Department of

Mathematics


Seminar Calendar
for Harmonic Analysis and Differential Equations events the year of Thursday, December 7, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2017          December 2017           January 2018    
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           1  2  3  4                   1  2       1  2  3  4  5  6
  5  6  7  8  9 10 11    3  4  5  6  7  8  9    7  8  9 10 11 12 13
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Tuesday, February 14, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, February 14, 2017

Gevrey smoothing of weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules

Tobias Ried (Karlsruhe Institute of Technology)

Abstract: We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \in L^1_2({\mathbb R}^d) \cap L \log L({\mathbb R}^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity. This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers. (Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)

Tuesday, April 4, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, April 4, 2017

Pointwise bounds for the 3-dimensional wave equation and spectral multipliers

Michael Goldberg (U. Cincinnati)

Abstract: The sine propagator for the wave equation in three dimensions, $\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}$, has an integral kernel $K(t,x,y)$ with the property $\int_{\mathbb R} |K(t, x, y)|dt = (2\pi|x-y|)^{-1}$. Finiteness comes from the sharp Huygens principle and power-law decay comes from dispersion. Estimates of this type are useful for proving ``reversed Strichartz" inequalities that bound a solution in $L^p_x L^q_t$ for admissible pairs $(p,q)$. We examine the propagator $\frac{\sin(t\sqrt{H})}{\sqrt{H}}P_{ac}(H)$ for operators $H = -\Delta + V$ with the potential $V$ belonging to the Kato-norm closure of test functions. Assuming zero is not an eigenvalue or resonance, the bound $\int_{\mathbb R} |K(t,x,y) \leq C|x-y|^{-1}$ continues to be true. Combined with a Huygens principle for the perturbed wave equation, this estimate suggests pointwise bounds for spectral multipliers of fractional integral or H\"ormander-Mikhlin type. This is joint work with Marius Beceanu (SUNY - Albany).

Tuesday, April 18, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, April 18, 2017

Ground states for nonlinear Schr\"odinger equation on a dumbbell graph

Jeremy Marzuola (University of North Carolina at Chapel Hill)

Abstract: With Dmitry Pelinovsky, we describe families of standing waves on a closed quantum graph in the shape of a dumbbell, namely having two loops connected by a link with Kirchhoff boundary conditions. We describe symmetry breaking bifurcations and prove a remarkable asymptotic property that is a bit surprising in terms of the energy minimizing solutions at a given mass.

Tuesday, April 25, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, April 25, 2017

Strichartz estimates for linear wave equations with moving potentials

Gong Chen (U Chicago)

Abstract: We will discuss Strichartz estimates for linear wave equations with several moving potentials in $\mathbb{R}^{3}$ (a.k.a. charge transfer Hamiltonians) which appear naturally in the study of nonlinear multisoliton systems. We show that local decay estimates systematically imply Strichartz estimates. To study local decay estimates, we introduce novel reversed Strichartz estimates along slanted lines and energy comparison under Lorentz transformations. As applications, we will also discuss related scattering problems and a construction of multisoliton in $\mathbb{R}^{3}$ with strong interactions.

Tuesday, May 9, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, May 9, 2017

Sharp Poincare' inequalities in a class of non-convex sets

Jeffrey Langford   [email]

Abstract: A classic result in spectral theory of PDEs by Payne and Weinberger states that the lowest nonzero Neumann eigenvalue of the Laplacian on a convex domain of diameter D can be estimated below by pi^2/D^2. I will discuss an analogue of this result for a particular class of non-convex domains.

Thursday, September 7, 2017

1:00 pm in 243 Altgeld Hall,Thursday, September 7, 2017

Dynamics near traveling waves of supercritical KDV equations

Zhiwu Lin (Georgia Institute of Technology)

Abstract: Consider generalized KDV equations with a power non-linearity (u^p)_x. These KDV equations have solitary traveling waves, which are linearly unstable when p>5 (supercritical case). Jointly with Jiayin Jin and Chongchun Zeng, we constructed invariant manifolds (stable, unstable and center) near the orbit of the unstable traveling waves in the energy space. These invariant manifolds are used to give a complete description of the local dynamics near unstable traveling waves. In particular, the global existence with orbital stability is shown on the center manifold of co-dimension two, while the exponential instability is proved for initial data not on the center manifold.

Tuesday, October 3, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, October 3, 2017

The Effectively Linear Behavior of the Nonlinear Schr\"odinger Equation

Katelyn Leisman (Illinois Math)

Abstract: The linear part of the Nonlinear Schr\"odinger Equation (NLS) ($iq_t=q_{xx}$) has dispersion relation $\omega=k^2$. We don't expect solutions to the fully nonlinear equation to behave nicely or have any kind of effective dispersion relation like this. However, I have seen that solutions to the NLS are actually weakly coupled and are often nearly sinusoidal in time with a dominant frequency, often behaving similarly to modulated plane waves. In fact, these highly nonlinear solutions eventually end up behaving more and more linearly.

