EveryTopic Seminar, Fall 2015

Thursday 2:15-3:05p, Goldsmith 300. (Refreshments at 2p in Goldsmith 100.)
Organizers: Carl Wang Erickson and Arunima Ray.

The EveryTopic Seminar is the Brandeis math department colloquium. Talks are 50 minutes long and aimed at a broad audience of mathematicians.

See (and add!) our google calendar.

Date Speaker (Affiliation) Title Abstract
Sep 3 Danny Ruberman (Brandeis) Configurations of embedded spheres Configurations of lines in the plane have been studied since antiquity. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects ("lines") and other objects ("points") can be realized by actual points and lines in a projective plane over a field. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. I will explain some joint work with Laura Starkston (Stanford) giving new topological restrictions on the realization of configurations of spheres in the complex projective plane.
Sep 17 Heather Macbeth (MIT) Geometric minimax problems I will discuss a general convexity principle for minimax problems, and relate it to classical results from the theory of zero-sum games. I will then describe some recent applications of this principle in differential geometry: to metrics with maximal first eigenvalue (due to Nadirashvili); to metrics with maximal first Steklov eigenvalue (due to Fraser and Schoen); and to conformal classes realizing the Yamabe invariant (due to myself).
Sep 24 David Hansen (Columbia) Arithmetic properties of Hurwitz numbers Hurwitz numbers are the "Q(i)-analogue" of Bernoulli numbers; they show a remarkable number of patterns and properties, and deserve to be better-known than they are. I'll discuss some old results on these numbers due to Hurwitz and Katz, and some newer results obtained by four Columbia undergraduates during an REU this past summer. No background knowledge will be assumed.
Oct 8 Gordana Todorov (Northeastern) Semi-invariant pictures and Maximal green sequences Abstract
Oct 22 Michael LaCroix (MIT) A Surprising Decomposition for the Singular Values of the Gaussian Unitary Ensemble The Gaussian Unitary Ensemble (GUE) consists of matrices with independent complex Gaussian entries, constrained by Hermitian symmetry. It is usually studied with respect to the statistics of its eigenvalues, but several seemingly unrelated properties of the GUE gain simultaneous explanations from the fact that the singular values of the ensemble can be decomposed as the union of two independent sets of singular values of ensembles of Laguerre type. These constituent matrices have nominal half-integer size, but can also have concrete sparse models, through which we can observe facts about the determinant of the GUE. In this talk, I will describe the decomposition, discuss some of the properties the decomposition explains, and outline how the decomposition is related to a combinatorial identity involving the enumeration of orientable maps.
Oct 29 Alexander Garver (Minnesota) Oriented Flip Graphs and Maximal Green Sequences The oriented exchange graph of a quiver (i.e. a directed graph) describes sequences of local transformations of quiver. The maximal directed paths in an oriented exchange graph of finite length are in bijection with maximal green sequences, which are of interest to representation theorists and string theorists. In addition, the class of oriented exchange graphs contains interesting families of partially ordered sets including the Tamari lattices. We model oriented exchange graphs combinatorially, using what we call an oriented flip graph, where quivers correspond to collections of noncrossing arcs in a disk and mutation corresponds to moving between such noncrossing collections. We show that oriented flip graphs are semidistributive, congruence uniform, and polygonal lattices. No background on quivers will be assumed. This is joint work with Thomas McConville.
Nov 5 Anand Patel (BC) Plane Cubics, Degenerate Polars, Degenerate Satellites, and Moduli spaces My intention is to talk about a very broad and active research area through a particular example: genus one curves with n marked points. I will show, through this example, how very classical algebraic geometry often manifests itself in the modern study of moduli spaces. Shiny new results will be discussed, but old gems will also appear, so this talk will be accessible to many. Many of the results are joint with Dawei Chen.
Nov 12 Yaiza Canzani (Harvard) Geometry and topology of the zero set of monochromatic random waves There are several questions about the zero set of Laplace eigenfunctions that have proved to be extremely hard to deal with and remain unsolved. Among these are the study of the size of the zero set, the study of the number of connected components, and the study of the topology of such components. A natural approach is to randomize the problem and ask the same questions for the zero sets of random linear combinations of eigenfunctions. In this talk I will present some recent results in this direction.
Nov 19 Carl Wang Erickson (Brandeis) Zeta-functions and Arithmetic One often hears that there is a relationship between the locations of the zeros of the Riemann zeta function and the distribution of the prime numbers among the natural numbers. Indeed, the Riemann Hypothesis predicts the location of the zeros of zeta, which has implications for the distribution of the prime numbers. The main goal of this talk is to show that this relationship may be made quite explicit through what is known as an "explicit formula." This will be done using basic tools of complex analysis. With what time remains, we will indicate how to generalize this setup to arithmetic objects other than the prime numbers.
Nov 30 (Monday) in Goldsmith 317 Lei Fu (Nankai) Monodromy of isolated singularities I will explain the concept of vanishing cycles. Using the l-adic Fourier transformation, I illustrate how to calculate the monodromy of vanishing cycles for some isolated singularities.
Dec 3 Sarah Bray (Tufts) Ergodic geometry for nonstrictly convex Hilbert geometries The geodesic flow of a compact hyperbolic Riemannian manifold is a classical object of study, and arguably an origin story for the study of ergodic theory and chaotic dynamical systems. Geodesic flows arising from natural generalizations of the notions of compact, Riemannian, and hyperbolic have entertained dynamicists since the 1930s, and research around the study of geodesic flows remains active today.

In this talk, I will introduce the audience to the world of Hilbert geometries, which can be thought of as a generalization of Riemannian geometries. In particular, compact, nonstrictly convex Hilbert geometries in three dimensions have a fascinating geometric irregularity which creates major barriers for the study of geodesic flows. Many classical techniques are no longer applicable, such as those used for nonuniformly hyperbolic systems, rank one manifolds, and CAT(0) spaces. Nonetheless, by pushing the limits of the geometry, we are able to prove many hyperbolic dynamical results. These results culminate in the construction of a measure of maximal entropy which is ergodic for the geodesic flow.

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