By Prof. Anil N. Hirani

Abstract 

This talk shows, on screen, two families of results in computational topology from about fifteen years ago, one with Tamal Dey and Bala Krishnamoorthy and the other with Nathan Dunfield. The first problem is finding the shortest cycle in a given homology class on a surface or a solid. It turns out to be solvable by linear programming, for a reason that is pleasant to explain. The second problem is finding the least area surface spanning a knot, the soap film the knot would hold if dipped in soap solution. Here the knot lies in a cube cut into tetrahedra, and the surface must be made of triangles of that cutting; the same linear programming idea, with more work, finds it. AI enters through the recent revival of the original code and the ease with which the old experiments could then be extended, including to knots whose soap film is not the surface of least genus. That ease suggests a potential for new AI-assisted research. A part of the talk is my thoughts on using AI in mathematics and computing, and on how it may change the sociology of mathematics research. The talk is built around live computer demonstrations and assumes no background beyond curiosity.

PhD opportunities

I am recruiting PhD students for my group at the University of Illinois Urbana-Champaign. I work at the meeting of geometry, topology and computation: taking differential forms, vector bundles and the partial differential equations written in them, and finding discrete versions on triangulated spaces that keep their geometric and topological structure. The questions that come up include which algebraic structures survive discretization, how to prove that the discrete equations converge, and, most recently, how geometry can inform machine learning for scientific computing. One funded research assistantship starts in summer 2027, on an NSF project on geometric scientific machine learning for partial differential equations; it can lean theoretical or computational. Students interested in other problems in my area are equally welcome. Come and talk to me after the talk or during my visit, or write to me a few lines on what interests you, with a CV if you have one. My homepage, https://hirani.github.io, has the details and my papers.