Last time we covered abelian theory over S^3 using standard path integral technique. Namely, using the fact that U(1)-theory contains only quadratic kinetic term we computed the two-point function as Green function of the curl operator, and, as we saw, the corresponding Wilson correlators are given by Gauss integral formula for the linking number of pair of knots in S^3 modulo k.

This time we will turn to the non abelian case SU(N) to recover the value of Jones polynomial at the primitive (N+k)-root of unity using axiomatic TQFT approach modulo some finite amount of data predicted by the connection of Chern-Simons to the conformal blocks of the underlying CFT.

This objective in its full generality is related to the question of separating NP and coNP. The ideal proof system defined by Grochow and Pitassi connects propositional proof complexity to algebraic circuit complexity. I will give an introduction to this proof system and then discuss a theorem from a paper with the title IPS Lower Bounds for Formulas and Sum of ROABPs, authored with Prerona Chatterjee, Utsab Ghosal, and Partha Mukhopadhyay.

Unlike college admissions where there is usually a fixed number of seats (called quotas), we consider the case when the quotas are flexible, but have a cost function associated with them.

Formally, the input consists of a bipartite graph between items and platforms. Each platform has a cost function, which is a function of the number of items assigned to the platform. The goal is to find an allocation between items and platforms where at least a given minimum number of items is matched, and subject to this constraint, the cost is minimized. Under the assumption that the cost functions are convex, we give an efficient (approximation) algorithm for this problem. The analysis exploits the connection between matchings and network flows and the properties of the network flow polytope.

This is a joint work with Anand Louis (IISc), Atasi Panda (IISc), and Harsh Sharma (CMI), and appears in AAAI 2026.