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.

We study random Peano paths on planar square grids that arise from fair random spanning trees. These are trees that are sampled in such a way as to have the same (if possible) edge probabilities. In particular, we are interested in identifying the scaling limit as the mesh-size of the grid tends to zero. It is known that if the trees are sampled uniformly, then the scaling limit exists and equals ${\rm SLE}_8$. We show that if we simply follow the same steps as in [lawler-schramm-werner2002], then fair Peano paths have a deterministic scaling limit. We end by describing a different approach that is conjectured to have a non-deterministic scaling.