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SUMMARY:Nelia Charalambous (University of Cyprus)
DTSTART;VALUE=DATE-TIME:20201217T143000Z
DTEND;VALUE=DATE-TIME:20201217T153000Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/1
DESCRIPTION:Title: The form spectrum of open manifolds\nby Nelia Charalambous (
University of Cyprus) as part of Virtual Mini-Workshop in Geometric Analys
is\n\n\nAbstract\nThe computation of the essential spectrum of the Laplaci
an requires the construction of a large class of test differential forms.
On a general open manifold this is a difficult task\, since there exists o
nly a small collection of canonically defined differential forms to work w
ith. In our work with Zhiqin Lu\, we compute the essential k-form spectrum
over asymptotically flat manifolds by combining two methods: First\, we i
ntroduce a new version of the generalized Weyl criterion\, which greatly r
educes the regularity and smoothness of the test differential forms\; seco
nd\, we make use of Cheeger-Fukaya-Gromov theory and Cheeger-Colding theor
y to obtain a new type of test differential forms at the ends of the manif
old. We also use the generalized Weyl criterion to obtain other interestin
g facts about the k-form essential spectrum over an open manifold.\n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/1/
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BEGIN:VEVENT
SUMMARY:Alexander Strohmaier (University of Leeds)
DTSTART;VALUE=DATE-TIME:20201217T154500Z
DTEND;VALUE=DATE-TIME:20201217T164500Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/2
DESCRIPTION:Title: The spectral shift function and a relative trace formula\nby
Alexander Strohmaier (University of Leeds) as part of Virtual Mini-Works
hop in Geometric Analysis\n\n\nAbstract\nSpectral theory of the Laplace op
erator on asymptotically Euclidean manifolds is described to a certain ext
ent by stationary scattering theory. I will define the spectral shift func
tion in this context and review some results for scattering of p-forms and
their application. \nIn the second part of the talk I will specialise to
obstacle scattering and explain a new trace formula and its relation to th
e spectral shift function. If there is time I will give some applications
in physics. (based on joint work with Y. Fang\, F. Hanisch and A. Waters)\
n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theodora Bourni (University of Tennessee)
DTSTART;VALUE=DATE-TIME:20201217T183000Z
DTEND;VALUE=DATE-TIME:20201217T193000Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/3
DESCRIPTION:Title: Ancient solutions to mean curvature flow\nby Theodora Bourni
(University of Tennessee) as part of Virtual Mini-Workshop in Geometric A
nalysis\n\n\nAbstract\nMean curvature flow (MCF) is the gradient flow of t
he area functional\; it moves the surface in the direction of steepest dec
rease of area. An important motivation for the study of MCF comes from it
s potential geometric applications\, such as classification theorems and g
eometric inequalities. MCF develops ``singularities'' (curvature blow-up)\
, which obstruct the flow from existing for all times and therefore unders
tanding these high curvature regions is of great interest. This is done b
y studying ancient solutions\, solutions that have existed for all times i
n the past\, and which model singularities. In this talk we will discuss t
heir importance and ways of constructing and classifying such solutions. I
n particular\, we will focus on ``collapsed'' solutions and construct\, in
all dimensions $n\\ge 2$\, a large family of new examples\, including bot
h symmetric and asymmetric examples\, as well as many eternal examples tha
t do not evolve by translation. Moreover\, we will show that collapsed so
lutions decompose ``backwards in time'' into a canonical configuration of
Grim hyperplanes which satisfies certain necessary conditions. This is joi
nt work with Mat Langford and Giuseppe Tinaglia.\n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/3/
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BEGIN:VEVENT
SUMMARY:Lazaro Recht (IAM)
DTSTART;VALUE=DATE-TIME:20201217T194500Z
DTEND;VALUE=DATE-TIME:20201217T204500Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/4
DESCRIPTION:Title: The Poincaré Half Space of a C* Algebra\nby Lazaro Recht (
IAM) as part of Virtual Mini-Workshop in Geometric Analysis\n\n\nAbstract\
nFor the abstract\, please look at the homepage of the event:\nhttps://mat
ematicas.uniandes.edu.co/es/workshop-geometric-analysis.\n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/4/
END:VEVENT
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SUMMARY:Florent Schaffhauser (Universidad de los Andes)
DTSTART;VALUE=DATE-TIME:20201218T143000Z
DTEND;VALUE=DATE-TIME:20201218T153000Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/5
