Representation of Quantum Affine sl_n and Connection with Affine Hecke Algebras and the Category O(gl_r), by: Dr. Vyjayanthi Chari
Abstract: The study of finite dimensional representations of quantum affine algebras goes back several decades. Many important results have been proved and deep techniques have been developed over the years; but many interesting natural questions remain unanswered. More recently the work of Hernandez and Leclerc on monoidal categorification has led to a renewed interest in the subject. The influential work of Kashiwara and his collaborators have led to further substantial developments in the subject.
In this talk, we will be guided by another remarkable connection, via affine Hecke algebras, with the category of smooth representations of GL_n(F) where F is a non-Archimedean field. We will discuss this connection, especially with the work of Lapid-Minguez. Motivated in part by their work, we introduce a family of modules for the quantum affine algebra and give an explicit determinantal formula for its character. As an application of our results we explain how to compute certain Kazhdan-Lusztig coefficients in the Bernstein-Gelfand-Gelfand category O.
Poster: can be accessed from here.
Generalized Willmore Energies, Elastic Surface Theory, and Biophysical Applications, by: Dr. Magdalena Daniela Toda
Abstract: Functionals involving surface curvature represent an extensive field of study across many sciences, including but not limited to mathematics, mechanics, physics and biology. Their extrema are crucial in the field of biophysics and have numerous applications to the shapes of biological cells, bio-membranes and protein folding. The talk will discuss Willmore energies and generalizations (in particular, p-Willmore energies and Helfrich-Canham energies). Some recent mathematical results of our team will be presented along with real-life applications and computational models.
Introduction to Soliton gasses, by: Dr. Kenneth McLaughlin
Abstract: In this presentation I will provide an introduction to solitons –fundamental solutions of nonlinear partial differential equations – with simple examples. Then we will consider more complicated solutions of the same pdes made up of many solitons, and construct a soliton gas by taking the limit as the number of solitons grows to infinity. A description of the history of the kinetic theory of soliton gasses will be encountered along the way. Time permitting, I will explain recent results in which random matrix theory techniques are being used to study fluctuations of random soliton gasses. The work presented will include joint work with Manuela Girotti (Emory University), Tamara Grava (SISSA and University of Bristol), Robert Jenkins (University of Central Florida), Alexander Minakov (Charles University, Prague), and Joseph Najnudel (Bristol).
Poster: can be accessed from here.
Semiclassical analysis of conformal blocks on the torus., by: Dr. Andrei Prokhorov
Abstract: Conformal blocks are the transcendental functions which are used to build the correlation functions of conformal field theory. We are interested in the geometric setup of the torus. Zamolodchikov recursion provides an effective procedure for calculation of conformal blocks, but it is not efficient for mathematical analysis. For certain values of parameters conformal blocks can be written as Dotsenko-Fateev multiple integral. For general values of parameters they can be written as expectation of integral of certain GMC measure. The semiclassical analysis describes behavior of conformal clocks as the central charge approaches infinity. According to Zamolodchikov conjecture, this limit is described in terms of Lame accessory parameters. We justify this conjecture using probabilistic methods. This is the joint work with Harini Desiraju and Promit Ghosal.
Poster: can be accessed from here.
Translation surfaces, billiards, and dynamics., by: Dr. Howard Masur
Abstract: This talk is intended as an introduction to the subject of translation surfaces and the dynamical system of linear flows on them. The study of linear flows on a translation surface began a number of years ago from the effort to understand the related question of billiards in polygons. A major part of this subject involves studying moduli spaces of translation surfaces and groups acting on them. I will introduce all of these objects and explain some of what is known and some of the open questions.
Poster: can be accessed from here.
Double affine Hecke algebras and their applications., by: Dr. Pavel Etingof
Abstract: Double affine Hecke algebras (DAHAs) were introduced by I. Cherednik 25 years ago to prove Macdonald's conjectures. A DAHA is the quotient of the group algebra of the elliptic braid group attached to a root system by Hecke relations. DAHAs and their degenerations are now central objects of representation theory. They also have numerous connections to many other fields -- integrable systems, quantum groups, knot theory, algebraic geometry, combinatorics, and others. In my talk, I will discuss the basic properties of double affine Hecke algebras and touch upon some applications.
