EveryTopic Seminar, Fall 2017

Thursday 2:00-3:00p, Goldsmith 317.
Organizers: Corey Bregman, Konstantin Matveev.

The EveryTopic Seminar is the Brandeis math department colloquium. Talks are 60 minutes long (sometimes longer with a break) and aimed at a broad audience of mathematicians.

Date Speaker (Affiliation) Title Abstract
Sep 14 Ying Zhou (Brandeis) Tame quivers have finitely many m-maximal green sequences Brustle-Dupont-Perotin proved that tame quivers admit finitely many maximal green sequences. We have shown that this result can be generalized to m-maximal green sequences. Furthermore we have defined the concepts of green sequence finiteness and almost morphism finiteness and have shown that quivers of finite and tame types have these properties. I'm going to start from some combinatorics to introduce all the concepts so that they are accessible.
Sep 28 Dmitry Kleinbock (Brandeis) Dynamical systems on arithmetic homogeneous spaces The theory of dynamical systems studies points moving along complicated trajectories. Very important classes of such systems come from actions of algebraic groups. I will talk about general theorems for such actions and discuss various connections to number theory. This can be viewed as an introduction to the next week's talk (Brandeis Thursday October 3) by Nattalie Tamam, Tel Aviv University.
Oct 3 (Brandeis Thursday) Nattalie Tamam (Tel Aviv University) Divergent trajectories in arithmetic homogeneous spaces of rational rank two In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.
Oct 11 (Brandeis Thursday) Dingxin Zhang (Brandeis University) Absolute values of Frobenius eigenvalues Given a "good" homogeneous polynomial P with integral coefficients, its zeros define a complex submanifold X in the projective space. A conjecture of Weil (now proven by Deligne) is that the Betti numbers of X, the Poincaré duality of the cohomology groups, etc., can be inferred from counting the number of solutions of P over a finite field. More precisely, Weil defined a formal power series as the generating function of the number of solutions of the P in F_{p^{n}}, and it turns out the (archimedean) absolute values of the "roots" and "poles" of this formal power series will reveal the topological information of X. I will talk about this circle of ideas, and also discuss some phenomena when we look at the p-adic (nonarchimedean) absolute values of these roots and poles.
Oct 19 Leonid Petrov (University of Virginia) Overview of random lozenge tilings I will discuss random lozenge tilings of polygons and other domains --- one of the most well-studied models of statistical mechanics. Due to free fermion structure, random lozenge tilings can be analyzed asymptotically in multiple different regimes with the help of exact formulas. I will describe existing results about random tilings, and point out new developments towards universality of these results.
Nov 2 Ira Gessel (Brandeis University) An introduction to lattice path enumeration Counting paths in a given region with a specified set of steps is one of the fundamental problems of enumerative combinatorics. I will discuss some of the fundamental methods and results in this area.
Nov 9 Corey Bregman (Brandeis University) Outer space, flat tori, and the period mapping Culler-Vogtmann's outer space parametrizes the set of marked, rank-n, metric graphs. It comes equipped with an action of Out(F_n), the outer automorphism group of a rank-n free group, and can therefore be seen as a free group analog of Teichmuller space. In the first half, we will provide an introduction to Outer space and the moduli space of rank n graphs. In the second half, we will define the period mapping, which is a continuous map from outer space to the space of positive definite quadratic forms. For each n>1, we explicitly describe the homotopy type of the fibers of this map.
Nov 16 Ruth Charney (Brandeis University) Searching for Hyperbolicity While groups are defined as algebraic objects, they can also be viewed as symmetries of geometric objects. This viewpoint gives rise to powerful tools for studying infinite groups. The work of Max Dehn in the early 20th century on groups acting on the hyperbolic plane was an early indication of this phenomenon. In the 1980’s, Dehn’s ideas were vastly generalized by Mikhail Gromov to a large class of groups, now known as hyperbolic groups. In recent years there has been an effort to push these ideas even further. If a group fails to be hyperbolic, might it still display some hyperbolic behavior? Might some of the techniques used in hyperbolic geometry still apply? The talk will begin with an introduction to some basic ideas in geometric group theory and Gromov’s notion of hyperbolicity, and conclude with a discussion of recent work on finding and encoding hyperbolic behavior in more general groups.
Nov 30 Wenyu Pan (Yale University) Local mixing and abelian covers of finite volume hyperbolic manifolds Abelian covers of finite volume hyperbolic manifolds are ubiquitous. We will discuss ergodic properties of the geodesic flow/ frame flow on such spaces. In particular, we will discuss the local mixing property of the geodesic flow/ frame flow, which we introduce to substitute the well-known strong mixing property in infinite volume setting. We will also discuss applications to measure classification problems and to counting and equidistribution problems. Part of the talk is based on the joint work with Hee Oh.
Dec 7 Shahriar Mirzadeh (Brandeis University) Dimension estimates for the set of points with non-dense orbit in homogeneous spaces In this talk we study the set of points in a homogeneous space whose orbit escapes the complement of a fixed compact subset. We find an upper bound for the Hausdorff dimension of this set. This extends the work of Kadyrov, where he found an upper bound for the Hausdorff dimension of the set of points whose orbit misses a fixed ball of sufficiently small radius in a compact homogeneous space. We can also use our main result to produce new applications to Diophantine approximation. This is joint work with Dmitry Kleinbock.

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