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BEGIN:VEVENT
SUMMARY:Jake Fillman (Texas State University)
DTSTART;VALUE=DATE-TIME:20200417T185000Z
DTEND;VALUE=DATE-TIME:20200417T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/1
DESCRIPTION:Title: Spe
ctra of Fibonacci Hamiltonians\nby Jake Fillman (Texas State Universit
y) as part of TAMU: Mathematical Physics and Harmonic Analysis Seminar\n\n
\nAbstract\nThe Fibonacci sequence is a prominent model of a 1D quasicryst
al. We will talk about some properties of continuum Schr\\"odinger operato
rs with potentials that are determined by the Fibonacci sequence. We show
that the spectrum is an (unbounded) Cantor set of zero Lebesgue measure an
d that the local Hausdorff dimension of the spectrum tends to one in the r
egimes of high energy and small coupling. We also show that multidimension
al Schr\\"odinger operators patterned on the Fibonacci sequence can exhibi
t the coexistence of two phenomena: (1) Cantor structure near the bottom o
f the spectrum and (2) an absence of gaps in the spectrum at high energies
. To prove (2)\, we develop an "abstract" Bethe--Sommerfeld criterion for
sums of extended Cantor sets\, which may be of independent interest. [Base
d on joint projects with David Damanik\, Anton Gorodetski\, and May Mei]\n
LOCATION:https://researchseminars.org/talk/MPHA/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Shipman (LSU)
DTSTART;VALUE=DATE-TIME:20200424T185000Z
DTEND;VALUE=DATE-TIME:20200424T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/2
DESCRIPTION:Title: Red
ucible and irreducible Fermi surfaces for periodic operators\nby Steph
en Shipman (LSU) as part of TAMU: Mathematical Physics and Harmonic Analys
is Seminar\n\n\nAbstract\nI will discuss new theorems concerning reducibil
ity of the Fermi surface for periodic Schrödinger operators. (1) Irreduc
ibility for a class of planar discrete graph operators\; (2) Reducibility
of multilayer graphs due to compatible asymmetries of the connecting edges
\; (3) Reducibility of multilayer graphs due to separability or bipartiten
ess of the layers. Parts of this work are in collaboration with Wei Li\,
Lee Fisher\, Karl-Michael Schmidt\, Ian Wood\, and Malcolm Brown.\n
LOCATION:https://researchseminars.org/talk/MPHA/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Milivoje Lukic (Rice University)
DTSTART;VALUE=DATE-TIME:20200501T185000Z
DTEND;VALUE=DATE-TIME:20200501T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/3
DESCRIPTION:Title: Sta
hl--Totik regularity for continuum Schr\\"odinger operators\nby Milivo
je Lukic (Rice University) as part of TAMU: Mathematical Physics and Harmo
nic Analysis Seminar\n\n\nAbstract\nThis talk describes joint work with Be
njamin Eichinger: a\ntheory of regularity for one-dimensional continuum Sc
hr\\"odinger\noperators\, based on the Martin compactification of the comp
lement of\nthe essential spectrum. For a half-line Schr\\"odinger operator
\n$-\\partial_x^2+V$ with a bounded potential $V$\, it was previously\nkno
wn that the spectrum can have zero Lebesgue measure and even zero\nHausdor
ff dimension\; however\, we obtain universal thickness statements\nin the
language of potential theory.\nNamely\, we prove that the essential spectr
um is not polar\, it obeys\nthe Akhiezer--Levin condition\, and moreover\,
the Martin function at\n$\\infty$ obeys the two-term asymptotic expansion
$\\sqrt{-z} +\n\\frac{a}{2\\sqrt{-z}} + o(\\frac 1{\\sqrt{-z}})$ as $z \\
to -\\infty$. The\nconstant $a$ in its asymptotic expansion plays the role
of a\nrenormalized Robin constant suited for Schr\\"odinger operators and
\nenters a universal inequality $a \\le \\liminf_{x\\to\\infty} \\frac 1x\
n\\int_0^x V(t) dt$. This leads to a notion of regularity\, with\nconnecti
ons to the exponential growth rate of Dirichlet solutions and\nthe zero co
unting measures for finite restrictions of the operator. We\nalso present
applications to decaying and ergodic potentials.\n
LOCATION:https://researchseminars.org/talk/MPHA/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Li (LSU)
DTSTART;VALUE=DATE-TIME:20200508T185000Z
DTEND;VALUE=DATE-TIME:20200508T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/4
DESCRIPTION:Title: Emb
edded eigenvalues of the Neumann Poincaré operator\nby Wei Li (LSU) a
s part of TAMU: Mathematical Physics and Harmonic Analysis Seminar\n\n\nAb
stract\nThe Neumann-Poincaré (NP) operator arises in boundary value probl
ems\, and plays an important role in material design\, signal amplificatio
n\, particle detection\, etc. The spectrum of the NP operator on domains w
ith corners was studied by Carleman before tools for rigorous discussion w
ere created\, and received a lot of attention in the past ten years. In th
is talk\, I will present our discovery and verification of eigenvalues emb
edded in the continuous spectrum of this operator. The main ideas are deco
upling of spaces by symmetry and construction of approximate eigenvalues.
