Seminars Sorted by Series
Analysis and Mathematical Physics
Continuous Symmetry Breaking: A Rigorous Approach
Sara Daneri
2:30pm|Simonyi Hall 101 and Remote Access
At the base of spontaneous pattern formation is universally
believed to be the competition between short range attractive and
long range repulsive forces. Though such a phenomenon is observed
in experiments and simulations, a rigorous understanding...
Global Well-Posedness of Stochastic Abelian-Higgs in Two Dimensions
Sky Cao
2:30pm|Simonyi Hall 101 and Remote Access
There has been much recent progress on the local solution theory
for geometric singular SPDEs. However, the global theory is still
largely open. In my talk, I will discuss the global well-posedness
of the stochastic Abelian-Higgs model in two...
Inertial Manifolds for the Hyperbolic Cahn-Hilliard Equation
Ahmed Bonfoh
2:30pm|Simonyi Hall 101 and Remote Access
An inertial manifold is a positively invariant smooth
finite-dimensional manifold which contains the global attractor and
which attracts the trajectories at a uniform exponential rate. It
follows that the infinite-dimensional dynamical system is...
Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties
Pavel Kurasov
2:30pm|Simonyi Hall 101 and Remote Access
Fourier Quasicrystals (FQ) are defined as crystalline measures
$$ \mu = \sum_{\lambda \in \Lambda} a_\lambda \delta_\lambda, \quad
\hat{\mu} = \sum_{s \in S} b_s \delta_s, $$ so that not only $ \mu
$ (and hence $ \hat{\mu} $) are tempered...
Serrin’s Overtermined Problem In Rough Domains
2:30pm|Simonyi Hall 101 and Remote Access
The classical Serrin’s overdetermined theorem states that a C^2
bounded domain, which admits a function with constant Laplacian
that satisfies both constant Dirichlet and Neumann boundary
conditions, must necessarily be a ball. While extensions of...
Evolution of Coherent Structures in Incompressible Flows
2:30pm|Simonyi Hall 101 and Remote Access
In this talk, we will explore recent developments in the study
of coherent structures evolving by incompressible flows. Our focus
will be on the behavior of fluid interfaces and vortex filaments.
We include the dynamics of gravity Stokes interfaces...
Duality of the Navier-Stokes Dynamics and Lack of Finite-Time Explosion
2:30pm|Simonyi Hall 101 and Remote Access
We reduce the solution of the Navier-Stokes (NS) equation in
infinite Euclidean space of arbitrary dimension to a
one-dimensional singular problem. Assuming rough initial data,
characterized by a Gaussian distribution around an arbitrary
velocity...
Absolute Continuity of the Robin Harmonic Measure On Rough Domains
Guy David
2:30pm|Simonyi Hall 101 and Remote Access
The question of asbolute continuity, with respect to the
reference measure, of the harmonic measure on a domain with rough
boundary has been the object of many important results. Here we ask
about the similar question, but where the Dirichlet...
2:30pm|Simonyi Hall 101 and Remote Access
2:30pm|Simonyi Hall 101 and Remote Access
2:30pm|Simonyi Hall 101 and Remote Access
2:30pm|Simonyi Hall 101 and Remote Access
2:30pm|Simonyi Hall 101 and Remote Access
Analysis Seminar
Orbit of the Diagonal of a Power of a Nilmanifold
Alexander Leibman
Let p_1,...,p_k be integer polynomials of one or several
variables. There is a relation between the density of polynomial
configurations a+p_1(n),...,a+p_k(n) in sets of integers and the
form of the closure of the diagonal of X^k under the...
Lior Siberman
2:00pm|West Bldg. Lecture Hall
Stationary Measures and Equidistribution on the Torus
10:30am|West Bldg. Lecture Hall
In this talk I will consider actions of non-abelian groups on
n-dimensional tori, explain the notions of stiffness and stationary
measures, and show how under fairly general assumptions stationary
measures can be classified. A key ingredient is a...
Expanders and Random Walks in SL(d,q)
A Hardy Field Extension of Szemeredi's Theorem
In 1975 Szemeredi proved that every subset of the integers with
positive density contains arbitrarily long arithmetic progressions.
Bergelson and Leibman showed in 1996 that the common difference of
the arithmetic progression can be a square, a cube...
On the Two dimensional Bilinear Hilbert Transform and Z^2 Actions
2:00pm|West Bldg. Lecture Hall
We investigate the Bilinear Hilbert Transform in the plane and
the pointwise convergence of bilinear averages in Ergodic theory,
arising from Z^2 actions. Our techniques combine novel one and a
half dimensional phase-space analysis with more...
