Svetlana Roudenko (Florida International University) Soliton stability in higher-dimensional generalization of KdV equation. Sponsored by the Meyer Fund
Tue, Jan. 14 2pm (MATH 350)
Lie Theory
Shawn Burkett (Kent State University) A Frobenius group analog for Camina triples Sponsored by the Meyer Fund
Jonathan Campbell (Duke University) Hilbert's Generalized Third Problem and Homotopy Theory
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We consider Zakharov-Kuznetsov (ZK) equation, a higher-dimensional generalization of the well-known Korteweg-de Vries (KdV) equation, which was introduced in physical context by V. Zakharov and E. Kuznetsov back in 1972. One of the questions they asked was existence of solitons in higher dimensions (the KdV is restricted as the one-dimensional model) and their stability. In this talk I will discuss behavior of solitary waves in the 3d ZK equation, proving that solutions in the energy space that are orbitally stable are also asymptotically stable, that is, as time goes to infinity, they converge to a rescaling and shift of the solitary wave Q(x-t,y,z) in a certain rightward moving window. This is a joint work with Luiz Gustavo Farah, Justin Holmer, and Kai Yang.
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Frobenius groups are an object of fundamental importance in finite group theory. As such, several generalizations of these groups have been considered. Some examples include: A Frobenius--Wielandt group is a triple $(G,H,L)$ where $H/L$ is almost a Frobenius complement for $G$; A Camina pair is a pair $(G,N)$ where $N$ is almost a Frobenius kernel for $G$; A Camina triple is a triple $(G,N,M)$ where $(G,N)$ and $(G,M)$ are almost Camina pairs. In this talk we discuss triples $(G,N,M)$ where $(G,N)$ and $(G,M)$ are almost Frobenius groups.
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Algebraic invariants of a manifold can often be formulated as constraint satisfaction problems, where local constraints are induced by the topology via the combinatorics of a triangulation of the manifold. There are many interesting complexity theoretic questions concerning such invariants that are basically wide open. In the first part of this talk, I’ll explain why I believe 3-dimensional manifolds are the most interesting from this perspective. I’ll then quickly discuss what is known about some important decision problems in 3-manifold topology (homeomorphism, unknot recognition, sphere recognition, etc), before turning to the things I think about the most: invariants of 3-manifolds coming from topological quantum field theories.
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In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory, and describe recent progress in this direction. Two n-dimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and re-assemble the pieces into $Q$. The scissors congruence problem, aka Hilbert Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent iff they have the same area. In three dimensions, there is volume AND another invariant, the Dehn Invariant. In higher dimensions, very little is known. I'll give an introduction to this very classical problem, and explain how homotopy theory can be used to get purchase on it. Prerequisites: The discussion of Hilbert's Third Problem and Dehn's invariant will be widely accessible. No knowledge of algebraic K-theory will be assumed. This is all joint work with Inna Zakharevich.