## September 2016

### Jane Hawkins, Ergodic Theory and Dynamical Systems Seminar

Title: Benford’s Law: Detecting fraud in business using ergodic theory Abstract: Benford’s Law is the observation that in many collections of numbers like mathematical tables or real-life data, the leading significant digits are not uniformly distributed, as might be expected, but are heavily skewed toward the smaller digits. Tax auditors and insurance companies use this to detect fraud. We discuss, using simple ideas in ergodic theory, what is behind Benford’s Law, and give a few extensions to mathematical (and other)…

Find out more »### You Qi, Geometric Methods in Representation Theory

Title: On the center of small quantum groups Abstract: We will report some recent progress on the problem of finding the center of small quantum groups. This will be based on joint work with Anna Lachowska.

Find out more »### Karl Petersen (UNC-CH), Ergodic Theory and Dynamical Systems Seminar

Title: Guessing the present from observations at random times in the future Abstract: In order to try to understand average information obtained from measurements made at random times, we consider very simple Markov chains, and even a switch of two points, observed only whenever a coin flip comes up heads. The basic ideas of entropy and conditioning that come up are explained as needed.

Find out more »### Jiuzu Hong (UNC-CH), Geometric Methods in Representation Theory Seminar

Title: Conformal blocks, Verlinde formula and diagram automorphisms Abstract: The Verlinde formula computes the dimension of conformal blocks associated to simple Lie algebras and Riemann surfaces. If the simple Lie algebra admits a nontrivial diagram automorphism, then this automorphism acts on the space of conformal blocks naturally. I will report a recent result on the analogue of Verlinde formula for the trace of this automorphism on conformal blocks. Motivated by this formula and Jantzen's twining formula, I will also give a conjecture…

Find out more »## October 2016

### Jane Hawkins (UNC-CH), Ergodic Theory and Dynamical Systems Seminar

Title: What is Ergodic Theory? Abstract:This will be an hour chalk talk where the basic ideas, theorems, and examples in ergodic theory are introduced with their physical roots and some current applications.

Find out more »### George Lusztig (MIT), Geometric Methods in Representation Theory

Title:Z/m graded Lie algebras and intersection cohomology Abstract: Let g be a semisimple Lie algebra with a given grading (g_i) where i runs over Z/m, a finite cyclic group. The variety of nilpotent elements in g_i (for fixed i) decomposes in finitely many orbits under the action of a certain algebraic group. The aim of the talk is the study of the intersection cohomology of the closure of any one of these orbits with coefficients in certain local systems. This…

Find out more »## November 2016

### Karl Petersen (UNC-CH), Ergodic Theory and Dynamical Systems Seminar

Title: A well-known but still fascinating example in ergodic theory: The Gauss Map Abstract: Defining Tx to be the fractional part of 1/x for x in the unit interval produces a map that is isomorphic to the shift map on continued fraction expansions. We review some interesting (long known) properties of this map. We will see that this map preserves a measure equivalent to Lebesgue measure, called the Gauss measure. The map is ergodic with respect to this measure,…

Find out more »### Michael Schuster (University of Georgia), Geometric Methods in Representation Theory

Title: Sub-cones of the additive eigencone Abstract: The additive eigencone is defined as the set of all solutions of the additive eigenvalue problem in linear algebra. Its interest lies in its close relationships with the representation theory of algebraic groups and cohomology of flag varieties. In this talk I will discuss special sub-cones of eigencones - called sub-eigencones - which satisfy a strong functoriality property. In particular, I will discuss a new family of examples of sub-eigencones arising from…

Find out more »### Chris Johnson (Wake Forest University), Ergodic Theory and Dynamical Systems Semianr

Title: From billiards in the T-fractal to non-singular transformations Abstract: The motion of an ideal point-mass in a self-similar polygonal region in the plane, called the T-fractal, naturally leads to the study of an infinite interval exchange. The self-similarity of this infinite interval exchange yields a finite affine interval exchange. Though such maps do not preserve Lebesgue measure, they are non-singular (i.e., preserve null sets) and many ergodic theoretic questions can still be asked about these maps. In this talk I will describe a…

Find out more »## December 2016

### Drew Ash (Davidson), Ergodic Theory and Dynamical Systems Semianr

Title: Odometers, Substitutions, and Bounded Topological Speedups Abstract: In dynamics, when we study a transformation of a space T:X->X, we often take the point of view that the action of T represents the passage of a unit of time, and thus by studying (X,T) we are studying how the space X evolves over time. In keeping with this view point, we can define a "speedup" of T - a transformation S:X->X which is obtained by applying a positive power p(x) of T to each point x.…

Find out more »## March 2017

### Olivier Debarre, Geometric Methods in Representation Theory

Title: Unexpected Isomorphisms between Hyperkahler Fourfolds Abstract: In 1985, Beauville and Donagi showed by an explicit geometric construction that the variety of lines contained in a Pfaffian cubic hypersurface in $P^5$ is isomorphic to a canonical desingularization of the symmetric self-product of a K3 surface (called its Hilbert square). Both of these projective fourfolds are hyperkähler (or symplectic): they carry a symplectic 2-form. In 1998, Hassett showed by a deformation argument that this phenomenon occurs for countably many families of…