On the Instability for 2D Fluids
For 2D Euler equation, we prove a double exponential lower bound
on the vorticity gradient. We will also discus some further results
on the singularity formation for other models.
On the Rigidity of Black Holes
Sergiu Klainerman
The classical result on the uniqueness of black holes in GR, due
to Hawking, which asserts that regular, stationary solutions of the
Einstein vacuum equations must be isometric to an admissible black
hole Kerr solution, has at its core a a highly...
The Defocusing Cubic Nonlinear Wave Equation in the Energy-Supercritical Regime
In this talk, we will present some recent results in the study
of the nonlinear wave equation with cubic defocusing nonlinearity,
describing the completion of a program to establish global
well-posedness and scattering in the energy-supercritical...
Around the Davenport-Heilbronn Function
The Davenport-Heilbronn function (introduced by Titchmarsh) is a
linear combination of the two L-functions with a complex character
mod 5, with a functional equation of L-function type but for which
the analogue of the Riemann hypothesis fails. In...
Tangent Cones to Calibrated Currents
Constante Bellettini
2:00pm|West Bldg. Lecture Hall
Calibrated currents are a particular class of volume-minimizers
and as such provide interesting explicit examples of solutions to
Plateau's problem. Their role goes however much beyond that: they
naturally appear when dealing with several geometric...
The Energy-Critical Defocusing NLS in Periodic Settings
I will discuss some recent work, joint with B. Pausader, on
constructing global solutions of defocusing energy-critical
nonlinear Schrodinger equations in periodic and semiperiodic
settings.
On the Ergodic Properties of Square-Free Numbers
I shall explain the structure of correlation functions for
square-free numbers and describe a 'natural' dynamical system
associated to them. Spectral analysis allows us to show that this
system is metrically isomorphic to a translation on a
compact...
Two-Point Problem for the Ideal Incompressible Fluid
Consider the flow of ideal incompressible fluid in a bounded 2-d
domain $M$ (say, $M= 3DT^2$, the 2-d torus). In the Lagrange
formulation, the flow is a geodesic $f_t$ on the group $SDif f(M)$
of volume-preserving diffeomorphisms of $M$ with respect...
On Zaremba's Conjecture on Continued Fractions
Zaremba's 1971 conjecture predicts that every integer appears as
the denominator of a finite continued fraction whose partial
quotients are bounded by an absolute constant. We confirm this
conjecture for a set of density one.
Reducibility for the Quasi-Periodic Liner Schrodinger and Wave Equations
Lars Hakan Eliasson
We shall discuss reducibility of these equations on the torus
with a small potential that depends quasi-periodically on time.
Reducibility amounts to "reduce” the equation to a time-independent
linear equation with pure point spectrum in which case...
A Centre-Stable Manifold for the Energy-Critical Wave Equation in $R^3$ in the Symmetric Setting
Consider the focusing semilinear wave equation in $R^3$ with
energy-critical non-linearity \[ \partial_t^2 \psi - \Delta \psi -
\psi^5 = 0,\ \psi(0) = \psi_0,\ \partial_t \psi(0) = \psi_1. \]
This equation admits stationary solutions of the form \[...
Global Existence of Surface Waves
Jalal Shatah
Various Approaches to Semiclassical Quantum Dynamics
George A. Hagedorn
I shall describe several techniques for finding approximate
solutions to the time-dependent Schr\"odinger equation in the
semiclassical limit. The first of these involves expansions in
"semiclassical wave packets" that are also sometimes called...
Nodal Lines of Maass Forms and Critical Percolation
We describe some results concerning the number of connected
components of nodal lines of high frequency Maass forms on the
modular surface. Based on heuristics connecting these to a critical
percolation model, Bogomolny and Schmit have conjectured...
Formation of Singularities in Fluid Interfaces
Charles Fefferman
The interface between water and vacuum (governed by the "water
wave equation"), and the interface between oil and water in sand
(governed by the "Muskat equation") can develop singularities in
finite time. Joint work with A. Castro, D. Cordoba, F...
Sub-Weyl Subconvexity and Short p-Adic Exponential Sums
Djordje Milicevic
One of the principal questions about L-functions is the size of
their critical values. In this talk, we will present a new
subconvexity bound for the central value of a Dirichlet L-function
of a character to a prime power modulus, which breaks a...
Hole Probability for Entire Functions Represented by Gaussian Taylor Series
We study the hole probability of Gaussian entire functions. More
specifically, we work with entire functions given by a Taylor
series with i.i.d complex Gaussian random variables and arbitrary
non-random coefficients. A 'hole' is the event where the...