DESCRIPTION:Title: Twisted local systems and equivariant harmonic maps\nby Flor
ent Schaffhauser (Universidad de los Andes) as part of Virtual Mini-Worksh
op in Geometric Analysis\n\n\nAbstract\nDiscrete subgroups of PSL(2\;R) ca
n be interpreted geometrically as hyperbolic 2-orbifolds. In the absence o
f torsion\, a finite-dimensional representation of such a group gives a ri
se to a local system on a surface. To classify the latter up to isomorphis
m (on a compact surface)\, it is useful to equip these objects with specia
l Hermitian metrics. Corlette's theory gives a construction of such metric
s in terms of equivariant harmonic maps\, going from the hyperbolic plane
to the symmetric space of a semisimple Lie group. In this talk\, we recall
the main features of this theory and discuss how to generalize it in orde
r to include discrete subgroups of PSL(2\;R) that are no longer torsion-fr
ee.\n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/5/
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BEGIN:VEVENT
SUMMARY:Ksenia Fedosova (University of Freiburg)
DTSTART;VALUE=DATE-TIME:20201218T154500Z
DTEND;VALUE=DATE-TIME:20201218T164500Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/6
DESCRIPTION:Title: On a generalization of transfer operators\nby Ksenia Fedosov
a (University of Freiburg) as part of Virtual Mini-Workshop in Geometric A
nalysis\n\n\nAbstract\nFor hyperbolic manifolds\, there exists a straightf
orward connection between the spectral and the geometric data. More precis
ely\, the lengths of its closed geodesics and the spectrum of its Laplace
operator acting on functions are connected by the Selberg trace formula\,
that can be considered a sibling of the Poisson summation formula. Selberg
trace formula provides the information on the eigenvalues of the Laplace
operator\, however\, completely ignoring its eigenfunctions.\n \nThere exi
sts a method\, originated from the classical statistical mechanics\, that
allows to obtain more information on the eigenfunctions. The method\, call
ed the transfer operator approach\, involves a construction of a so-called
transfer operator from a certain discretisation of the geodesic flow on t
he manifold. For a modular surface\, this transfer operator is ultimately
connected to a Gauss map. One can show that the 1-eigenfunctions of this o
perator correspond via a certain integral transform to the eigenfunctions
of the Laplace operator. The integral transform mirrors the Eichler-Shimur
a-Manin isomorphism.\n \nIn this talk\, inspired by Bismut's hypoelliptic
Laplacians\, we try to construct an analogue of the transfer operator\, us
ing the Brownian paths on the manifold instead of the geodesics. We obtain
an operator\, whose 1-eigenfunctions turn out to be the boundary forms of
eigenfunctions of the Laplace operator.\n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rafe Mazzeo (Stanford University)
DTSTART;VALUE=DATE-TIME:20201218T183000Z
DTEND;VALUE=DATE-TIME:20201218T193000Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/7
DESCRIPTION:Title: ALG spaces and Hitchin systems\nby Rafe Mazzeo (Stanford Uni
versity) as part of Virtual Mini-Workshop in Geometric Analysis\n\n\nAbstr
act\nAn ALG space is a 4-dimensional hyperkaehler manifold with a very spe
cial asymptotic structure. I will survey some known results about their ge
ometry and topology and some recent results by others about their moduli.
These spaces can also arise as moduli spaces for solutions of the Hitchin
equations. The precise correspondence between these two rather different
pictures is a special case of Boalch’s `modularity conjecture’. This t
alk will focus mostly on describing the various ingredients and techniques
that go into this\, leading to a description of some recent progress obta
ined in collaboration with Fredrickson\, Swoboda and Weiss.\n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raquel Perales (IMATE-UNAM)
DTSTART;VALUE=DATE-TIME:20201218T194500Z
DTEND;VALUE=DATE-TIME:20201218T204500Z
DTSTAMP;VALUE=DATE-TIME:20220528T194842Z
UID:VMWinGeomAnalysis/8
DESCRIPTION:Title: Limits of manifolds with boundary\nby Raquel Perales (IMATE-
UNAM) as part of Virtual Mini-Workshop in Geometric Analysis\n\n\nAbstract
\nI will discuss available convergence results for manifolds with boundary
. \nIn particular\, we will focus on intrinsic flat and Gromov-Hausdorff c
onvergence results. \nWe will first consider convergence of sequences of m
anifolds with Ricci curvature and mean curvature bounds and we will finali
ze with a convergence result for sequences of the form $(M\,g_j)$\, $j=0\
,1\,...$\, that satisfy $d_{g_0} \\leq d_{g_j}$\, among other conditions
\, and where we are able to show that the limit equals $(M\,g_0)$.\n
LOCATION:https://researchseminars.org/talk/VMWinGeomAnalysis/8/
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