This is based on two works with Stephen Shipman and Karl-Mikael Perfekt.\n
LOCATION:https://researchseminars.org/talk/MPHA/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Jauslin (Princeton University)
DTSTART;VALUE=DATE-TIME:20200515T185000Z
DTEND;VALUE=DATE-TIME:20200515T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/5
DESCRIPTION:Title: A s
imple equation to study interacting Bose gasses\nby Ian Jauslin (Princ
eton University) as part of TAMU: Mathematical Physics and Harmonic Analys
is Seminar\n\n\nAbstract\nIn this talk\, I will discuss a partial differen
tial equation introduced by\n Lieb in 1963 in the context of studying int
eracting Bose gasses. I will first\n discuss how this equation can be use
d to accurately compute physically\n relevant quantities related to the B
ose gas\, such as the ground state energy\n and condensate fraction. I wi
ll then present a construction of the solutions\n to the equation\, and d
iscuss some of their properties.\n
LOCATION:https://researchseminars.org/talk/MPHA/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Kachkovskiy (MSU)
DTSTART;VALUE=DATE-TIME:20200522T185000Z
DTEND;VALUE=DATE-TIME:20200522T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/6
DESCRIPTION:Title: On
spectral band edges of discrete periodic Schrodinger operators\nby Ily
a Kachkovskiy (MSU) as part of TAMU: Mathematical Physics and Harmonic Ana
lysis Seminar\n\n\nAbstract\nWe consider discrete Schrodinger operators on
$\\ell^2(\\mathbb Z^d)$\, periodic with respect to some lattice $\\Gamma$
in $\\mathbb Z^d$ of full rank. Our main goal is to study dimensions of l
evel sets of spectral band functions at the energies corresponding to thei
r extremal values (the edges of the bands).Suppose that $d\\ge 3$ and the
dual lattice $\\Gamma’$ does not contain the vector $(1/2\,…\,1/2)$. T
hen the above mentioned level sets have dimension at most $d-2$.\n\nSuppos
e that $d=2$ and the dual lattice does not contain vectors of the form $(1
/p\,1/p)$ and $(1/p\,-1/p)$ for all $p\\ge 2$. Then the same statement hol
ds (in other words\, the corresponding level sets are finite modulo $\\mat
hbb Z^d$).For all lattices that do not satisfy the above assumptions\, the
re are known counterexamples of level sets of dimensions $d-1$.\n\nPart of
the argument also implies a discrete Bethe-Sommerfeld property: if $d\\ge
2$ and the dual lattice does not contain the vector $(1/2\,…\,1/2)$\, t
hen\, for sufficiently small potentials (depending on the lattice)\, the s
pectrum of the periodic Schrodinger operator is an interval. Previously\,
this property was studied by Kruger\, Embree-Fillman\, Jitomirskaya-Han\,
and Fillman-Han. Our proof is different and implies some new cases.\n\nThe
talk is based on joint work with in progress with N. Filonov.\n
LOCATION:https://researchseminars.org/talk/MPHA/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christoph Fischbacher (UC Irvine)
DTSTART;VALUE=DATE-TIME:20200529T185000Z
DTEND;VALUE=DATE-TIME:20200529T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/7
DESCRIPTION:Title: Log
arithmic lower bounds for the entanglement entropy of droplet states for
the XXZ model on the ring\nby Christoph Fischbacher (UC Irvine) as pa
rt of TAMU: Mathematical Physics and Harmonic Analysis Seminar\n\n\nAbstra
ct\nWe study the free XXZ quantum spin model defined on a ring of size L a
nd \nshow that the bipartite entanglement entropy of eigenstates belonging
to \nthe first energy band above the vacuum ground state satisfy a \nloga
rithmically corrected area law. This is joint work with Ruth Schulte \n(LM
U).\n
LOCATION:https://researchseminars.org/talk/MPHA/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Kuchment (TAMU)
DTSTART;VALUE=DATE-TIME:20200731T185000Z
DTEND;VALUE=DATE-TIME:20200731T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/8
DESCRIPTION:Title: Spe
ctral properties of periodically perforated spaces\nby Peter Kuchment
(TAMU) as part of TAMU: Mathematical Physics and Harmonic Analysis Seminar
\n\n\nAbstract\nWe study spectra of Schr\\"odinger operators with periodic
\npotentials in R^n with periodic perforations. We prove analytic \ndepen
dence on the shape of the perforation and absolute continuity of \nthe spe
ctrum.\n
LOCATION:https://researchseminars.org/talk/MPHA/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Casey Rodriguez (MIT)
