Fall 2025

Putnam Postmortem

Speaker: Noam Elkies

Abstract: Join our own Professor Noam Elkies to discuss a selection of this year’s Putnam problems.

On Knot Theory and Protein Folding

Speaker: Zach Buller

Abstract: Knot theory is the topological study of closed loops, and protein folding is the biochemical study of how proteins form their functional shapes. In this talk, I investigate the intersection of these two fields: how do knotted proteins fold, and what can we learn about their folding intermediates? How can we reconcile pure mathematical formalism with messy computational data? We build algebraic and observational knot invariants from the ground up and I present my senior thesis research on molecular dynamics simulation of a protein predicted by AI to fold into a 7-1 knot.

Partitions, Posets, and Lie algebras

Speaker: Ari Krishna

Abstract: Finite Young's lattices L(m,n) (partitions in an m x n box) have striking properties such as rank-symmetry, rank-unimodality, and the strong Sperner property. Stanley conjectured that these admit a symmetric chain order. After a general discussion of posets and their properties of interest, I present a geometric model that identifies L(m,n) with a natural ordering on the integer points of a dilated simplex that arises as a weight diagram of type A_n in representation theory. With this, we also exhibit a visual reconstruction of Lindström’s symmetric chain decomposition for L(3,n). This work is joint with Robert W. Donley, Ammara Gondal, and Terrance Coggins.

Preventing Period-Doubling in Discrete-Time Systems

Speaker: Grayson Kemplin

Abstract: We embark on a research problem in bifurcation theory, a subfield in dynamical systems studying fundamental changes to the behavior of a system on varying a parameter. In many instances of physical systems, the onset of period-doubling bifurcation presents an issue [including a structural instability of a building or health incident]. We examine how we can measure the robustness of a feedback control mechanism on such problems using a variety of theoretical and applied tools. No background in dynamical systems is supposed.

Machine Learning for Knot Theory

Speaker: Luke Zhu

Abstract: Traditionally, machine learning — due to its ability to detect structure in large datasets — has largely been a tool for applied mathematics.  Recent work has repurposed these ML tools to reveal new relationships in pure mathematics, aiding human intuition and even guiding conjecture generation. We discuss how knot theory presents itself as a playground for these ML methods, and experiment with invariant prediction, classification tasks, and even applications to biopolymer identification.

Nontrivial Holomorphic Structures on the Quantum Sphere

Speaker: Lillian MacArthur

Abstract: The area of Noncommutative Geometry is quite new, and even newer is the concept of a complex structure in Noncommutative Geometry. In previous literature, various complex structures have been defined within the noncommutative framework, with various results found, including the Quantum Sphere. However, all previous investigations into the structure of the Quantum Sphere have always found its algebraic and analytic properties to be not too dissimilar from the classical Riemann Sphere. Here, we demonstrate that in regard to holomorphic structures that can be defined on the space, the Quantum Sphere is indeed very different from the Riemann Sphere.

Jobs Panel

Interested in learning about careers you could pursue which would use the quantitative skills you've developed studying mathematics? Come to our jobs panel to ask questions and hear from Harvard alums about their chosen career paths.