Posted April 19, 2024
Combinatorics Seminar Questions or comments?
2:00 pm – 3:00 pm Lockett 233
Ryan Martin, Iowa State University
Counting cycles in planar graphs
Basic Tur\'an theory asks how many edges a graph can have, given certain restrictions such as not having a large clique. A more generalized Tur\'an question asks how many copies of a fixed subgraph $H$ the graph can have, given certain restrictions. There has been a great deal of recent interest in the case where the restriction is planarity. In this talk, we will discuss some of the general results in the field, primarily the asymptotic value of ${\bf N}_{\mathcal P}(n,H)$, which denotes the maximum number of copies of $H$ in an $n$-vertex planar graph. In particular, we will focus on the case where $H$ is a cycle. It was determined that ${\bf N}_{\mathcal P}(n,C_{2m})=(n/m)^m+o(n^m)$ for small values of $m$ by Cox and Martin and resolved for all $m$ by Lv, Gy\H{o}ri, He, Salia, Tompkins, and Zhu. The case of $H=C_{2m+1}$ is more difficult and it is conjectured that ${\bf N}_{\mathcal P}(n,C_{2m+1})=2m(n/m)^m+o(n^m)$. We will discuss recent progress on this problem, including verification of the conjecture in the case where $m=3$ and $m=4$ and a lemma which reduces the solution of this problem for any $m$ to a so-called ``maximum likelihood'' problem. The maximum likelihood problem is, in and of itself, an interesting question in random graph theory. This is joint work with Emily Heath and Chris (Cox) Wells.
Posted January 17, 2024
Last modified March 4, 2024
Control and Optimization Seminar Questions or comments?
11:30 am – 12:20 pm Zoom (click here to join)
Tobias Breiten, Technical University of Berlin
On the Approximability of Koopman-Based Operator Lyapunov Equations
Computing the Lyapunov function of a system plays a crucial role in optimal feedback control, for example when the policy iteration is used. This talk will focus on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup in a specially weighted Lp-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay, which justifies finite rank approximations with numerical methods. The usefulness for numerical computations will also be demonstrated with two short examples. This is joint work with Bernhard Höveler (TU Berlin).
Posted September 24, 2023
Last modified March 3, 2024
Algebra and Number Theory Seminar Questions or comments?
3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Ana Bălibanu, Louisiana State University
TBA
Posted February 1, 2024
Last modified February 11, 2024
Geometry and Topology Seminar Seminar website
3:30 pm Lockett 233
Jean-François Lafont, The Ohio State University
TBA
Posted April 16, 2024
Faculty Meeting Questions or comments?
3:00 pm – 4:00 pmMeeting with Dean Cynthia Peterson
Posted April 19, 2024
Combinatorics Seminar Questions or comments?
2:00 pm – 3:00 pm Zoom (Please email zhiyuw at lsu.edu for Zoom link)
Peter Nelson, University of Waterloo
Infinite matroids on lattices
There are at least well-studied ways to extend matroids to more general objects - one can allow the ground set to be infinite, or instead define the concept of independence on a lattice other than a set lattice. I will discuss some nice ideas that arise when combining these two generalizations. This is joint work with Andrew Fulcher.