Seminar time: Thursdays 1:50-2:45pm.
Location: Dulles Hall DU12.
February 6: Yewen Sun (The Ohio State University)
Title: A symmetric multivariate Elekes-Rónyai theorem
Abstract: The Elekes-Rónyai theorem is a well-known result in additive combinatorics. Their work has been significantly generalized and improved by many mathematicians, in part due to its connections to other areas such as geometry and model theory. In 2020, Ray and Tov extended the Elekes-Rónyai theorem to higher dimensions. Jing, Roy, and Tran later proved a symmetric version of the theorem in 2022. In this talk, we will discuss our recent work generalizing the symmetric Elekes-Rónyai theorem to higher dimensions. We also proved an Erdős-Szemerédi-type theorem for two polynomials in higher dimensions, generalizing a result by Jing, Roy, and Tran. The key ingredient in our proofs is a variation of a theorem by Elekes, Nathanson, and Ruzsa.
February 13: Jimmy He (The Ohio State University)
Title: Symmetries of periodic and free boundary measures on partitions
Abstract: The periodic and free boundary q-Whittaker measures are probability measures on partitions defined in terms of q-Whittaker functions and an additional parameter $u$ controlling behavior of the system at the boundary. I will explain a hidden distributional symmetry of this model which exchanges the $u$ and $q$ parameters, as well as related results on Hall-Littlewood measures. As a special case, we recover identities of Imamura–Mucciconi–Sasamoto. This is joint work with Michael Wheeler.
March 13: Spring Break, No Seminar
March 27: Bogdan Ion (University of Pittsburgh)
Title: Motivic Chern classes of Schubert varieties in the affine Grassmannian
Abstract: I will report on ongoing joint work with Leonardo Mihalcea and Changjian Su that aims to describe the equivariant Euler characteristic of the motivic Chern classes associated to Schubert varieties in the affine Grassmannian. In type A these give a distinguished basis for the ring of symmetric Laurent polynomials with coefficients rational functions in two equivariant parameters. Certain limits with respect to these parameters are identified with affine Demazure characters, and with Hall-Littlewood polynomials. The relationship with the Macdonald polynomials (which share the same limiting behavior) is still mysterious.
April 17: Yifan Jing (The Ohio State University)
Title: Measure growth in groups
Abstract: Let G be a locally compact group. A central problem in additive combinatorics is to provide lower bounds on the size of the product set AB, where A and B are subsets of G, and AB denotes the set of pairwise products. This problem is deeply connected to various areas, including harmonic analysis, convex geometry, the geometry of numbers, and random walks on groups. In this talk, we will survey some classical results and highlight recent developments in the field.