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for events the day of Tuesday, September 7, 2021.

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Tuesday, September 7, 2021

11:00 am in 347 Altgeld Hall,Tuesday, September 7, 2021

Degenerate Turan problems for graphs

Tao Jiang   [email] (Miami University of Ohio)

Abstract: In Turan type extremal problems, we want to determine how dense a graph or hypergraph is without containing a particular subgraph or family of subgraphs. Such problems are central to extremal graph theory, because solving them requires one to thoroughly investigate the interaction of global graph parameters with local structures. Efforts in solving these problems have spurred the developments of some powerful tools in extremal graph theory, such as the regularity method, probabilistic and algebraic methods.

While Turan problems have satisfactory solutions for non-bipartite graphs, the problem is still generally wide-open for bipartite graphs with many intriguing conjectures and results. In this talk, we will discuss some conjectures on Turan problems for bipartite graphs and some recent progress on them. Time permitting, we will also discuss a colored variant of the Turan problem.

There will also be a presentation on "Tao Jiang's favorite problems" (September 9, 11 - 11:50 a.m., Room 141 Altgeld Hall.

1:00 pm in 245 Altgeld Hall,Tuesday, September 7, 2021

Weighted Turan Numbers and Maximum Crossing Numbers of Trees

Sean English (UIUC)

Abstract: In extremal graph theory, the most natural question to consider involves finding the most edges in an $n$-vertex graph that does not contain any copy of some small forbidden graph $F$. We will explore a generalization of this to edge weighted graphs in which the edge weights are induced by a vertex weighting according to some rule. We will solve this problem for cliques when the rule involves weighting each edge by the product or the minimum of the weights of the endpoints.

The main motivation for the study of such problems is in applications to other combinatorial problems. In particular, we will use the product weighting to solve an extremal problem in which the $n$-vertex host graph is not complete, and we will use the minimum edge weighting to solve a problem involving the maximum rectilinear crossing number of trees.

This project was joint work with Patrick Bennett and Maria Talanda-Fisher.

2:00 pm in 347 Altgeld Hall,Tuesday, September 7, 2021

Information Projections On Banach Spaces With Applications To KL Weighted Control And A Feynman-Kac Formula For ODEs

Zachary Selk (Purdue University)

Abstract: In this talk, we discuss a portmanteau theorem establishing the equivalence between information projections on a Banach space, constrained Kullback-Leibler weighted control, finding the mode of a measure through Onsager-Machlup formalism and in the classical Wiener space case, an Euler-Lagrange equation. As one example of an application of our theorem, we discuss a Feynman-Kac type formula, showing that the solution to a second order linear ODE (or system of ODEs) is the mode of a particular diffusion. Our portmanteau theorem along with our Feynman-Kac result provides numerics and insight for solving these ODEs. Joint work with William Haskell and Harsha Honnappa.