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Title: The conjugacy problem in \(\mathrm{GL}(n,\mathbb{Z})\) Speaker: Tommy Hofmann, TU Kaiserslautern Time: 3:00pm‐4:00pm Place: CMC 130
We consider the problem of deciding whether two matrices are conjugate. If the coefficient ring is a field, this problem can be easily solved by using the Jordan normal form or the rational canonical form. For more general coefficient rings, the situation becomes increasingly challenging, both from a theoretical and a practical viewpoint.
In this talk, we show how the conjugacy problem for integer matrices can be efficiently decided using techniques from group and number theory.
This is joint work with Bettina Eick and Eamonn O'Brien.
Title: Tug-of-war games and Biased Infinity Laplacian Boundary Problem on finite graphs Speaker: Zoran Šunić, Hofstra University Time: 3:00pm‐4:00pm Place: CMC 130
We provide an algorithm, running in polynomial time in the number of vertices, computing the unique solution to the Biased Infinity Laplacian Boundary Problem on finite graphs.
The problem is, on the one hand, motivated by problems in auction theory, and on the other, it forms a basis for a numerical method for certain partial differential equations. We will discuss neither of these in depth. The following probabilistic/graph theoretic interpretation suffices for our purposes.
Let \(G\) be a finite graph with boundary \(B\) (any subset of vertices) and boundary condition \(g: B \to R\) (any real-valued function defined on the boundary). We may think of \(g\) as the pay-off function for a random-turn two-player zero-sum game played on \(G\) as follows. In the beginning a token is placed at a non-boundary vertex. At every step, one of the players randomly (decided by a biased coin) gets the right to move and then chooses (not randomly!) a neighboring vertex to which the token is moved. The game ends when the token reaches a boundary vertex, say \(b\), at which point Player I wins the amount \(g(b)\) from Player II.
A solution to the Boundary Problem is the value of the game, that is, a function \(p: V(G) \to R\) such that, for every vertex \(v\) in \(V(G)\), \(p(v)\) is the expected pay-off for Player I under optimal strategy by both players when the game starts with the token at \(v\).
The algorithm is based on an adjusted (biased) notion of a slope of a function on a path in a graph.
Title: On solutions of integrable equations with self-consistent sources Speaker: Yehui Huang, North China Electric Power University Time: 3:00pm‐4:00pm Place: CMC 130
In this talk, I will introduce a kind of integrable extensions of integrable equations generated from self-consistent sources. By generalized Darboux transformations, I will present a solution formula with multiple parameters for the nonlinear Schrödinger equation with self-consistent sources and the parity-time symmetric nonlinear Schrödinger equation with self-consistent sources. After taking special choices for the seed solution and the eigenfunctions, different types of exact solutions are derived, which include solitons, rational solutions and rogue wave solutions.