


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 UrbanaChampaign, 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 controltype) 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 KalmanBucy, 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 KalmanBucy 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/ekzuhivgrs
Host: J. K. Verma
Speaker: Lisa Seccia
Affiliation: University of Genoa, Genoa, Italy
Title: Weaklyclosed graphs and Fpurity 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 weaklyclosed graphs (or cocomparability graphs). Building on known results about Knutson ideals of generic matrices, we characterize weaklyclosed graphs as the only graphs whose binomial edge ideals are Knutson ideals (associated with a certain polynomial f). In doing so, we reprove Matsuda's theorem about the Fpurity of binomial edge ideals of weaklyclosed graphs in prime characteristic and we extend it to generalized binomial edge ideals.
Lastly, we will discuss some open conjectures on the Fpurity of binomial edge ideals and on the relation between Knutson ideals and compatible ideals.