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Prasad Senesi, The Catholic University of America
Highest-weight representations play a prominent role in the representation theory of Lie algebras and quantum groups. Particular examples of highest-weight representations of certain infinite-dimensional Lie algebras called the Weyl modules (for loop and quantum algebras) were introduced by Chari and Pressley in 2000. In this introductory talk, we proceed by example from the classical structure and representation theory of the special linear algebra in dimensions 2 and 3, to that of the corresponding loop algebras and quantum groups. Along the way, the utility of highest-weight representations, and of the (local and global) Weyl Modules, in all of these settings will be described. We will conclude with a discussion of the Yangian, its relation to the quantum loop algebra, and some recent work concerning its global Weyl modules. This is joint work with Bach Nguyen (Temple University) and Matt Lee (University of Illinois at Chicago).
David Harbater, University of Pennsylvania
Local-global principles have long played an important role in number theory and in the study of curves over finite fields, beginning with the Hasse-Minkowski theorem on quadratic forms. After reviewing the classical situation, this talk will discuss local-global principles that have recently been found to hold in the context of certain "higher dimensional" fields, using new methods.
Tatyana Shcherbina, Princeton University
Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.
Ivan Levcovitz, Technion
Associated to any simplicial graph K is the right-angled Coxeter group (RACG) whose presentation consists of an order 2 generator for each vertex of K and relations stating that two generators commute if there is an edge between the corresponding vertices of K. RACGs contain a rich class of subgroups including, up to commensurability, hyperbolic 3-manifold groups, surface groups, free groups, Coxeter groups and right-angled Artin groups to name a few. I will describe a procedure which associates a cube complex to a given subgroup of RACG. I will then present some results regarding structural and algorithmic properties of subgroups of RACGs whose proofs follow from this viewpoint. This is joint work with Pallavi Dani.
Lise-Marie Imbert-Gerard, University of Maryland College Park
Laurent Stolovitch, Université Nice Sophia Antipolis
We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (in particular are convergent). The result is obtained by using the recent CR - DS technique, connecting degenerate CR-manifolds and Dynamical Systems, and employing subsequently the multisummability theory of divergent power series used in the Dynamical Systems theory. This is a joint work with I. Kossovskiy and B. Lamel.
Elie Abdo, Temple University
Abstract: Several theorems that hold in the theory of one complex variable cannot be generalized to the theory of several complex variables, one of them is the Riemann Mapping Theorem. In fact, invariant metrics are important tools that can be used to show that the open bi-disc and the open unit-ball in $\mathbb{C}^2$ are not biholomorphic. In this talk, we introduce some invariant metrics, the Kobayashi, Caratheodory and Sibony metrics, list some of their properties, and do some interesting examples.
There are no conferences this week.