DTSTART;VALUE=DATE-TIME:20200807T203000Z
DTEND;VALUE=DATE-TIME:20200807T213000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/9
DESCRIPTION:Title: The
Radiative Uniqueness Conjecture for Bubbling Wave Maps\nby Casey Rodr
iguez (MIT) as part of TAMU: Mathematical Physics and Harmonic Analysis Se
minar\n\n\nAbstract\nWe will discuss the finite time breakdown of solution
s to a canonical example of a geometric wave equation: energy critical wav
e maps. Breakthrough works of Krieger-Schlag–Tataru\, Rodnianski-Sterben
z and Raphael–Rodnianski produced examples of wave maps that develop sin
gularities in finite time. These solutions break down by concentrating ene
rgy at a point in space (via bubbling a harmonic map) but have a regular l
imit\, away from the singular point\, as time approaches the final time of
existence. The regular limit is referred to as the radiation. This mechan
ism of breakdown occurs in many other PDE including energy critical wave e
quations\, Schrodinger maps and Yang-Mills equations. A basic question is
the following:\n\nCan we give a precise description of all bubbling singul
arities for wave maps with the goal of finding the natural unique continua
tion of such solutions past the singularity?\n\nIn this talk\, we will dis
cuss recent work (joint with J. Jendrej and A. Lawrie) which is the first
to directly and explicitly connect the radiative component to the bubbling
dynamics by constructing and classifying bubbling solutions with a simple
form of prescribed radiation. Our results serve as an important first ste
p in formulating and proving the following Radiative Uniqueness Conjecture
for a large class of wave maps: every bubbling solution is uniquely chara
cterized by its radiation\, and thus\, every bubbling solution can be uniq
uely continued past blow-up time while conserving energy.\n
LOCATION:https://researchseminars.org/talk/MPHA/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Avy Soffer (Rutgers)
DTSTART;VALUE=DATE-TIME:20200814T185000Z
DTEND;VALUE=DATE-TIME:20200814T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/10
DESCRIPTION:Title: Ev
olution of NLS with Bounded Data\nby Avy Soffer (Rutgers) as part of T
AMU: Mathematical Physics and Harmonic Analysis Seminar\n\n\nAbstract\nWe
study the nonlinear Schrodinger equation (NLS) with bounded initial data w
hich does not vanish at infinity. Examples include periodic\, quasi-period
ic and random initial data. On the lattice we prove that solutions are pol
ynomially bounded in time for any bounded data. In the continuum\, local e
xistence is proved for real analytic data by a Newton iteration scheme. Gl
obal existence for NLS with a regularized nonlinearity follows by analyzin
g a local energy norm (arXiv:2003.08849 [math.AP]\, J.Stat.Phys\, 2020).\
nThis is a joint work with Ben Dodson and Tom Spencer.\n
LOCATION:https://researchseminars.org/talk/MPHA/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Tovbis (University of Central Florida)
DTSTART;VALUE=DATE-TIME:20200821T185000Z
DTEND;VALUE=DATE-TIME:20200821T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/11
DESCRIPTION:Title: So
liton and breather gases for the focusing Nonlinear Schrödinger equation
(fNLS): spectral theory and possible applications\nby Alexander Tovbis
(University of Central Florida) as part of TAMU: Mathematical Physics and
Harmonic Analysis Seminar\n\n\nAbstract\nIn the talk we introduce the ide
a of an "integrable gas" as a collection of large random ensembles of spec
ial localized solutions (solitons\, breathers) of a given integrable syste
m. These special solutions can be treated as "particles". Known laws of pa
irwise elastic collisions allow one to write the heuristic "equation of st
ate" for the gas of such particles.\n\nIn this talk we consider soliton an
d breather gases for the fNLS as special thermodynamic limits of finite ga
p (nonlinear multi phase wave) fNLS solutions. In this limit the rate of g
rowth of the number of bands is linked with the rate of (simultaneous) shr
inkage of the size of individual bands. This approach leads to the derivat
ion of the equation of state for the gas and its certain limiting regimes
(condensate\, ideal gas limits)\, as well as construction of various inter
esting examples. We also discuss the recent progress and perspectives of f
