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Wednesday February 24
12:00-1:00
Room: Zoom
Cris Negron (UNC)
Title: Quantum groups
Abstract: In this talk I will introduce the (well-established) notion of a tensor category, via a series of basic examples. We will then focus in on quantum groups, which provide the most important source of examples of tensor categories. I will describe two particular problems for quantum groups: one coming from mathematical physics (conformal field theory), and one coming from the boundary between representation theory and homotopy theory. We mean to highlight here the general importance of quantum groups, both in terms of their applications to other fields and in terms of their rich ``internal" structures. I will discuss some recent progress on these problems, which constitutes joint work with Terry Gannon and Julia Pevtsova. This talk should be accessible to mathematicians of varied backgrounds.
Wednesday March 10
12:00-1:00
Room: Zoom
Helen Wong (Claremont McKenna College)
Title: Kauffman bracket skein algebra
Abstract: Although the definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial for knots in S^3, it was later discovered that the skein algebra is closely tied to the hyperbolic geometry of the surface. In particular, the skein algebra is a quantization of the SL_2(C)-character variety of the surface, which contains representations of the fundamental group to SL_2(C). In this talk, we’ll give an introduction to the skein algebra, especially its interesting algebraic properties and its surprising ties to hyperbolic geometry and quantization theory.
Thursday March 25
12:00-1:00
Room: Zoom
Edray Goins (Pomona College)
Title: A Survey of Diophantine Equations
Abstract: There are many beautiful identities involving positive integers. For example, Pythagoras knew 3^2 + 4^2 = 5^2 while Plato knew 3^3 + 4^3 + 5^3 = 6^3. Euler discovered 59^4 + 158^4 = 133^4 + 134^4, and even a famous story involving G.H. Hardy and Srinivasa Ramanujan involves 1^3 + 12^3 = 9^3 + 10^3. But how does one find such identities?
Around the third century, the Greek mathematician Diophantus of Alexandria introduced a systematic study of integer solutions to polynomial equations. In this talk, we'll focus on various types of so-called Diophantine Equations, discussing such topics as Pythagorean Triples, Pell's Equations, Elliptic Curves, and Fermat's Last Theorem.
Wednesday April 7th
12:00-1:00
Room: Zoom
Nicolle Gonzalez (UCLA)
Title: Categorification: a revelation of hidden shadows
Abstract: In recent years there has been a lot of activity revolving around a series of celebrated conjectures that relate Khovanov-Rozansky homology, a homological invariant for knots and links, to Hilbert schemes. In an effort to prove these conjectures, unexpected and deep ties to the category of Soergel bimodules and the combinatorics of Catalan numbers and Macdonald polynomials have been discovered. Central to this story is the idea of diagrammatic categorification, an innovative process by which classical algebraic and topological objects are replaced by a richer categorical structure that is defined in terms of planar diagrams. I will explain what categorification is and how seemingly disparate areas like low dimensional topology, algebraic geometry, representation theory, and algebraic combinatorics come together to present a unified picture. I will then describe how joint work with Matt Hogancamp on a skein theoretic formulation and subsequent categorification of the Dyck path algebra and its polynomial representation can shed light on some of these mysterious conjectured connections.