Spring 2024
Time & Location: All talks are on TBA in Gibson Hall 126A at 3:00 P.M. unless otherwise noted.
Organizer: Mahir Can
April 5
Title: Poisson geometry of cluster algebras and their quantization
Bach Nguyen
Abstract: The relationship between Poisson geometry and cluster algebra was first studied by M. Gekhtman, M. Shapiro, and A. Vainshtein. Following their work, we study the global geometry picture of the affine Poisson varieties associated to a cluster algebra and its quantization, root of unity quantum cluster algebra. In particular, we prove that the spectrum of the upper cluster algebra, endowed with the GSV Poisson structure, has a Zariski open orbit of symplectic leaves and give an explicit description of it. Our result provides a generalization of the Richardson divisor of Schubert cells in flag varieties. Further, we describe the fully Azumaya loci of the root of unity upper quantum cluster algebras, using the theory of Poisson orders. This classifies their irreducible representations of maximal dimension. This is a joint work with Greg Muller, Kurt Trampel and Milen Yakimov.
Location: Gibson Hall 126A
Time: 3:00
April 12
Title: Maximal Linear Sections of Grassmann and Schubert Varieties and Linear Error Correcting Codes
Sudhir Ghorpade - IIT Bombay
Abstract: Consider the Grassmann variety with its canonical Plucker embedding, or more generally a Schubert variety in a Grassmannian with its nondegenerate embedding in a subspace of the Plucker projective space. We can cut it by linear subspaces of a fixed dimension of the ambient projective space, and ask which of the linear sections are ”maximal”. The term ”maximal” can be interpreted in several ways and we will be particularly interested in maximality with respect to the number of rational points over a given finite field. In general, this is an open problem. This problem is also closely related to questions in the study of linear error correcting codes. We will quickly outline the relevant background, explain the connection with coding theory, and then describe some of the known results and problems.
Location: Gibson Hall 126A
Time: 3:00