uture work\, as well as some possible applications.\n
LOCATION:https://researchseminars.org/talk/MPHA/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seonghyeon Jeong (MSU)
DTSTART;VALUE=DATE-TIME:20200911T185000Z
DTEND;VALUE=DATE-TIME:20200911T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/12
DESCRIPTION:Title: St
rong MTW type condition to local Holder regularity in generated Jacobian e
quations\nby Seonghyeon Jeong (MSU) as part of TAMU: Mathematical Phy
sics and Harmonic Analysis Seminar\n\n\nAbstract\nn this talk\, we present
a proof of local Holder regularity of solutions to generated Jacobian equ
ations as a generalization of optimal transport case\, which is proved by
George Loeper. We compare structures of generated Jacobian equations with
optimal transport\, and point out differences with difficulties which the
differences can cause. For local Holder regularity theory\, we use (G3s) c
ondition and solution in Alexandrov sense. (G3s) is a strict positiveness
type condition on MTW tensor associated to the generating function G\, and
Alexandrov solution is a solution that satisfies pullback measure conditi
on. (G3s) is used to show a quantitative version of (glp)\, which gives so
me room for G-subdifferentials of solutions. Then the inequality for Holde
r regularity is shown by comparing volumes of G-subdifferentials using the
fact that our solutions is in Alexandrov sense.\n
LOCATION:https://researchseminars.org/talk/MPHA/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Breuer (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20201001T150000Z
DTEND;VALUE=DATE-TIME:20201001T160000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/13
DESCRIPTION:Title: Pe
riodic Jacobi Matrices on Trees\nby Jonathan Breuer (Hebrew University
of Jerusalem) as part of TAMU: Mathematical Physics and Harmonic Analysis
Seminar\n\n\nAbstract\nThe theory of periodic Jacobi matrices on the line
is extremely rich and very well studied. Viewing the line as a regular tr
ee of degree 2 leads to a natural generalization to periodic Jacobi matric
es on general trees. This family of operators\, which is at least as rich
(by definition)\, but considerably less well understood\, is at the center
of this talk. We review some of the few known results\, present some exam
ples\, and discuss open problems and directions for future research. The t
alk is based on joint work with Nir Avni and Barry Simon.\n
LOCATION:https://researchseminars.org/talk/MPHA/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Delio Mugnolo (University of Hagen)
DTSTART;VALUE=DATE-TIME:20201008T150000Z
DTEND;VALUE=DATE-TIME:20201008T160000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/14
DESCRIPTION:Title: Bi
-Laplacians on graphs: self-adjoint extensions and parabolic theory\nb
y Delio Mugnolo (University of Hagen) as part of TAMU: Mathematical Physic
s and Harmonic Analysis Seminar\n\n\nAbstract\nElastic beams have been stu
died by means of hyperbolic equations driven by bi-Laplacian operators sin
ce the early 18th century: several properties of the corresponding parabol
ic equation on the whole Euclidean space have been discovered since the 19
60s by Krylov\, Hochberg\, and Davies\, among others. On a bounded domain
or a metric graph\, the bi-Laplacian is generally not anymore acting as a
squared operator\, though: this strongly affects the features of associate
d PDEs.\n\nI am going to present a full characterization of self-adjoint e
xtensions of the bi-Laplacian\, focusing on a class of realizations that e
ncode dynamic boundary conditions. Maximum principles of parabolic equatio
ns will also be discussed: after a transient time\, I am going to show tha
t solutions often display Markovian features.\n\nThis is joint work with F
ederica Gregorio.\n
LOCATION:https://researchseminars.org/talk/MPHA/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiaqi Yang (GeorgiaTech)
DTSTART;VALUE=DATE-TIME:20200925T185000Z
DTEND;VALUE=DATE-TIME:20200925T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/15
DESCRIPTION:Title: Pe
rsistence of Invariant Objects in Functional Differential Equations close
to ODEs\nby Jiaqi Yang (GeorgiaTech) as part of TAMU: Mathematical Phy
sics and Harmonic Analysis Seminar\n\n\nAbstract\nWe consider functional d
ifferential equations which are perturbations of ODEs in $\\mathbb{R}^n$.
