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Schedule of talks:

Abstracts

(L. Levine): Aligning AI with human values is a pressing unsolved problem. How can mathematicians contribute to solving it? We can start by clarifying the terms: What are values? What does it mean to "align" an AI to a given set of values? And how would one verify that a given AI is aligned? These hard questions, plus the annoying little issue of whose values to prioritize, led researchers at leading AI labs to aim instead for "intent alignment": AI that (wants to) do what its developers intend. Can intent alignment deliver a good future, where humans thrive alongside AI that's much smarter than us? I'll argue that intent alignment might be extremely hard to achieve, and that it's neither necessary nor sufficient for a good future. Our best shot at a good future is to not build superhuman AI. Not building superhuman AI is a coordination problem: it would require international treaties, monitoring, regulation. Coordination problems are hard (but not as hard as any form of AI alignment!). Coordination failures happen, so it would be wise to have a backup plan. Returning to value alignment as a possible path forward, I'll describe the math behind EigenBench, a pagerank-inspired approach to the problem of measuring AI values.

(V. Gorin): Dunkl differential-difference operators are one-parameter deformations of the usual derivative. 
First studied for their remarkable commutativity and their role in the Calogero-Moser-Sutherland 
quantum many-body system, they have recently found surprising applications in random matrix theory.
I will show how Dunkl operators can be used in the asymptotic analysis of random matrix eigenvalues, 
where Catalan numbers, the semicircle law, the Airy-beta line ensemble, and Tracy-Widom distributions 
naturally emerge.

(T. Lidman): A mathematical knot is simply a closed loop in space, but they show up in a variety of settings, ranging from algebraic geometry to biology.  We will discuss some foundational questions in knot theory, what tools can be used to study knots, and how they relate to various aspects of topology in dimensions 3 and 4. 

(G. Panova): Representation theoretic multiplicities are at the heart of many open problems in algebraic combinatorics. At the same time these quantities appear in Geometric Complexity Theory in the search for multiplicity obstructions for separating computational complexity classes like VP vs VNP. Most recently they have also been considered in quantum computing. In this talk we will introduce the objects and problems, explain how formalization through computational complexity theory could answer some of the open problems in the negative. We will also explain their role in GCT and quantum computing with a mixture of positive and negative answers.

(X. Zhou): The volume spectrum is an analogue of the Laplace spectrum, defined through the area functional together with homological or cohomological relations in the space of hypercycles. In this talk, I will introduce several versions of the volume spectrum and highlight its applications in geometric variational theory. Specifically, I will discuss how the spectrum can be used to address existence problems for minimal surfaces and constant mean curvature surfaces. Finally, I will present recent progress on extending these ideas to a higher codimensional problem.

(J. Avigad): New technologies for reasoning and discovery are bound to have a profound effect on mathematical practice. Proof assistants are already changing the nature of collaboration, communication, and curation of mathematical knowledge. Automated reasoning tools are used to find mathematical objects with specified properties or rule out their existence, and to decide or verify mathematical claims. Machine learning and neural methods can discover patterns in mathematical data, explore complex mathematical spaces, and generate mathematical objects of interest. Neurosymbolic theorem provers, now capable of solving the most challenging competition problems, combine aspects of all of these technologies. It is helpful to keep in mind that the phrase "AI for mathematics" encompasses several distinct technologies that overlap and interact in interesting ways. In this talk, I will survey the landscape, describe a few landmark applications to mathematics, and encourage you to join me in thinking about how mathematicians can shepherd mathematics through this era of technological changes.

(J. Rosenberg): The scalar curvature function is the simplest curvature invariant of a (closed) Riemannian manifold.  In dimension 2, it is just twice the Gaussian curvature, and if it doesn't change sign, it must have the same sign as the Euler characteristic.  But in dimensions 3 and up, there is no obstruction to negative scalar curvature, though there are many obstructions to positive scalar curvature, mostly related to the (generalized) index theory of the Dirac operator on spinors.  We review this theory and discuss a "generalized scalar curvature" on spin^c manifolds, which is connected in a similar way to the (generalized) index theory of the spin^c Dirac operator.  This is joint work with Boris Botvinnik and Paolo Piazza.

(S. Krushkal):

(J. Etnyre):

(L. Lin):

(J. Rivera-Letelier):

(T. Kucherenko):

(D. Barrett):

Past Ohio State University Mathematics Department Colloquia

• 2024-2025

• 2023-2024

• 2022-2023

• 2020-2021

• 2019-2020

• 2018-2019

• 2017-2018

• 2016-2017

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