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

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)

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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 $k=\alpha (\Delta )n$ 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 $\Delta$. 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 ${O}_{\Delta}(n\phantom{\rule{0.167em}{0ex}}\mathrm{log}\phantom{\rule{0.167em}{0ex}}n)$.

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