Seminar Calendar
for Harmonic Analysis and Differential Equations events the next 12 months of Wednesday, August 1, 2012.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
      July 2012             August 2012           September 2012
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6  7             1  2  3  4                      1
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Tuesday, September 11, 2012

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hal,  Tuesday, September 11, 2012
 Del Edit Copy
Submitted by vzh.
 Jeremy Louis Marzoula   [email] (UNC Math)Quasilinear Schroedinger EquationsAbstract: In this talk, we will discuss joint works with Jason Metcalfe and Daniel Tataru on short time local well-posedness in low-regularity Sobolev spaces for quasilinear Schrödinger equations. Such results are refinements of the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove existence of solutions in very high regularity spaces. Our arguments however are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the same authors related to local smoothing estimates.

Tuesday, November 13, 2012

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, November 13, 2012
 Del Edit Copy
Submitted by palbin.
 Pierre Albin   [email] (UIUC Math)Inverse boundary problems for systems in two dimensionsAbstract: Inverse problems typically consist in trying to recover data about the interior of a domain from observations made at the boundary. For instance, if you know the boundary values of solutions to a Schrodinger equation or a Dirac equation on a surface, can you recover the operators in the interior? I will describe joint work with Colin Guillarmou, Leo Tzou, and Gunther Uhlmann in which we show that, except for an obvious gauge obstruction, the answer is yes.

Tuesday, November 27, 2012

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, November 27, 2012
 Del Edit Copy
Submitted by laugesen.
 Dirk Hundertmark   [email] (Karlsruhe Institute of Technology)Ludicrous speed of decay of solitons in Hertzian chains Abstract: A Hertzian chain is a model for granular matter. It describes a chain of beads which are just touching each other. The system is modeled by an advance delay equation and the discrete nature of this equation makes is somewhat harder to study than the usual continuum models. It turns out, both experimentally and theoretically, that there are solitary pulses in this system, which are highly localized. The existence of such pulses was also shown rigorously and it was also shown that there are solitary pulses which decay at a double exponential rate, i.e., the asymptotic profile of the pulse goes to zero like exp(-exp(x)). We give an argument that every solitary pulse must decay at this ludicrously fast rate. (Which reminds us of... http://www.youtube.com/watch?v=mk7VWcuVOf0 )

Tuesday, December 4, 2012

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, December 4, 2012
 Del Edit Copy
Submitted by laugesen.
 Richard Laugesen   [email] (UIUC Math)Sharp Spectral Bounds on Starlike DomainsAbstract: If one knows geometric properties of a domain, such as its area or perimeter, then what inequalities can one deduce on the eigenvalues of the Laplacian for that domain? For example, the fundamental tone (first eigenvalue) and spectral functionals such as the spectral zeta function and heat trace have long been known to be extremal when the domain is a ball, provided the volume of the domain is fixed. We prove complementary bounds in the opposite direction (again sharp for the ball) by introducing an additional geometric quantity that measures the "deviation of the domain from roundness". An intriguing role in the proof is played by volume-preserving diffeomorphisms that are not the identity map. [Joint work with B. Siudeja, U. of Oregon]

Tuesday, February 12, 2013

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, February 12, 2013
 Del Edit Copy
Submitted by tzirakis.
 Hsi-Wei Shih (University of Minnesota)Global spacetime bounds for the energy-critical nonlinear wave equation in high spatial dimensionsAbstract: I will present my recent result on global spacetime bounds for the energy-critical nonlinear wave equation in all space dimensions greater or equal than 3. This extends work in the case d = 3 of Tao, who proved a certain global Strichartz norm has an explicit double-exponential bound in the energy. Our work uses Tao's argument extensively, but differs in important points too. Besides applying also in high dimensions, we get a sharper bound (though still double-exponential) in the case d = 3. These explicit bounds are interesting not only for their intrinsic value, but also through their connection with regularity problems for slightly energy-supercritical nonlinear wave equations.

Tuesday, March 12, 2013

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, March 12, 2013
 Del Edit Copy
Submitted by tzirakis.
 Nikos Tzirakis (UIUC)Smoothing properties and applications for solutions of periodic dispersive PDEAbstract: In this talk we demonstrate different smoothing phenomena for a large class of periodic dispersive PDE. Examples include the nonlinear Schroedinger equation (NLS), the KdV equation and the Zakharov system. These smoothing estimates provide short answers to some basic questions that arise in the modern theory of dispersive PDE. Applications include: Dispersive quantization and Talbot effect, existence of global attractors for dispersive PDE with dissipation, structure of the global attractor, global well-posedness for initial data with infinite energies, weak turbulence phenomena. The work is joint with B. Erdogan.

