Talk
Speaker: Dr. Aniruddha Seal, Department of Mathematics, IISc, Bangalore
Host: Neela Nataraj
Title: Numerical Analysis of Finite Element Methods for Time-Fractional Problems
Time, day and date: 3:15:00 PM – 4:15:00 PM, Wednesday, January 28
Venue: Online (https://us06web.zoom.us/j/6950684207?pwd=VkNmYzFBd0c0QWx3azhJODJ0QUp1Zz09&omn=81584666321 Meeting ID: 695 068 4207 Passcode: 802747)
Abstract: Fractional differential equations arise naturally in models of anomalous diffusion, phase separation, and nonlocal transport phenomena. Their inherent nonlocality poses significant analytical and numerical challenges, particularly in the construction of stable and accurate discretization schemes. Time-fractional models, in particular, are widely used to describe processes with memory effects and nonlocal temporal behavior, and require careful numerical treatment supported by rigorous mathematical analysis. 
In this talk, we present finite element formulations based on C0 interior penalty and weak Galerkin frameworks for time-fractional problems. We discuss the ideas used to establish stability and convergence of the presented schemes, and incorporate numerical experiments that support the theoretical results. The talk highlights how careful mathematical analysis and appropriate discretization strategies lead to reliable and accurate simulation of nonlocal time-dependent problems.

Real Algebraic Geometry and Combinatorial Optimization Seminar
Speaker: Luca Wellmeier, UiT - The Arctic University of Norway
Host: Sudhir R. Ghorpade
Title: Hierarchies in Polynomial Optimization
Time, day and date: 5:15:00 PM - 6:15:00 PM, Wednesday, January 28
Venue: Ramanujan Hall
Abstract: Polynomial optimization provides a unifying framework for problems arising in combinatorics, quantum information, power systems, and robotics, among many more - thereof also many well-known NP-hard problems. This motivates the study of systematic relaxations rather than exact solution methods. The goal of this talk is to give a brief introduction to the Moment-SOS hierarchy, a sequence of semidefinite relaxations rooted in real algebraic geometry. From one viewpoint, polynomial nonnegativity is approximated by membership in the sum-of-squares cone, connecting the theory to classical work on Hilbert's 14th problem and Positivstellensätze. From the dual viewpoint, the same hierarchy arises from truncated moment problems and the reconstruction of measures. This duality yields a theoretically rich class of tractable algorithms for computing tight bounds on polynomial optimization problems, which have been proven useful in many domains.