The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.
The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.
Graeme Milton, University of Utah.
Joel Hass, UC Davis and IAS.
Xiaojun Huang, Rutgers University.
Howard Stone, Princeton University.
Nicholas Ercolani, University of Arizona.
Charles Van Loan, Cornell University.
Milen Yakimov, LSU.
Daniel Groves, University of Illinois at Chicago.
No colloquium today due to department faculty meeting.
Dan Margalit, Georgia Tech University
To each homeomorphism of a surface we can associate a real number, called the entropy, which encodes the amount of mixing being effected. This number can be studied from topological, geometrical, dynamical, analytical, and algebraic viewpoints. We will start by explaining Thurston’s beautiful insight for how to compute the optimal entropy within a homotopy class and explain a new, fast algorithm based on his ideas, which is joint work with Balazs Strenner and Oyku Yurttas. We will also discuss some classical results and recent work with Ian Agol, Benson Farb, and Chris Leininger on the problem of understanding homeomorphisms with small entropy. One theme is that algebraic complexity and geometric complexity both imply dynamical complexity.
Sarah Witherspoon, Texas A&M University
The classical Poincare-Birkhoff-Witt (PBW) Theorem sheds light on the structure of Lie algebras: These are, by definition, nonassociative rings, and the PBW Theorem states that nonetheless, a Lie algebra embeds into an associative ring, namely its universal enveloping algebra, that behaves in many ways like a polynomial ring (and this can be made precise). Many other rings share this advantageous property. In particular, they have PBW bases, which greatly facilitate their study. In this talk, we will first recall Lie algebras and the classical PBW Theorem. Then we will mention some more recent appearances of PBW-type theorems in the contexts of quantum groups, symplectic reflection algebras, graded Hecke algebras, and generalizations.
Julia Hartmann, University of Pennsylvania
Differential Galois theory is an algebraic theory for linear differential equations, in analogy to classical Galois theory. It was proposed by Picard and Vessiot more than a hundred years ago and then developed by Kolchin. Patching techniques have been used in inverse Galois theory and more recently in other areas of algebra and arithmetic geometry. The talk gives an introduction to differential Galois theory and to patching. Using patching methods, we will deduce new properties of differential Galois extensions over function fields of Riemann surfaces.
Samuel Taylor, Yale University
Gromov’s notion of a hyperbolic group encompasses many of the beautiful features of negative curvature classically found in the fundamental group of a negatively curved Riemannian manifold. As these features are desirable from geometric, algorithmic, and dynamical points of view, we should strive to understand which groups are hyperbolic, as well as what techniques from this theory can be used to study broader classes of geometrically significant groups.
This talk will address these problems, focusing on the geometry of group extensions. We will also discuss how probabilistic techniques can be used to show that these negative curvature features are, in fact, pervasive.
There will be tea at 3:30.
David Fisher, Indiana University
Lattices in higher rank simple Lie groups, like SL(n,R) for n>2, are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds. After providing some history and motivation, I will discuss a very recent result, proving many cases of the main conjecture. While avoiding technical matters, I will try to describe some of the novel flavor of the proof. The proof has many surprising features, including that it uses hyperbolic dynamics to prove an essentially elliptic result, that it uses results on homogeneous dynamics, including Ratner's measure classification theorem, to prove results about inhomogeneous system and that it uses analytic notions originally defined for the purposes of studying the K theory of C^* algebras. This is joint work with Aaron Brown and Sebastian Hurtado.
Steven Heilman, UCLA
Given votes for candidates, what is the best way to determine the winner of the election, if some of the votes have been corrupted or miscounted? As we saw in Florida in 2000, where a difference of 537 votes determined the president of the United States, the electoral college system does not seem to be the best voting method. We will survey some recent answers to the above question along with some open problems. These results use tools from probability, from discrete Fourier analysis and from the geometry of the Gaussian measure on Euclidean space. Answering the above voting question reveals unexpected connections to Khot's Unique Games Conjecture in theoretical computer science and to Bell's inequality from quantum mechanics. We will discuss these connections and present recent results and open problems.
Priyam Patel, UC Santa Barbara
Abstract: Peter Scott’s famous result states that the fundamental groups of hyperbolic surfaces are subgroup separable, which has many powerful consequences. For example, given any closed curve on such a surface, potentially with many self-intersections, there is always a finite cover to which the curve lifts to an embedding. It was shown recently that hyperbolic 3-manifold groups share this separability property, and this was a key tool in Ian Agol's resolution to the Virtual Haken and Virtual Fibering conjectures for hyperbolic 3-manifolds.
I will begin this talk by giving some background on separability properties of groups, hyperbolic manifolds, and these two conjectures. There are also a number of interesting quantitative questions that naturally arise in the context of these topics. These questions fit into a recent trend in low-dimensional topology aimed at providing concrete topological and geometric information about hyperbolic manifolds that often cannot be gathered from existence results alone. I will highlight a few of them before focusing on a quantitative question regarding the process of lifting curves on surfaces to embeddings in finite covers.
Arjun Krishnan, University of Utah
First-passage percolation is a random growth model on the cubic lattice Z^d. It models, for example, the spread of fluid in a random porous medium. This talk is about the asymptotic behavior of the first-passage time T(x), which represents the time it takes for a fluid particle released at the origin to reach a point x on the lattice.
The first-order asymptotic --- the law of large numbers --- for T(x) as x goes to infinity in a particular direction u, is given by a deterministic function of u called the time-constant. The first part of the talk is about a new variational formula for the time-constant, which results from a connection between first-passage percolation and stochastic homogenization for discrete Hamilton-Jacobi-Bellman equations.
The second-order asymptotic of the first-passage time describes its fluctuations; i.e., the analog of the central limit theorem for T(x). In two dimensions, the fluctuations are (conjectured to be) in the Kardar-Parisi-Zhang (KPZ) or random matrix universality class. We will present some new results (with J. Quastel) that proves the KPZ universality conjecture in the intermediate disorder regime.
William Perkins, Birmingham University
Gibbs measures (also known as Markov random fields or probabilistic graphical models) arise in many mathematical and scientific disciplines including probability, statistical physics, and machine learning. The central object in understanding the behavior of a given Gibbs measure is the partition function, the normalizing constant of the probability distribution. I will describe two new methods for approximating partition functions of Gibbs measures, using the Potts model from statistical physics as a running example. I will present applications to the stochastic block model of community detection, random graph coloring, and extremal combinatorics.
Tian Yang, Stanford University
Character varieties of a surface are central objects in several beaches of mathematics, such as low dimensional topology, algebraic geometry, differential geometry and mathematical physics. On the character varieties, there is a tautological action of the mapping class group -- the group of symmetries of the surface, which is expected to be ergodic in certain cases. In this talk, I will review related results toward proving the ergodicity and introduce two long standing and related conjectures: Goldman's Conjecture and Bowditch’s Conjecture. It is shown by Marche and Wolff that the two conjectures are equivalent for closed surfaces. For punctured surfaces, we disprove Bowditch's Conjecture by giving counterexamples, yet prove that Goldman's Conjecture is still true in this case.