Arctic Curve and Some Other Unsolved Problems in Statistical Physics, by: Dr. Pavel Bleher
Abstract: We will discuss some challenging, unsolved problems in statistical physics. This will include:
The distribution of the Lee-Yang zeroes of the partition function.
Double scaling limits and phase transitions in the dimer model.
Arctic curve and emptiness formation probability in the six-vertex model, and others.
The talk will be oriented to a general audience.
How to Make the Slope Constant in a Dynamical System, by : Dr. Michal Misiurewicz
Abstract: Suprisingly many models allow some kind of reduction to iterations of a continuous map. The complexity of such a system is measured by its topological entropy. To simplify the system even further, one can attempt to produce a map of constant (absolute value of the) slope, equal to the exponential of the entropy. In the late 1970's Milnor and Thurston proved that such simplification is always possible when the map is piecewise monotone with finitely many pieces and the entropy is positive. I will describe the ideas of a proof, possible generalizations, and the problems that arise when we assume that the number of pieces of monotonicity is infinite.
You Didn’t Think I Could Solve It, but I Can! A Survey of the Modern Theory of Integrable Systems, by: Dr. Alexander Its
Abstract: The term “Integrable Systems” usually refers to mathematical objects, most often differential equations, with special symmetry properties which allow to study them in a very detailed way and sometimes even to solve them in a closed form. The class of integrable systems includes several fundamental equations of nature, and the mathematical foundations of integrable systems go back to classical works of Liouville, Gauss, and Poincaré. In our days, the theory of integrable systems has become an expanding area which plays an increasingly important role as one of the principal sources of new analytical and algebraic ideas for many branches of modern mathematics and theoretical physics. In this talk, a brief history and state-of-the-art of the theory of integrable system together with the place the integrable systems occupy in the general area of mathematics will be presented.
Dynamics: from the torus to the Cantor set, by: Dr. Bruce Kitchens
Abstract : The torus can be thought of as an abelian group using coordinate-wise addition modulo 1 as the group operation. It has a topological and differentiable structure. The usual two-dimensional Lebesgue measure is invariant under the group operation and is the topological group’s Haar measure. There are continuous group automorphisms of the torus and they can be described by 2 × 2 matrices having integer entries and determinant ±1. The automorphisms preserve Lebesgue measure. We will examine the dynamics of one of these. It is the prototype for the Anosov diffeomorphisms of manifolds. There is a very nice geometric construction of a Markov partition for the map. It allows one to relate the dynamics of the map on the torus to a simply described homeomorphism of the Cantor set. The homeomorphism on the Cantor set is a topological Markov shift and can be described using the matrix that described the map on the torus. The advantage of this construction is that it allows one to easily analyze many important dynamical properties of the original map on the torus. The properties include periodic points, invariant measures and entropy.
Transitionally Commutative Structures on Rank 2 Real Vector Bundles, by: Dr. Bernardo Villarreal
Abstract: A transitionally commutative structure (tc structure) is a lift of the classifying map of the bundle to the classifying space for commutativity denoted BcomG. In this talk I will focus on G=O(2), by constructing explicit tc structures on the trivial bundle over the 2 sphere, and showing that (surprisingly) the tautological bundle over the Grassmannian of 2-planes in R^n (n>3) does not admit any tc structure.
This is part of joint work with O. Antolín-Camarena and S. Gritschacher, and part of work in progress with D. Ramras.
The Proof of Shapiro-Shapiro Conjecture, by: Dr. Evgeny Mukhin
Abstract: I will recall how a study in the area of mathematical physics unexpectedly led to a proof of a long standing conjecture in real algebraic geometry. This talk is based on several papers written together with Prof. Tarasov (IUPUI) and Prof. Varchenko (UNC, Chapel HIll) a few years ago.
