$\, then $ \\displaystyle |\\gamma_n|\\\,\\leq \\\,\\frac{\\delta}{2n(n+1
)}$ for $n\\ge 4$.\n\n\nThis talk is based on the following article.\n\n
S. Ponnusamy\, N. L. Sharma and K.-J. Wirths\,\nLogarithmic coefficients p
roblems in families related to starlike and convex functions\, . Aust. Mat
h. Soc.\, 109(2) (2019)\, 230--249.\n
LOCATION:https://researchseminars.org/talk/CAvid/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Fasondini (University of Leicester)
DTSTART;VALUE=DATE-TIME:20231017T130000Z
DTEND;VALUE=DATE-TIME:20231017T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/108
DESCRIPTION:Title:
Complex-plane singularity dynamics for blow up in a nonlinear heat equatio
n: analysis and computation\nby Marco Fasondini (University of Leicest
er) as part of CAvid: Complex Analysis video seminar\n\nLecture held in N/
A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Toby Driscoll (University of Delaware)
DTSTART;VALUE=DATE-TIME:20231024T130000Z
DTEND;VALUE=DATE-TIME:20231024T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/109
DESCRIPTION:Title:
AAA for rational interpolation on continuua\nby Toby Driscoll (Univers
ity of Delaware) as part of CAvid: Complex Analysis video seminar\n\nLectu
re held in N/A.\n\nAbstract\nThe AAA algorithm of Nakatsukasa\, Sète\, an
d Trefethen has rapidly risen to prominence as a fast and powerful way to
approximate functions in the complex plane. As originally presented\, AAA
incrementally constructs an approximation based on a fixed initial discret
ization\, which is not ideal in cases where a good initial distribution of
nodes may be difficult to discern. By also incrementally adding nodes fro
m the domain based on the latest residual\, the algorithm can be adapted t
o work well automatically even when singularities are very close to or eve
n on the approximation interval. This capability has been released as a Ju
lia software package\, and another package is in development to use these
approximations for computing conformal maps to simply- and doubly-connecte
d domains.\n
LOCATION:https://researchseminars.org/talk/CAvid/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luna Lomonaco (Institute of Pure and Applied Mathematics)
DTSTART;VALUE=DATE-TIME:20231031T130000Z
DTEND;VALUE=DATE-TIME:20231031T140000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/110
DESCRIPTION:Title:
Mating quadratic maps with the modular group\nby Luna Lomonaco (Instit
ute of Pure and Applied Mathematics) as part of CAvid: Complex Analysis vi
deo seminar\n\nLecture held in N/A.\n\nAbstract\nHolomorphic correspondenc
es are multi-valued maps defined by polynomial relations $P(z\,w)=0$. We c
onsider a specific 1-(complex)parameter family of (2:2) correspondences (e
very point has 2 images and 2 preimages)\nwhich encodes both the dynamics
of a quadratic rational map and the dynamics of the modular group. We show
that the connectedness locus for this family is homeomorphic to the parab
olic Mandelbrot set\, itself homeomorphic to the Mandelbrot set. Joint wor
k with S. Bullett.\n
LOCATION:https://researchseminars.org/talk/CAvid/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Buckingham (University of Cincinnati)
DTSTART;VALUE=DATE-TIME:20231107T140000Z
DTEND;VALUE=DATE-TIME:20231107T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/111
DESCRIPTION:Title:
Asymptotics of Rational Painlevé V Functions\nby Robert Buckingham (U
niversity of Cincinnati) as part of CAvid: Complex Analysis video seminar\
n\nLecture held in N/A.\n\nAbstract\nThe Painlevé functions are a family
of ordinary differential equations with myriad applications to mathematica
l physics and probability. The rational solutions of these equations have
drawn attention for the remarkable geometric structure of their zeros and
poles. We study the family of rational solutions of the Painlevé-V equa
tion built from Umemura polynomials. We derive a new Riemann-Hilbert repr
esentation and use it to obtain the boundary of the pole region and the la
rge-degree behavior in the pole-free region. This is joint work with Matt
hew Satter of the University of Cincinnati.\n
LOCATION:https://researchseminars.org/talk/CAvid/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liz Vivas (Ohio State University)
DTSTART;VALUE=DATE-TIME:20231114T140000Z
DTEND;VALUE=DATE-TIME:20231114T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/112
DESCRIPTION:Title:
Dimension results on generalized Bergman spaces\nby Liz Vivas (Ohio St
ate University) as part of CAvid: Complex Analysis video seminar\n\nLectur
e held in N/A.\n\nAbstract\nWiegerinck proved that the Bergman space over
any domain in the complex plane is either trivial or infinite dimensional.
