Undergraduate

Spring 2025: Jordan Ellenberg (University of Wisconsin Madison), Machine Learning as a Tool for Pure Mathematics, video recording.

• Time and place: February 28, 2025, 12-1pm, Behrakis 310.

Abstract: This talk will cover Funsearch and PatternBoost, two machine learning protocols designed to support research in pure mathematics. It will be a fairly non-technical discussion on both the math and CS sides. The focus will be on which types of math problems are currently most accessible to machine learning methods, and on some of the obstacles that make these problems different from "standard" machine learning problems. A key challenge discussed will be the difficulty in defining and measuring "success" in this context.

References: 

• Mathematical discoveries from program search with large language models, Bernardino Romera-Paredes, Mohammadamin Barekatain, Alexander Novikov, Matej Balog, M. Pawan Kumar, Emilien Dupont, Francisco J. R. Ruiz, Jordan S. Ellenberg, Pengming Wang, Omar Fawzi, Pushmeet Kohli, Alhussein Fawzi, Nature volume 625, pages 468–475 (2024)

•  PatternBoost: Constructions in Mathematics with a Little Help from AI, François Charton, Jordan S. Ellenberg, Adam Zsolt Wagner, Geordie Williamson, arXiv:2411.00566.

•  Int2Int: a framework for mathematics with transformers, François Charton, arXiv:2502.17513. 

Github repositories:

• The Github repo you can use to try funsearch yourself: https://github.com/kitft/funsearch.

• The Github repo you can use to try PatternBoost yourself: https://github.com/zawagner22/transformers_math_experiments.

• The Github repo you can use to try Int2Int yourself (not mentioned in the talk, but in the same vein): https://github.com/f-charton/Int2Int.

Spring 2018: Aaron Lauda (University of Southern California). Poster.

• Time and rooms:

Wednesday, February 21, 2018, 4-5:30pm. Room: Behrakis Health Sciences Center -- BK 010 (BK is building 26 on this map). Lecture 1.

Thursday, February 22, 2018, .4:30-5:30pm. Room: Lake Hall -- LA 509 (LA is building 34 on this map). Colloquium.

Friday, February 23, 2018, 4-5:30pm. Room: Behrakis Health Sciences Center -- BK 010. Lecture 2.

• Abstract: 

Lecture 1:  Diagrammatic algebra

In this talk we will introduce a calculus of planar diagrams that can be used to represent algebraic structures in a wide variety of contexts.  We will start by introducing a diagrammatic framework for studying linear algebra.  In this framework, familiar notions such as trace and dimension take on a diagrammatic meaning.  We will see how the notion of duality transforms algebraic notions into intuitive manipulations of diagrams.  Finally, we will see how this diagrammatic reformulation of linear algebra can be used to study invariants of tangled pieces of string (knot theory).

Lecture 2: Higher categories and two-dimensional topology 

In the second talk we will introduce the world of higher categories.  We define the notion of a 2-category and explore some examples that lurk in the background of several advanced undergraduate courses.  Our perspective is that 2-categories are really just a framework for studying an enhanced version of the diagrammatic calculus from the previous lecture.  We will see how the notion of duality generalizes to the notion of adjunction.  By interpreting the resulting diagrams as actual surfaces, we uncover a deep connection between adjunctions (or adjoints) and surfaces.  In the process, we will rediscover that certain 2-dimensional surfaces can be completely described using an algebraic structure called Frobenius algebras.  Mathematicians usually call this process of turning topology into algebra a "Topological Quantum Field Theory".  Our investigations of adjoints in 2-categories will illuminate a simple example of a 2-dimensional (planar) TQFT. 

Colloquium: A new look at quantum knot invariants

The Reshetikhin-Turaev construction associates knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison's novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (skew Howe) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the existence of an (a,q)-super polynomial conjectured by physicists (joint with Garoufalidis and Le), and leads to a new elementary approach to link homology theories categorifying RT-invariants (joint with Queffelec and Rose).