NODE Group Wiki

All meetings take place on Wednesdays, 9:30 - 10:20 am in Burt Hall 204 unless otherwise noted.

“Vertex-Minimal Subgraphs with K_n Minors”, David Hathaway
Let h(G) be the Hadwiger number of G, the largest n such that G has a K_n minor, and define h_n(G) = min{|V(H)| : H\subset G, h(H)=n}, defined for any n = 1, 2, …, h(G). We seek to study the structure of any given graph through this sequence of orders and its long-term behavior. As a result, the primary focus will be on infinite graphs, in particular those with infinite Hadwiger number. We shall investigate lower bounds originating from basic structural details or from partial knowledge of early elements in the sequence. Along the way, we'll work many examples of interesting finite and infinite graphs.

Generalized Nash equilibrium problems describe strategic interactions in which each player’s feasible choices may depend on the other players’ decisions. In this talk, I will present an algorithm for games with polynomial objectives and constraints that are linear in each player’s own strategy. The key idea is to use Carathéodory’s theorem and partial Lagrange multiplier expressions to decompose the Karush–Kuhn–Tucker (KKT) set into simpler subsets, leading to a collection of polynomial optimization problems. I will explain how the algorithm identifies equilibrium candidates and verifies them in the nonconvex setting, and discuss conditions under which it can find all equilibria or detect their nonexistence. Time permitting, I will discuss global optimization methods for solving the polynomial optimization problems arising in this approach.

This summer I started collecting some notes on this fascinating topic. I will start from the beginning and lay down the basic definitions, which means that I won't get to the Burton-Pemantle Theorem in one hour. However, that is the goal and hopefully we will get there in subsequent talks. After that we can start looking at the many applications of this matrix.