This is a singular perturbation problem even for small perturbations. We p
rove that for small enough perturbations\, some invariant objects of the u
nperturbed ODEs persist and depend on the parameters with high regularity.
We formulate a-posteriori type of results in the case when the unperturbe
d equations admit periodic orbits. The results apply to state-dependent de
lay equations and equations which arise in the study of electrodynamics. T
he proof is constructive and leads to an algorithm. This is a joint work w
ith Joan Gimeno and Rafael de la Llave.\n
LOCATION:https://researchseminars.org/talk/MPHA/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rodrigo Matos (Texas A&M University)
DTSTART;VALUE=DATE-TIME:20201002T185000Z
DTEND;VALUE=DATE-TIME:20201002T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/16
DESCRIPTION:Title: Dy
namical Contrast on Highly Correlated Anderson-type models\nby Rodrigo
Matos (Texas A&M University) as part of TAMU: Mathematical Physics and Ha
rmonic Analysis Seminar\n\n\nAbstract\nWe present examples of random Schö
dinger operators obtained in a similar fashion but exhibiting distinct tra
nsport properties. The models are constructed by connecting\, in different
ways\, infinitely many copies of the one dimensional Anderson model. \nSp
ectral aspects of the models will also be presented. In particular\, we ob
tain a physically motivated example of a random operator with purely absol
utely continuous spectrum where the transient and recurrent components coe
xist. This can be interpreted as a sharp phase transition within the absol
utely continuous spectrum. Time allowing\, I will discuss some tools relat
ed to harmonic analysis\, including a version of Boole's equality which\,
to the best of our knowledge\, is new. Based on joint work with Rajinder M
avi and Jeffrey Schenker.\n
LOCATION:https://researchseminars.org/talk/MPHA/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Farhan Abedin (Michigan State University)
DTSTART;VALUE=DATE-TIME:20201030T185000Z
DTEND;VALUE=DATE-TIME:20201030T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/18
DESCRIPTION:Title: He
le-Shaw Flow and Parabolic Integro-Differential Equations\nby Farhan A
bedin (Michigan State University) as part of TAMU: Mathematical Physics an
d Harmonic Analysis Seminar\n\n\nAbstract\nI will present a regularization
result for a special case of the two-phase Hele-Shaw free boundary proble
m (a.k.a. interfacial Darcy flow)\, which models the evolution of two immi
scible fluids flowing in the narrow gap between two parallel plates and su
bject to an external pressure source. Assuming that the fluid interface is
given by the graph of a function\, recent work of Chang-Lara\, Guillen\,
and Schwab establishes the equivalence between the Hele-Shaw free boundary
problem and a first-order parabolic integro-differential equation. By exp
loiting this equivalence and using available regularity theory for nonloca
l parabolic equations\, we show that if the gradient of the graph of the f
luid interface has a Dini modulus of continuity for all times\, then the g
radient must be Holder continuous. This is joint work with Russell Schwab
(MSU).\n
LOCATION:https://researchseminars.org/talk/MPHA/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Kitagawa (Michigan State University)
DTSTART;VALUE=DATE-TIME:20201023T185000Z
DTEND;VALUE=DATE-TIME:20201023T195000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/19
DESCRIPTION:Title: On
free discontinuities in optimal transport\nby Jun Kitagawa (Michigan
State University) as part of TAMU: Mathematical Physics and Harmonic Analy
sis Seminar\n\n\nAbstract\nIt is well known that regularity results for th
e optimal transport (Monge-Kantorovich) problem require rigid geometric re
strictions. In this talk\, we consider the structure of the set of ``free
discontinuities'' which arise when transporting mass from a connected doma
