Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, April 30, 2019.

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Tuesday, April 30, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, April 30, 2019

Topological Restrictions on Anosov Representations

Richard Canary (U Michigan)

Abstract: The theory of Anosov representations was introduced by Francois Labourie in his study of Hitchin representations. They have emerged as the natural analogue, for higher rank Lie groups, of Fuchsian representations, or more generally convex compact representations into rank one Lie groups. We will give a gentle introduction to Anosov representations, followed by a discussion of topological restrictions on the groups which admit Anosov representations into SL(d,R). For example, we will see characterizations of groups admitting Anosov representations into SL(3,R) and SL(4,R) and restrictions on the cohomological dimension for all values of d. (This is joint work with Kostas Tsouvalas.)

1:00 pm in 347 Altgeld Hall,Tuesday, April 30, 2019

Blow-up and Soliton Stability in KdV-type equations

Svetlana Roudenko (Florida International University)

Abstract: While the KdV equation and its generalizations with higher power nonlinearities (gKdV) have been long studied, a question about existence of blow-up solutions for higher power nonlinearities has posed lots of challenges and far from being answered. One of the main obstacles is that unlike other dispersive models such as the nonlinear Schrodinger or wave equations, the gKdV equation does not have a suitable virial quantity which is the key to prove the finite time blow-up. Partially, the question of existence and formation of singularities intertwines with the soliton stability or actually the instability, which may lead to a blowup. Only at the dawn of this century the groundbreaking works of Martel and Merle showed the existence of finite-time blow-up solutions for the quintic (critical) gKdV equation, as well as the asymptotic stability of solitons in the subcritical gKdV equation. We consider a higher dimensional extension of the gKdV equation, called generalized Zakharov-Kuznetsov (gZK) equation (the gKdV is limited as a spatially one-dimensional model), and investigate stability of solitons and the existence of blow-up solutions. We positively answer the question of existence of blowup in the two dimensional version of critical Zakharov-Kuznetsov equation and also obtain the asymptotic stability in the subcritical setting. We will discuss some of the important ingredients to obtain these results, including the Liouville-type theorem, which uses time-decay estimates, a la virial type quantity and spectral properties associated to it (this is a joint work with Luiz Farah, Justin Holmer and Kai Yang).

4:00 pm in 2 Illini Hall,Tuesday, April 30, 2019

Amenable first order theories

Anand Pillay (Notre Dame Math)

Abstract: (Joint with Hrushovski and Krupinski.) We extract from the properties (extreme) amenability of automorphisms groups of omega-categorical theories, the notion of (extreme) amenability of a first order theory T, which is much less restrictive than the automorphism property. I discuss some results around the Lascar group of such a theory, some proofs of which use versions of continuous logic.