| Date | Speaker | Topic | Abstract |
|---|---|---|---|
| Aug 30, 2016 | Zhexiu Tu | The art of the matroid | In this talk, we will start from linear spaces and introduce different kinds of matroids, not only in the realm of normal matroids, but oriented matroids, valuated matroids and so on. We then generalize the notion of all these matroids, reaching the objects called matroids over hyperfields, which, once defined, facilitate the creation of more matroids. In general, you will learn how to build that matroid you want. |
| Sep 6, 2016 | Gautam Gopal Krishnan | Finite Gelfand Pairs | Gelfand pairs arise in the study of Riemannian symmetric spaces and the examples that arise in this way are the most widely studied and best understood. They also arise from the study of semidirect products of compact Lie groups with two-step nilpotent Lie groups. Recently there has been some work on understanding finite Gelfand pairs associated to Heisenberg groups over finite fields. We will see what these finite Gelfand pairs are and how their properties mimic the other known examples. All the required terminology will be introduced in the talk and no previous knowledge of Gelfand pairs is necessary. |
| Sep 13, 2016 | Ian Pendleton | Toric symplectic and origami manifolds | Toric symplectic manifolds are a special subset of symplectic manifolds classified by a nice set of polytopes in $R^n$. I'll give an introduction to symplectic and toric symplectic manifolds, and then talk about a generalization of symplectic manifolds called origami manifolds. Toric origami manifolds are classified by "origami templates," which in 2 dimensions resemble the folded paper art for which they are named. |
| Sep 20, 2016 | Anwesh Ray | The algebra of Zeta values | The Riemann Zeta Function $ζ(s)$ encodes the arithmetic properties of prime numbers, its conjectured non-vanishing outside the line $1/2+iy$ has far reaching consequences in mathematics. The values of $ζ(m)= ∑∞k=1k−m$ at integers $m>1$ are of much interest. After describing some of the basic properties of the zeta function and some generalizations we will talk about Kontsevich's formulation of multiple zeta values as iterated integrals and the its implications to realizing the algebraic relations of multi zeta values. The goal of the talk is to bring to light the interesting phenomenon that a generalization of a zeta function to more variables can be more transparent and significant to a general mathematical audience (than the classical zeta function). No familiarity with number theory or the zeta function is assumed in this talk. |
| Sep 27, 2016 | Lila Greco | $p$ -rotor walk on $Z^d$ | Simple random walk on Zd is defined as follows: at every step, a particle moves from its current position to a neighboring lattice point, with each neighbor being equally likely. In contrast, rotor walk on $Z^d$ is a deterministic walk. Each lattice point has a rotor which points to one of its neighbors and a rotor mechanism which determines how the rotor rotates; at every step of rotor walk, the rotor at the particle's current position rotates according to the rotor mechanism, and the particle follows the rotor to its next position. Simple random walk is well-understood, but rotor walk is difficult to study because all randomness lies in the initial conditions. In this talk, we consider $p$-rotor walk, which bears similarities to rotor walk but introduces some randomness at each step. I will present new results which are the result of joint work with Swee Hong Chan and Boyao Li under Lionel Levine. |
| Oct 4, 2016 | David Mehrle | Categorifying $sl(2)$ | Categories are big. Entire fields of mathematics can happen entirely inside a single category. But the study of categories themselves is more like algebra, and in many ways generalizes it. For example, a category with one object is a monoid. Categori cation is the opposite process — promoting an algebraic object to a category in such a way that its original structure can be recovered. I will explain a particularly neat way to categorify the Lie Algebra $sl(2)$ and its representations by means of pretty pictures. |
| Oct 18, 2016 | Dylan Peifer | The Grobner walk | Grobner bases are the foundation of computational algebra, where computing a Grobner basis of an ideal is often the first step in determining properties of ideals and algebraic varieties. Unfortunately, the Grobner basis and the work required to compute it depend greatly on the chosen monomial order, with some desired monomial orders leading to difficult or impossible computations. The Grobner walk is an algorithm that sidesteps this problem by first computing the Grobner basis of an ideal I with respect to an easy monomial order and then moving in small steps towards the hard monomial order while making small adjustments to the Grobner basis for I. The end result is a Grobner basis for I with respect to the hard monomial order that has been computed using dramatically less time and memory. In this talk we will review the concept of a Grobner basis and present the steps involved in the basic Grobner walk. |
| Oct 25, 2016 | Joseph Gallagher | Irrational geometry | Some of the most beautiful proofs of geometric constructions being impossible are based on subtle number theoretic properties of geometrical invariants. We will tour some examples, such as the resolution of Hilbert's third problem by Dehn, the impossibility of squaring the circle, and various questions associated to hyperbolic geometry. |
| Nov 1, 2016 | Frederik De Keersmaeker | Symplectic capacities | Symplectic manifolds carry a natural volume that is preserved by symplectic maps between them. Gromov proved in 1985 that one cannot embed a ball into a cylinder symplectically unless the radius of the ball is less than or equal to the radius of the cylinder. It is called the non-squeezing theorem for obvious reasons: if you squeeze a ball with a large radius into a cylinder with a small radius, you satisfy the volume constraint but the embedding cannot be symplectic. Symplectic capacities are notions of volume more suited to the symplectic world. One example is Gromov width: the radius of the largest ball (Darboux chart) you can symplectically embed into the manifold. I will talk about some interesting features of symplectic capacities, after a brief introduction to symplectic manifolds. |
| Nov 8, 2016 | Balázs Elek | The Leech lattice | We will explore one of the most exceptional objects in mathematics, the Leech lattice. We will construct it using 26-dimensional Minkowski spacetime, then discuss its awesomeness. Applications include stacking spheres efficiently, sending messages to distant spacecrafts, maximizing kissing numbers and winning a bottle of Jack Daniels. |
| Nov 15, 2016 | Teddy Einstein | Free groups and the topology of finite graphs | Stallings discovered that many properties of free groups can be proved elegantly by studying the topology of finite graphs. In this talk, I will explain some of Stallings' tools and techniques including folding and core graphs. I will also explain short proofs of some remarkable results such as Marshall Hall's theorem which shows that every finitely generated subgroup of a free group is the intersection of the finite index subgroups containing it. |
| Nov 22, 2016 | Daoji Huang | What is a gauge | Coordinates are bridges between geometry and "numbers". If we want to assign coordinates to a family of objects parametrized by some space, we get a family of coordinate systems that depend on locations in the parameter space. This is pretty much what a gauge is. A gauge transformation is a change of coordinates applied to each such coordinate system in the family. A gauge theory is for systems to which gauge transformations can be applied, and very often we care about quantities that are invariant under gauge transformations. This talk will be focused on basic mathematical ideas behind classical gauge theory, illustrated with simple conceptual examples in geometry and combinatorics. |
| Nov 29, 2016 | Hannah Keese | Knot invariants and representation theory | One of the central problems in knot theory is determining when two knots are the same, and despite the seeming simplicity of the problem it can be difficult and time-consuming to resolve. The main approach to this question is the study of knot invariants of varying complexity, such as knot groups and polynomial invariants. In recent years, there has been much interest in homological knot invariants, whose extra structure make them interesting from the perspective of representation theory. We will discuss some well-known examples of knot invariants and their links to the representation theory of Lie algebras. Further knot-related puns may be attempted but are (k)not guaranteed. |