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Events for May – September

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Wed, May. 1 3:30pm (MATH 3…	Math Phys Howy Jordan (CU Boulder, Department of Mathematics) Infinite Dimensional Control Theory
Thu, May. 2 11:15am (MATH …	Algebraic Geometry Luca Schaffler (Universita Roma Tre) X Given the moduli space of hyperplanes in projective space, V. Alexeev constructed a family of compactifications parametrizing stable hyperplane arrangements with respect to given weights. In particular, there is a toric compactification that generalizes the Losev–Manin compactification for the moduli of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. In particular, we prove that in dimensions two the wall crossing corresponds to blowing up at the identity of the dense torus of the generalized Losev–Manin space. As an application, we show that any Q-factorialization of this blow-up is not a Mori dream space for a sufficiently high number of lines. This is joint work in progress with Patricio Gallardo. An explicit wall crossing for the moduli space of hyperplane arrangements Sponsored by the Meyer Fund
Thu, May. 2 12:15pm (Math …	Algebraic Geometry Farbod Shokrieh (University of Washington) X I will describe some connections between arithmetic of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For example, we give formulas for (non-archimedean) canonical local heights in terms of tropical geometry. Our formulas extend classical computations of local height functions due to Néron (involving Néron models) and, in the case of elliptic curves, due to Tate (involving Bernoulli polynomials). Based on joint work with Robin de Jong. Canonical local heights and tropical theta functions Sponsored by the Simons Foundation
Thu, May. 2 3:35pm (MATH 3…	Probability Vishesh Jain (University of Illinois at Chicago) X Markov chains provide a natural approach to sample from various distributions on the independent sets of a graph. For the uniform distribution on independent sets of a given size $k$ in a graph, perhaps the most natural Markov chain is the so-called ``down-up walk''. The down-up walk, which essentially goes back to the foundational work of Metropolis, Rosenbluth, Rosenbluth, Teller and Teller on the Markov Chain Monte Carlo method, starts at an arbitrary independent set of size $k$ , and in every step, removes an element uniformly at random and adds a uniformly random legal choice. Davies and Perkins showed that there is a critical $k = α(Δ)n$ such that it is hard to (approximately) sample from the uniform distribution on independent sets for the class of graphs $G$ with $n$ vertices and maximum degree at most $Δ$ . They conjectured that for $k$ below this critical value, the down-up walk mixes in polynomial time. I will discuss a resolution of this conjecture, which additionally shows that the down-up walk mixes in (optimal) time $OΔ(nn)$ . Based on joint work with Marcus Michelen, Huy Tuan Pham, and Thuy-Duong Vuong. Optimal mixing of the down-up walk on fixed-size independent sets
Tue, May. 7 10am (MATH 350)	Lie Theory Lucas Gagnon (York University) X A long-standing conjecture of Stanley and Stembridge asks whether certain symmetric function invariants of graphs are “e-positive,” i.e. expand positively in the basis of elementary symmetric functions. Efforts to resolve Stanley—Stembridge have raised the wider question of whether other symmetric functions are e-positive, particularly for symmetric invariants of other interesting objects. This talk will describe an instance of e-positivity arising in the study of modules of the unipotent upper triangular groups over a finite field. I use a collection of novel module homomorphisms to prove e-positivity, and these maps implicate a well-known algorithm from the study of Iwahori—Hecke algebras. Throughout, I will connect these results to previously known connections between the unipotent upper triangular group and symmetric functions in non commuting variables. Positivity in unipotent symmetric function invariants and Hecke traces Sponsored by the Meyer Fund
Tue, May. 7 12:10pm (MATH …	Kempner Luis David Garcia Puente (Colorado College) X Let G be a finite, connected graph. An arithmetical structure on G is an assignment of positive integers to the vertices such that, at each vertex, its label divides the sum of the labels at adjacent vertices (counted w/ multiplicity). Arithmetical structures naturally generalize the notion of the Laplacian matrix of a graph. They also appear in the context of arithmetical geometry. In this talk, I will present a general introduction to the topic, we will focus on the combinatorics of arithmetical structures on certain graph families including some counting results that relate arithmetical structures on a path graph and Catalan numbers. We will also discuss some results about the associated finite Abelian group of an arithmetical structure. What is an arithmetical structure? Sponsored by the Meyer Fund
Thu, Jun. 27 5pm (MATH 220)	Probability Stanislav Volkov (Lund University) X I will present some generalizations of the one-dimensional forest fire model with ignition occurring only at zero, which was analyzed in particular by Volkov (2009). In more recent work, we allow the rates at which the trees grow to depend on their location, and the fire can spread using long-range connections. We establish that the expected time required for the fire to reach a site at a distance $x$ from the origin is at most of the order $log x)log 2 + o(1)$ . The talk is based on a recent joint paper with Mikhail Menshikov (Durham) and the late Francis Comets (Paris). Generalized one-dimensional forest fire model

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