Mathematics Colloquium

Date and time: Friday, 13 January, 2023, 4 pm
Venue: Ramanujan Hall
Host: Mayukh Mukherjee

Speaker: Prashant G. Mehta
Affiliation:  University of Illinois at Urbana-Champaign, USA

Title:  A variational formulation of nonlinear filtering
Abstract. There is a certain magic involved in recasting the equations in Physics in variational
terms. The most classical of these ‘magics’ is the Lagrange’s formulation of the Newtonian mechanics. An accessible take on all this and more appears in the February 19, 2019 issue of The New Yorker magazine. My talk is concerned with a variational (optimal control-type) formulation of the problem of nonlinear filtering/estimation. Such formulations are broadly referred to as duality between optimal estimation and optimal control. The first duality principle appears in the original (1961) paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a linear quadratic optimal control problem.

In my talk, I will describe a newly discovered generalization of the Kalman-Bucy duality to nonlinear filtering. As an application, I will discuss some comparisons between the stochastic stability of a Markov process and the (filter) stability of a conditioned process. Either of these is shown to arise from assuming a respective Poincaré inequality (PI). This is joint work with Jin Won Kim. The talk is based on the following papers:

https://arxiv.org/abs/2208.06586 and https://arxiv.org/abs/2208.06587
 

Virtual Commutative algebra seminar

Date and Time: Friday, 13 January 2023, 5:30 pm 
Gmeet link: https://meet.google.com/ekz-uhiv-grs
Host: J. K. Verma

Speaker: Lisa Seccia 
Affiliation: University of Genoa, Genoa, Italy

Title: Weakly-closed graphs and F-purity of binomial edge ideals
Abstract: Herzog et al. characterized closed graphs as the graphs whose binomial edge ideals have a quadratic Groebner basis. In this talk, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs). Building on known results about Knutson ideals of generic matrices, we characterize weakly-closed graphs as the only graphs whose binomial edge ideals are Knutson ideals (associated with a certain polynomial f). In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly-closed graphs in prime characteristic and we extend it to generalized binomial edge ideals. 
Lastly, we will discuss some open conjectures on the F-purity of binomial edge ideals and on the relation between Knutson ideals and compatible ideals.