Seminars Sorted by Series

Analysis and Mathematical Physics

May
24
2024

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...

May
31
2024

Analysis and Mathematical Physics

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...

Jun
14
2024

Analysis and Mathematical Physics

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...

Oct
08
2024

Analysis and Mathematical Physics

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...

Oct
15
2024

Analysis and Mathematical Physics

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...

Oct
29
2024

Analysis and Mathematical Physics

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...

Nov
05
2024

Analysis and Mathematical Physics

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...

Nov
12
2024

Analysis and Mathematical Physics

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...

Analysis Seminar

Feb
27
2008

Analysis Seminar

Orbit of the Diagonal of a Power of a Nilmanifold
Alexander Leibman
2:00pm|S-101

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...

Mar
12
2008

Analysis Seminar

Constructing Wild Groups
Lior Siberman
2:00pm|West Bldg. Lecture Hall
Apr
02
2008

Analysis Seminar

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...

Apr
23
2008

Analysis Seminar

A Hardy Field Extension of Szemeredi's Theorem
2:00pm|S-101

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...

May
14
2008

Analysis Seminar

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...

Oct
21
2011

Analysis Seminar

On the Instability for 2D Fluids
3:00pm|S-101

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.

Oct
25
2011

Analysis Seminar

On the Rigidity of Black Holes
Sergiu Klainerman
2:30pm|S-101

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...

Nov
01
2011

Analysis Seminar

The Defocusing Cubic Nonlinear Wave Equation in the Energy-Supercritical Regime
3:15pm|S-101

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...

Nov
10
2011

Analysis Seminar

Around the Davenport-Heilbronn Function
3:00pm|S-101

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...

Nov
17
2011

Analysis Seminar

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...

Nov
29
2011

Analysis Seminar

The Energy-Critical Defocusing NLS in Periodic Settings
2:30pm|S-101

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.

Dec
06
2011

Analysis Seminar

On the Ergodic Properties of Square-Free Numbers
2:30pm|S-101

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...

Dec
13
2011

Analysis Seminar

Two-Point Problem for the Ideal Incompressible Fluid
2:30pm|S-101

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...

Feb
14
2012

Analysis Seminar

On Zaremba's Conjecture on Continued Fractions
2:30pm|S-101

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.

Feb
21
2012

Analysis Seminar

Reducibility for the Quasi-Periodic Liner Schrodinger and Wave Equations
Lars Hakan Eliasson
2:30pm|S-101

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...

Feb
28
2012

Analysis Seminar

A Centre-Stable Manifold for the Energy-Critical Wave Equation in $R^3$ in the Symmetric Setting
2:00pm|S-101

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 \[...

Mar
06
2012

Analysis Seminar

Various Approaches to Semiclassical Quantum Dynamics
George A. Hagedorn
2:00pm|S-101

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...

Mar
20
2012

Analysis Seminar

Nodal Lines of Maass Forms and Critical Percolation
2:00pm|S-101

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...

Mar
27
2012

Analysis Seminar

Formation of Singularities in Fluid Interfaces
Charles Fefferman
2:00pm|S-101

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...

Apr
17
2012

Analysis Seminar

Sub-Weyl Subconvexity and Short p-Adic Exponential Sums
Djordje Milicevic
2:00pm|S-101

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...

Oct
09
2012

Analysis Seminar

Hole Probability for Entire Functions Represented by Gaussian Taylor Series
3:00pm|S-101

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...

Oct
19
2012

Analysis Seminar

Uniqueness and Nondegeneracy of Ground States for Non-Local Equations
Rupert Frank
3:15pm|S-101

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.

Oct
23
2012

Analysis Seminar

The Strauss Conjecture on Black Holes
Mihai Tohaneanu
3:15pm|S-101

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 $...

Oct
30
2012

Analysis Seminar

The Incompressible Euler Equations in Lagrangian Coordinates, with Applications to Analyticity of Fluid Particle Trajectories and to Numerical Simulations
Uriel Frisch
3:15pm|S-101

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...

Nov
09
2012

Analysis Seminar

Three Projection Operators in Several Complex Variables
Elias Stein
3:15pm|S-101

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...

Nov
13
2012

Analysis Seminar

A Non-Commutative Analog of the 2-Wasserstein Metric for which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy
Eric Carlen
3:15pm|S-101

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...

Nov
30
2012

Analysis Seminar

Magnetic Vortices, Nielsen-Olesen-Nambu Strings and Theta Functions
Israel M. Sigal
3:15pm|S-101

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...

Dec
07
2012

Analysis Seminar

Nonlinear Long-Range Resonant Scattering and Kink Dynamics
Avy Soffer
3:15pm|S-101

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...

Jan
15
2013

Analysis Seminar

Dispersive Estimates for Schroedinger's Equation with a Time-Dependent Potential
3:15pm|S-101

I present some new dispersive estimates for Schroedinger's equation with a time-dependent potential, together with applications.

Jan
22
2013

Analysis Seminar

Hamiltonian Evolution Equations -- Where They Come From, What They Are Good For
Juerg Froehlich
3:15pm|S-101

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...

Jan
22
2013

Analysis Seminar

Sphere Packing Bounds Via Spherical Codes
Henry Cohn
4:30pm|S-101

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...

Jan
29
2013

Analysis Seminar

Toeplitz Matrices and Determinants Under the Impetus of the Ising Model
Percy Deift
3:15pm|S-101

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...

Feb
01
2013

Analysis Seminar

Large Data Dynamics for Nonlinear Dispersive PDEs
3:30pm|S-101

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...

Feb
19
2013

Analysis Seminar

New Approximations of the Total Variation, and Filters in Image Processing
Haim Brezis
3:15pm|S-101

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.

Mar
19
2013

Analysis Seminar

Dynamics of Gibbs Measure Evolution for the Radial Nonlinear Schr\"odinger and Wave Equations on the Ball
3:15pm|S-101

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...