Tuesday, October 10, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, October 10, 2017

Recent progress on the pointwsie convergence problem of Schrodinger equations

Xiaochun Li (Illinois Math)

Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We prove that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method.

Friday, October 13, 2017

1:00 pm in 345 Altgeld Hall,Friday, October 13, 2017

Variation of inverse cascade spectrum for gravity waves due to condensate

Alexander O. Korotkevich   [email] (University of New Mexico)

Abstract: During most of numerical experiments in wave turbulence of gravity waves we operate on a discrete wavenumbers grid. As a result, if we consider formation of inverse cascade, propagation of action flux to the small wavenumbers is arrested at some scale due to inefficiency of resonant four-waves interactions. It results in formation of srong long wave background, which we call condensate using analogy with Bose-Einstein condensation in Statistical Physics. As it is shown in a long numerical experiment, inverse cascade spectrum in the presence of such a condensate has a different power than predicted by the Theory of Wave Turbulence. We propose some preliminary explanation to this interesting phenomenon.

Tuesday, October 17, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, October 17, 2017

On Singularity Formation in General Relativity

Xinliang An (U Toronto Math)

Abstract: In the process of gravitational collapse, singularities may form, which are either covered by trapped surfaces (black holes) or visible to faraway observers (naked singularities). In this talk, with three different approaches coming from hyperbolic PDE, quasilinear elliptic PDE and dynamical system, I will present results on four physical questions: i) Can “black holes” form dynamically in the vacuum? ii) To form a “black hole”, what is the least size of initial data? iii) Can we find the boundary of a “black hole” region? Can we show that a “black hole region” is emerging from a point? iv) For Einstein vacuum equations, could singularities other than black hole type form in gravitational collapse?

Tuesday, October 31, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, October 31, 2017

Efficient computational methods for the shallow water equations and their applications

Yulong Xing (Ohio State University)

Abstract: Nonlinear shallow water equations (SWEs) with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas, with applications arising from ocean, hydraulic engineering, and atmospheric modeling. Development of efficient and accurate numerical algorithms for SWEs has been an active research area in the past few decades. In this presentation, we will talk about efficient finite element methods developed for the SWEs and their applications in computational hydrology.

Tuesday, November 7, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, November 7, 2017

Mathematics of collective behavior and self-organized dynamics

Roman Shvydkoy (University of Illinois at Chicago)

Abstract: We discuss a new class of hydrodynamic models arising in studies of self-organized dynamics of agents. These models belong to the class of fractional parabolic equations with (critical) drift and present unique challenges from the analytical prospective. We will show how these models can be used to prove two observable phenomena: alignment (such as consensus of opinions) and flocking (such as swarming of animals).

Tuesday, November 14, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, November 14, 2017

Almost sure scattering for the energy-critical NLS

Jason Murphy   [email] (Missouri S&T Math)

Abstract: We consider the defocusing energy-critical nonlinear Schrödinger equation in four space dimensions with radial (i.e. spherically-symmetric) initial data below the energy space. In this setting, the problem is known to be ill-posed. Nonetheless, we can show that for suitably randomized radial initial data, one obtains global well-posedness and scattering almost surely. This is joint work with R. Killip and M. Visan.

Tuesday, November 28, 2017

1:00 pm in 347 Altgeld Hall,Tuesday, November 28, 2017

Decoupling for Parsell-Vinogradov Systems

Ruixiang Zhang (IAS)

Abstract: Parsell-Vinogradov (P-V) systems are "high dimensional" generalizations of the relevant system in Vinogradov's Mean Value Theorem (VMVT) and can be generalized to the so-called translation-dilation-invariant (TDI) systems. For P-V systems, there is an easy lower bound for the number of integer solutions inside a box. It is conjectured that this lower bound is also an upper bound with at most an $N^{\varepsilon}$-loss. In the VMVT case this was proved by Wooley and Bourgain-Demeter-Guth. Two parallel theories, namely efficient congruencing and decoupling, have been developed to attack counting questions for such systems. I'll present the decoupling approach in this talk, starting with the Bourgain-Demeter-Guth work on the one-dimensional system in VMVT case. Then I'll explain some new phenomena in high dimensional setting. We will see how the new difficulties are largely related to combinatorics and explain how we can address them. We hope the framework here will also work for general TDI systems. This is joint work in progress with Shaoming Guo.

Tuesday, December 5, 2017

1:00 pm in 347 Altgeld Hall ,Tuesday, December 5, 2017

A Spherical Maximal Function along the Primes

Theresa Anderson (University of Wisconsin-Madison)

Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior.  The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example.  In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to.  We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory.  This is joint work with Cook, Hughes, and Kumchev.