in to a disconnected one\, and show regularity of this set\, along with a
stability result under suitable perturbations of the target measure. These
are based on a non-smooth implicit function theorem for convex functions\
, which is of independent interest. This talk is based on joint work with
Robert McCann (Univ. of Toronto).\n
LOCATION:https://researchseminars.org/talk/MPHA/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Egger (Technion)
DTSTART;VALUE=DATE-TIME:20201029T150000Z
DTEND;VALUE=DATE-TIME:20201029T160000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/20
DESCRIPTION:Title: We
ll-defined spectral position for Neumann domains\nby Sebastian Egger (
Technion) as part of TAMU: Mathematical Physics and Harmonic Analysis Semi
nar\n\n\nAbstract\nA Laplacian eigenfunction on a two-dimensional Riemanni
an manifold provides a natural partition generated by specific gradient fl
ow lines of the eigenfunction. The restricted eigenfunction onto the parti
tion's components satisfies Neumann boundary conditions and the components
are therefore coined 'Neumann domains'. Neumann domains represent a compl
ementary path to the famous nodal-domain partition to study elliptic eigen
functions where the latter is associated with the Dirichlet Laplacian. A v
ery basic but fundamental property of nodal domains is that the restricted
eigenfunction onto a nodal domain always gives the ground-state of the Di
richlet Laplacian. That feature becomes significantly more complex for Neu
mann domains due to the presence of possible cusps and cracks. In this tal
k\, we focus on this problem and show that the spectral position for Neuma
nn domains is well-defined. Moreover\, we provide explicit examples of Neu
mann domains displaying a fundamentally different behavior in their spectr
al position than their nodal-domain counterparts.\n
LOCATION:https://researchseminars.org/talk/MPHA/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Kennedy (University of Lisbon)
DTSTART;VALUE=DATE-TIME:20201119T160000Z
DTEND;VALUE=DATE-TIME:20201119T170000Z
DTSTAMP;VALUE=DATE-TIME:20240715T183742Z
UID:MPHA/21
DESCRIPTION:Title: Sp
ectral partitions of metric graphs\nby James Kennedy (University of Li
sbon) as part of TAMU: Mathematical Physics and Harmonic Analysis Seminar\
n\n\nAbstract\nWe introduce a theory of partitions of metric graphs via sp
ectral-type functionals\, inspired by the theory of spectral minimal parti
tions of domains but also with a view to understanding how to detect "clus
ters" in metric graphs.\n\nThe goal is to associate with any given partiti
on a spectral energy built around eigenvalues of differential operators li
ke the Laplacian\, and then minimize (or maximize) this energy over all ad
missible partitions. Since metric graphs are essentially one-dimensional m
anifolds with singularities (the vertices)\, the range of well-posed probl
ems is much greater than on domains. We first sketch a general existence t
heory for optimizers of such partition functionals\, and discuss a number
of natural functionals and optimization problems.\n\nWe also illustrate ho
w changing the functionals and the classes of partitions under considerati
on -- for example\, imposing Dirichlet versus standard conditions at the c
ut vertices or considering min-max versus max-min type functionals -- may
lead to qualitatively different optimal partitions which seek out differen
t features of the graph.\n\nFinally\, we show how for many problems the op
timal energies behave very similarly to the eigenvalues of the Laplacian (
with Dirichlet or standard vertex conditions)\, in terms of Weyl asymptoti
cs\, upper and lower bounds\, and interlacing inequalities.\n\nThis is bas
ed on joint works with Matthias Hofmann\, Pavel Kurasov\, Corentin Léna\,
Delio Mugnolo and Marvin Plümer.\n
LOCATION:https://researchseminars.org/talk/MPHA/21/
END:VEVENT
END:VCALENDAR