The following is the list of speakers in Graduate Students Seminar (GSS) and the dates that they are going to give a talk. If you are are one of the future speakers please send the abstract of your talk to shahmir@brandeis.edu . All talks are at Fridays at 2 p.m in 317.9/9/16 Yan Zuang Abstract: This talk will be an introduction to Eulerian polynomials in permutation enumeration. By strapping jet packs to mountain goats placed on a permutation visualized as a mountain range, we will prove the gamma-nonnegativity of Eulerian polynomials as well as a unexpected formula relating Eulerian polynomials and the distribution of peaks over the symmetric group. Moreover, we will prove a new refinement of this formula which relates Eulerian polynomials to the joint distribution of peaks and descents, and time permitting, discuss other connections to my research.9/16/16 McKee Krumpak Abstract: Morse homology is an important tool used in low dimensional topology. The ideas of Morse homology were generalized in infinite dimensions by Floer to produce the Floer homology invariants that are studied today. This talk will be a short exposition of the basics of Morse homology with some indication of how this can be adapted in the context of gauge theory.9/23/16 Devin Murray Title: The Contracting Boundary of CAT(0) GroupsAbstract: CAT(0) groups are an important class of groups which generalizes the notion of the fundamental group of a non-positively curved manifold. I will define what a CAT(0) group is and a few notions of the boundary. I will also talk a little bit about the basic ideas of geometric group theory and how this construction fits into the field and why it's important. My hope is that the talk will be approachable to everyone. 9/30/16 Mattew Garcia Title: A primer on smooth topos theoryAbstract: Starting with the definition of a special class of categories, topoi, that behave like the category of sets and very often are derived from a topology of some kind, we will build to a set of desirable conditions that a topos should have, resulting in a so-called "smooth topos", if we wish to embed the category of smooth manifolds within. One of the advantages of doing so is the presence of infinitesimal manifolds in a smooth topos. We will try to highlight the geometric and intuitive gains afforded by such an embedding.10/7/16 Abhishek Gupta Title: Quadratic and other reciprocity laws.Abstract: Quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. In this talk, we shall explain and prove this theorem. We shall then mention some other reciprocity laws, which are generalizations of quadratic reciprocity.10/14/16 Duncan Levear Title: Introducing The Geometry of NumbersAbstract: Geometrical arguments have been used since the ancient Greeks to prove facts about real numbers. In this talk, we will use geometry to study Z, the set of integers. We will define the linkage, lattices, and prove a geometrically flavored theorem with a dazzling number theoretic consequence. Restricting to one dimension, we will show that lattices in R are its only discrete subgroups, and that the rest are dense. The proof will exhibit the pervasive chaos of irrational numbers. 10/21/16 Eric Hanson Title: A Closer Look at InfinityAbstract: The nature of infinity is something that is extremely difficult to pin down. In this talk, we will begin by examining the physical universe to see if we can find evidence of a physical infinity. We will then transition to a more abstract setting and examine the infinities which appear in thoughts and mathematics. Finally, we will explore some fun and surprising paradoxes caused by the concept of infinity. Inspiration for this talk comes from the book 'Infinity and the Mind' by Rudy Rucker.10/28/16 Charlotte Morris-Wright Title: Covering Spaces and Tilings in Hyperbolic SpaceAbstract: In this talk, we will dive down the rabbit hole and discuss some of the crazy and counterintuitive ideas that come from disregarding Euclid's parallel postulate. I will begin by reviewing some classical hyperbolic geometry including the upper half plane and Poincare disk models. I will then illustrate some important properties of hyperbolic space that distinguish it from Euclidean space. Finally, I will discuss how tilings of hyperbolic space relate to the universal covering spaces of surfaces. 11/4/16 Abhishek Gupta Title: Spectral SequencesAbstract: Spectral sequences are an important tool in homological algebra and algebraic topology for computing homology groups. In this talk, we shall introduce these objects and look at some examples. If time permits, we will construct the spectral sequence of a filtered complex. 11/11/16 Shahriar Mirzadeh
11/18/16 Nick Wadleig Title: A zero-one law for improvements to dirichlet's theoremAbstract: Dirichlet's theorem states that for a real number x, |xq+p| ≤ t, |q| < t has nontrivial integer solutions for all t > 1. Davenport and Schmidt have observed that if 1/t is replaced with c/t, c<1, almost no x has the property that there exist solutions for sufficiently large t. Replacing c/t with an arbitrary function, it's natural to ask when precisely does the set of such x drop to a null set. We give an answer using dynamics of continued fractions. Joint work with Dmitry Kleinbock. 12/2/16 Cristobal Lemus Vidales |