Find out more »### Nancy Rodriguez (UNC-CH), Ergodic Theory & Dynamical Systems Seminar

Title: Periodic Cycles of Social Activity Abstract: In this talk I will begin by introducing a 2x2 dynamical system with a time-periodic source term, which can be seen as a toy model for the dynamics social outbursts. It consists of an explicit field measuring the level of activity and an implicit field measuring the effective tension. The system can be used to represent a general type of phenomena in which one variable exhibits self-excitement once the other variable has reached a…

Find out more »### AWM–Math Funding Workshop for Graduate Students

Alyssa Spoonts from the UNC Graduate Funding Information Center (GFIC) will lead this workshop designed to help grad students in Mathematics find and apply for funding. Topics that will be covered include: using funding databases to find funding opportunities, dos and don'ts of successful applications, and understanding the funding process. Also, she will briefly talk about finding postdoc opportunities, and finding funding for international students.

Find out more »## April 2017

### Curtis Porter (NCSU), Geometric Methods in Representation Theory

Title: Straightening out degeneracy in CR Geometry: When can it be done? Abstract: CR geometry studies boundaries of domains in C^n and their generalizations. A central role is played by the Levi form L of a CR manifold M, which measures the failure of the CR bundle to be integrable, so that when L has a nontrivial kernel of constant rank, M is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold N, then we…

Find out more »### Lev Rozansky (UNC-CH), Phyiscally Inspired Mathematics Seminar

Title: Flag varieties, Hilbert schemes on C^2 and link homology Abstract: This is a joint work with A. Oblomkov. We construct a categorical action of an ordinary and affine braid group on matrix factorizations over varieties constructed out of Hilbert schemes and flag varieties. The `trace' of the action of an ordinary braid yields the HOMFLY-PT homology of the link presented by its closure. Our construction should clarify some properties of this homology which look mysterious within the Soergel bimodule based…

Find out more »### William Minicozzi (MIT), PDE Mini-School

Title Lecture 3: Level set method for motion by mean curvature Abstract: Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to a degenerate elliptic nonlinear pde. A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid…

Find out more »### Reuven Hodges, Geometric Methods in Representation Theory

Title: Levi subgroup actions on Schubert varieties in the Grassmannian Abstract: Let L be the Levi part of the stabilizer in GL_N(C) (for left multiplication) of a Schubert variety X(w) in the Grassmannian. For the induced action of L on C, the homogeneous coordinate ring of X(w) (for the Plucker embedding), I will give a combinatorial description of the decomposition of C into irreducible L-modules. Using this combinatorial description, I give a classification of all Schubert varieties X(w) in the…

Find out more »## September 2021

### Calvin McPhail-Snyder, Duke – Geometric Methods in Representation Theory Seminar

Title: Quantum invariants from unrestricted quantum groups Abstract: Quantum groups are a central part of the construction of quantum invariants of knots, links, and 3-manifolds. Existing work focuses mainly on the case where the quantization parameter q is generic, or on the semisimplified theory at q a root of unity. In this talk, I will discuss how to construct invariants of knots (and links) using the non-semisimple part of unrestricted quantum sl_2 at a root of unity. These “holonomy invariants”…

Find out more »## October 2021

### Calvin McPhail-Snyder, Duke – Geometric Methods in Representation Theory Seminar

Title: Quantum invariants from unrestricted quantum groups Abstract: Quantum groups are a central part of the construction of quantum invariants of knots, links, and 3-manifolds. Existing work focuses mainly on the case where the quantization parameter q is generic, or on the semisimplified theory at q a root of unity. In this talk, I will discuss how to construct invariants of knots (and links) using the non-semisimple part of unrestricted quantum sl_2 at a root of unity. These “holonomy invariants” turn out…

Find out more »### Olivia Dumitrescu, UNC-Chapel Hill – Geometric Methods in Representation Theory Seminar

In-Person Phillips Hall, Room 385

Find out more »## November 2021

### David Nadler, UC Berekley – Geometric Methods in Representation Theory Seminar

Location: Phillips Hall, Room 385 Time: 2:30 – 3:30 pm Speaker: David Nadler, UC Berkeley Title: Betti Geometric Langlands Abstract: I'll introduce Betti Geometric Langlands through some key objects, conjectures and results. Then I'll discuss recent and ongoing work with Zhiwei Yun devoted to constructing some of its expected topological field theory structures.

Find out more »### Xuqiang Qin, UNC-Chapel Hill – Geometric Methods in Representation Theory Seminar

Mode: In-Person Location: Phillips Hall 385 Title: Minimal instantons on Fano threefolds and compactifications of their moduli spaces Abstract: Instanton bundles were first introduced on P^3 as stable rank 2 bundles E with c1(E)=0 and H^1(E(-2))=0. Torsion free generalizations and properties of moduli spaces of instanton bundles have been widely studied. Faenzi and Kuznetsov generalized the notion of instanton bundles to other Fano threefolds. In this talk, we look at semistable sheaves of rank 2 with Chern classes c1 =…

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