In this talk I will discuss various generalizations and open questions re
lated to this theorem. I will survey the case of the complex plane being r
eplaced by C^n as well as a domain in a compact Riemann Surface.\n\nThe ta
lked is based in joint work with A-K. Gallagher and P. Gupta.\n
LOCATION:https://researchseminars.org/talk/CAvid/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guang-Yuan Zhang (Tsinghua University)
DTSTART;VALUE=DATE-TIME:20231128T140000Z
DTEND;VALUE=DATE-TIME:20231128T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/114
DESCRIPTION:Title:
The precise form of Ahlfors' Second Fundamental Theorem of covering surfac
es\nby Guang-Yuan Zhang (Tsinghua University) as part of CAvid: Comple
x Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nA simply con
nected covering surface $\\Sigma =\\left( f\,\\overline{\\Delta }%\n\\righ
t) $ over the unit Riemann sphere $S$ is an orientation-preserving\,\ncont
inuous\, open and finite-to-one mapping (OPCOFOM) $f$ from the closed\nuni
t disk $\\overline{\\Delta }$ into the sphere $S$. Here open means that $f
$\ncan be extended continuous and open to a neighborhood of $\\overline{\\
Delta }.\n$ We denote by $\\mathbf{F}$ all simply connected surfaces.\n\nL
et $E_{q}=\\left\\{ a_{1}\,a_{2}\,\\dots \,a_{q}\\right\\} $ be a set on t
he unit\nRiemann sphere consisting of $q$ distinct points with $q>2.$ \nAh
lfors' second\nfundamental theorem (SFT) states that there exists a positi
ve number $h$\ndepending only on $E_{q}\,$ such that for any surface $\\Si
gma =\\left( f\,%\n\\overline{\\Delta }\\right) \\in \\mathbf{F}\,$\n\\[\n
\\left( q-2\\right) A\\left( \\Sigma \\right) <4\\pi \\overline{n}\\left(
\\Sigma\n\\right) +hL\\left( \\partial \\Sigma \\right) \,\n\\]\nwhere $\\
Delta $ is the unit disk\, $A\\left( \\Sigma \\right) $ is the spherical\n
area of $\\Sigma $\, $L\\left( \\partial \\Sigma \\right) $ is the spheric
al\nlength of the boundary curve $\\partial \\Sigma =\\left( f\,\\partial
\\Delta\n\\right) \,$ and $\\overline{n}\\left( \\Sigma \\right) =\\#f^{-1
}(E_{q})\\cap\n\\Delta .$\n\nIf we define $R\\left( \\Sigma \\right) =R\\l
eft( \\Sigma \,E_{q}\\right) $ to be\nthe error term in Ahlfors' SFT\, say
\,\n\\[\nR\\left( \\Sigma \\right) =\\left( q-2\\right) A\\left( \\Sigma \
\right) -4\\pi\n\\overline{n}\\left( \\Sigma \\right) \,\n\\]\nthen Ahlfor
s' SFT reads\n\\[\nH_{0}=\\sup_{\\Sigma \\in \\mathbf{F}}\\left\\{ \\frac{
R(\\Sigma )}{L(\\partial\n\\Delta )}:\\Sigma =\\left( f\,\\overline{\\Delt
a }\\right) \\right\\} <+\\infty .\n\\]\nWe call $H_{0}=H_{0}(E_{q})$ Ahlf
ors' constant for simply connected\nsurfaces.\n\nIn this talk\, I will int
roduce my recent work which identify the precise\nbound $H_{0}=H_{0}(E_{q}
).$\n
LOCATION:https://researchseminars.org/talk/CAvid/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Faouzi Thabet (University of Gabès)
DTSTART;VALUE=DATE-TIME:20231212T140000Z
DTEND;VALUE=DATE-TIME:20231212T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/116
DESCRIPTION:Title:
Trajectories of Particular Quadratic Differentials on the Riemann Sphere\nby Faouzi Thabet (University of Gabès) as part of CAvid: Complex Anal
ysis video seminar\n\nLecture held in N/A.\n\nAbstract\nIn this lecture\,
we give some basics of the theory of Quadratic\nDifferentials on the Riema
nn Sphere. In the first part\, the focus will be on\nthe investigation of
the existence of finite critical trajectories\, and the\ndescription of th
e critical graph of some quadratic differentials related to\nsolutions as
Cauchy transform of a signed measure of an algebraic quadratic\nequation a
s the form : $p\\left( z\\right) \\mathcal{C}^{2}\\left( z\\right)\n+q\\le
ft( z\\right) \\mathcal{C}\\left( z\\right) +r=0\,$ for some polynomials $
p\,$\n$q$ and $r.$ As an example\, we study the large-degree analysis of t
he\nbehaviour of the generalized Laguerre polynomials $L_{n}^{(\\alpha )}$
when\nthe parameters are complex and depend on the degree $n$ linearly.\n
\nIn the second part\, we describe the critical graph of a polynomial quad
ratic\ndifferential related to the Schr\\"{o}dinger equation with cubic po
tential.\n
LOCATION:https://researchseminars.org/talk/CAvid/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva A. Gallardo Gutiérrez (ICMAT)
DTSTART;VALUE=DATE-TIME:20231219T140000Z
DTEND;VALUE=DATE-TIME:20231219T150000Z
DTSTAMP;VALUE=DATE-TIME:20231209T120128Z
UID:CAvid/117
DESCRIPTION:by Eva A. Gallardo Gutiérrez (ICMAT) as part of CAvid: Comple
x Analysis video seminar\n\nLecture held in N/A.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CAvid/117/
END:VEVENT
END:VCALENDAR