Uniqueness and Nondegeneracy of Ground States for Non-Local Equations
Rupert Frank
We consider the non-local and non-linear equation
$(-\Delta)^sQ+Q-Q^{\alpha+1}= 0$ involving the fractional Laplacian
$(-\Delta)^s$ with $0 < s <1$. We prove uniqueness of energy
minimizing solutions for the optimal range of $\alpha$'s. As a
technical key result, we show that the associated linearized
operator is nondegenerate, in the sense that its kernel is spanned
by $\nabla Q$. This solves an open problem posed by Weinstein and
by Kenig, Martel and Robbiano.
The talk is based on joint work with E. Lenzmann and L.
Sylvestre.
The Strauss Conjecture on Black Holes
Mihai Tohaneanu
The Strauss conjecture for the Minkowski spacetime in three
dimensions states that the semilinear equation \[\Box u=|u|^p,\
u(0) =\epsilon f,\ \partial_t u(0) = \epsilon g\] has a global
solution for all $f$ and$g$ smooth, compactly supported and
$...
The Incompressible Euler Equations in Lagrangian Coordinates, with Applications to Analyticity of Fluid Particle Trajectories and to Numerical Simulations
Uriel Frisch
As is well known, Eulerian simulations with a very small spatial
mesh using an explicit scheme also require very small time steps,
because the latter must be smaller than the time required to travel
accross the mesh at the maximum flow velocity...
Three Projection Operators in Several Complex Variables
Elias Stein
I will report on recent joint work with L. Lanzani on three
basic projection operators, each associated to an appropriate
domain in C^n. These are: variants of Cauchy-Fantappie integrals;
the Cauchy-Szego projection: and the Bergman projection. The...
A Non-Commutative Analog of the 2-Wasserstein Metric for which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy
Eric Carlen
The Fermionic Fokker-Planck equation is a quantum-mechanical
analog of the classical Fokker-Planck equation with which it has
much in common, such as the same optimal hypercontractivity
properties. In this paper we construct a Riemannian metric
on...
Magnetic Vortices, Nielsen-Olesen-Nambu Strings and Theta Functions
Israel M. Sigal
The Ginzburg-Landau theory was first developed to explain
magnetic and other properties of superconductors, but had a
profound influence on physics well beyond its original area. It had
the first demonstration of the Higgs mechanism and it became
a...
Nonlinear Long-Range Resonant Scattering and Kink Dynamics
Avy Soffer
We study the nonlinear Klein-Gordon equation, in one dimension,
with a qudratic term and variable coefficient qubic term. This
equation arises from the asymptotic stability theory of the kink
solution.Our main result is the global existence and...
Dispersive Estimates for Schroedinger's Equation with a Time-Dependent Potential
I present some new dispersive estimates for Schroedinger's
equation with a time-dependent potential, together with
applications.
Hamiltonian Evolution Equations -- Where They Come From, What They Are Good For
Juerg Froehlich
Several examples of Hamiltonian evolution equations for systems
with infinitely many degrees of freedom are presented. It is
sketched how these equations can be derived from some underlying
quantum dynamics ("mean-field limit") and what kind of...
Sphere Packing Bounds Via Spherical Codes
Henry Cohn
We develop a simple geometric variant of the
Kabatiansky-Levenshtein approach to proving sphere packing density
bounds. This variant gives a small improvement to the best bounds
known in Euclidean space (from 1978) and an exponential improvement
in...
Toeplitz Matrices and Determinants Under the Impetus of the Ising Model
Percy Deift
This is the second of two talks in which the speaker will
discuss the development of the theory of Toeplitz matrices and
determinants in response to questions arising in the analysis of
the Ising model of statistical mechanics. The first talk
will...
Large Data Dynamics for Nonlinear Dispersive PDEs
We will discuss recent work on wave evolutions for large data.
Particular emphasis will be placed on concentration compactness
ideas. Amongst others, we will describe a result for wave equations
from R^3 minus the unit ball into the sphere S^3 where...
New Approximations of the Total Variation, and Filters in Image Processing
Haim Brezis
I will present new results concerning the approximation of the
BV-norm by nonlocal, nonconvex, functionals. The original
motivation comes from Image Processing. Numerous problems remain
open. The talk is based on a joint work with H.-M. Nguyen.
Dynamics of Gibbs Measure Evolution for the Radial Nonlinear Schr\"odinger and Wave Equations on the Ball
In this talk, we present recent works with Jean Bourgain on
global well-posedness for the radial nonlinear Schr\"odinger and
wave equations set on the unit ball in $\mathbb{R}^N$ with
supercritical data chosen randomly in the support of the...