Tuesday, April 2, 2013

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, April 2, 2013
 Del Edit Copy
Submitted by berdogan.
 Gang Zhou (UIUC Math)Some Hamiltonian Models of FrictionAbstract: I will present some mathematical results on some models describing the motion of a tracer particle through a Bose-Einstein condensate. In the limit of a very dense, very weakly interacting Bose gas and for a very large particle mass, the dynamics of the coupled system is determined by classical non-linear Hamiltonian equations of motion. The particle's motion exhibits deceleration corresponding to friction (with memory) caused by the emission of Cerenkov radiation of gapless modes into the gas. These are joint works with D.Egli, J.Froehlich, A.Soffer, and I.M. Sigal.

Tuesday, April 9, 2013

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, April 9, 2013
 Del Edit Copy
Submitted by ekirr.
 Andrew Comech   [email] (Texas A&M Math)Linear instability of solitary waves in nonlinear Dirac equationAbstract: We study the linear instability of solitary wave solutions to the nonlinear Dirac equation (known to physicists as the Soler model). That is, we linearize the equation at a solitary wave and examine the presence of eigenvalues with positive real part. We show that the linear instability of the small amplitude solitary waves is described by the Vakhitov-Kolokolov stability criterion which was obtained in the context of the nonlinear Schroedinger equation: small solitary waves are linearly unstable in dimensions 3, and generically linearly stable in 1D. A particular question is on the possibility of bifurcations of eigenvalues from the continuous spectrum; we address it using the limiting absorption principle and the Hardy-type estimates. The method is applicable to other systems, such as the Dirac-Maxwell system. Some of the results are obtained in collaboration with Nabile Boussaid, Universite de Franche-Comte, and Stephen Gustafson, University of British Columbia.

Tuesday, April 16, 2013

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, April 16, 2013
 Del Edit Copy
Submitted by berdogan.
 Alp Eden (Bosphorus Univ. (visiting Indiana Univ.))Generalized Davey-Stewartson Equations: a surveyAbstract: The Davey-Stewartson system plays a special role among the 2+1 integrable systems, especially because the usual 2+1 nonlinear cubic Schrodinger equation is not integrable in two space dimensions. The Davey-Stewartson equations can also be derived as a model for the weakly non-linear packets of waves that travel in one direction but the amplitude modulates in two directions. It was observed by Babaoglu and Erbay that these derivations did not consider the effect of the second spatial coordinate. They have derived a system of three non-linear evolution equations for the amplitudes of a short transverse wave, a long transverse wave and a long longitudinal wave that travel in a quadratically non-linear elastic medium. They have called these equations as Generalized Davey-Stewartson equations. Depending on the underlying physical parameters the classification of this system can range from the purely elliptic to purely hyperbolic. My talk will be basically on the purely elliptic case with some results in the elliptic-hyperbolic-hyperbolic case at the end.

Tuesday, April 23, 2013

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, April 23, 2013
 Del Edit Copy
Submitted by zrapti.
 Panos Kevrekidis   [email] (UMass Amherst, Mathematics)Existence, Stability and Dynamics of Solitary Waves and Vortices in Superfluids: From Theory to ExperimentsAbstract: In this talk, we will present an overview of recent theoretical, numerical and experimental work concerning the static, stability, bifurcation and dynamic properties of coherent structures that can emerge in one-and higher-dimensional settings within Bose-Einstein condensates at the coldest temperatures in the universe (i.e., at the nanoKelvin scale). We will discuss how this ultracold quantum mechanical setting can be approximated at a mean-field level by a deterministic PDE of the nonlinear Schrodinger type and what the fundamental nonlinear waves of the latter are, such as dark solitons and vortices. Then, we will try to go to a further layer of simplified description via nonlinear ODEs encompassing the dynamics of the waves within the traps that confine them, and the interactions between them. Finally, we will attempt to compare the analytical and numerical implementation of these reduced descriptions to recent experimental results and speculate towards a number of interesting future directions within this field.