Mathematics
Seminar
The ubiquitous nature of the Yang-Baxter equation
SPEAKER | Valentin Buciumas
INSTITUTE | Stanford University
DATE | March 23(Wed), 2016
TIME | 13:30
PLACE | 1423
Keyword |
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ABSTRACT | The Yang-Baxter equation (YBE) was used by Baxter to evaluate the partition functions of certain statistical-mechanical models like the well-known six vertex model. The search for solutions of the YBE then led to the discovery of quantum groups by Drinfeld and Jimbo independently. Besides its central role in the theory of quantum groups, the YBE is useful in other areas like topology, combinatorics and number theory.
The main purpose of this talk is to present a diverse range of instances in mathematics where the YBE proves to be useful. I will start by explaining what the YBE is and how it is used in computing partition functions of lattice models. I will then discuss what quantum groups are, how they provide solutions to the YBE, and how one can obtain knot invariants like the Jones polynomial by using their representation theory (the YBE plays a critical role here). I will present a proof due to Kuperberg of the Alternating Sign Matrix theorem that uses the YBE. Finally, I will discuss joint work with Brubaker and Bump on computing Whittaker functions on the n-fold metaplectic cover of GL(r, F) as partition functions of a certain lattice model. The solution to the (parametrized) YBE that allows us to compute the partition function in the last example comes from a quantum affine super group: $U_q(\hat{\mathfrak{gl}}(n|1))$.