Solutions of the cubic Fermat equation in algebraic number fields, Part II, by: Dr. Patrick Morton
Abstract: I will discuss my work in trying to prove that the cubic Fermat equation x^3 + y^3 = z^3 has nontrivial solutions in every quadratic field K=Q(sqrt(-d)) in which -d < 0 and d = 2 (mod 3). This was conjectured by Aigner in 1955, but is still an open problem. I will show how to find solutions in a large number of these quadratic fields (at least 50% of them) using modular functions and Galois theory. In particular, I will show how these techniques lead to the solution (-4+29sqrt(-17))^3 + (-4-29sqrt(-17))^3 = 70^3 in the field K = Q(sqrt(-17)).
Spray Geometry, by: Dr. Zhongmin Shen
Abstract: A spray on a manifold can be viewed as a collection of systems of second-order ODEs. The solutions are called the geodesics of the spray. A spray can also be viewed as a collection of parametric curves (called geodesics) in the manifold with the following properties: 1) for every tangent vector v at a point p, there is a unique geodesic c(t) with c(0)=p, c’(0)=v; 2) for any geodesic c(t) and any positive number k > 0, c(kt) is still a geodesic. No distance measure is associated with the spray. Every Riemannian metric determines a spray. In this talk, I will introduce, curvatures for sprays and discuss their geometric meaning. Basic knowledge on manifolds, vector fields, and differential forms on manifold are required.
Contact Embeddings in Dimension Three, by: Dr. Olguta Buse
Abstract: In joint work with D. Gay, we introduce the concepts of capacity and shape for a three dimensional contact manifold relative to a transversal knot. We will explain the connection with the existing literature and provide our main computation for the shape in the case of lens spaces L(p,q). The main tool used here are rational surgeries which will be explained through their toric interpretations based on the continuous fraction expansions of p/q. We will discuss possible parallels with the study of ellipsoid embeddings in four dimensions.
Quantum integrable models, what mathematics are they about?, by: Dr. Vitaly Tarasov
Abstract : After a brief gentle introduction into quantum mechanics and integrability, I will try to explain what kind of problems in linear algebra, representation theory (it is "in between" abstract algebra and linear algerba), combinatorics, complex analysis, etc. arise under the umbrella of quantum integrable models. I hope to keep the content within general mathematical background, but it would be useful to know in advance the basic idea of a tensor product.
Untangling the double twist, by: Dr. Daniel Ramras, Nov 10, 2016
Abstract : It has long been known to mathematicians and physicists that while one full turn of an object in 3-space causes tangling, two full turns can be untangled. Physical demonstrations of this fact include Dirac's belt trick and the Indonesian candle dance. Mathematically, this is a topological feature of the rotation group SO(3). In this talk, we will explore a geometrically defined untangling procedure, leading to interesting conclusions about the minimum complexity of untanglings. Along the way, we'll see quaternions, degrees of continuous mappings, and animations of our geometric untangling. This is based on joint work with David Pengelley.
Bridging Complex and Functional Analysis, by: Dr. Carl Cowen, March 7, 2016
Abstract: The impression one gets from undergraduate and graduate classes is that mathematics is divided into separate realms: "algebra", "analysis","applied math", "topology", ... In reality, nothing could be further from the truth! In many cases, the greatest progress in creation of new mathematics comes from bridging the gap between two or more "branches" of mathematics! The goal of this talk is to illustrate how some bridges between complex analysis and functional analysis, with occasional glimpses of algebraic geometry, can result in progress toward solution of some hard problems! Very often, using the tools and results of different fields, one can get new insights into problems than can be seen as in the overlap between these fields. Some illustrations of success in solving some problems and current work in attacking hard problems will be given.
Relative Amenabilty and Relative Property A for Discrete groups, by: Dr. Ron Ji, March 7, 2016
Abstract: A discrete group G is said amenable if there is a G-invariant mean on groups include finite groups, abelian groups, solvable groups and groups of subexponential growth. Non-abelian free groups are not amenable. Yet free groups from many points of view are nice groups. We will discuss a generalization of the amenability, namely Property A that will include all free groups and in general include Gromov’s word hyperbolic groups. A relative version of amenability, Property A and their applications will be discussed.