In 2025 Fall, the Math Department Colloquium Series will be held on Wednesdays 3:30 pm at CU 218. Tea and refreshments available at 3:00 p.m. in the Assmus Conference Room (CU 212).
If you have any questions, please reach out to the organizers Taeho Kim (tak422 AT lehigh DOT edu) (On leave) and Ao Sun (aos223 AT lehigh DOT edu).
Date: 09/17/2025
Speaker: Rob Neel (Lehigh University)
Title: Geometric couplings and sub-Riemannian diffusions
Abstract: We recall the notion of a coupling of two stochastic processes and its application to showing convergence to equilibrium. We then describe the classical applications to Riemannian geometry, and how these natural constructions fail in sub-Riemannian geometry, even for the simplest case of the Heisenberg group. After reviewing the situation, we describe an improvement and extension of constructions by Banerjee-Gordina-Mariano and Bénéfice of non-Markovian reflection couplings on sub-Riemannian model spaces. Moreover, this construction is relatively simple and geometrically appealing, being based on global symmetries of the underlying spaces. This talk is based on joint work with Liangbing Luo.
Date: 10/1/2025 (postponed)
Speaker: Yu Deng (UChicago)
Title: TBD
Abstract: TBD
Date: 10/8/2025
Speaker: Mona Merling (UPenn)
Title: Algebraic K-theory and manifolds
Abstract: In mathematics, we often break complex objects into simpler pieces and then understand how they can be reassembled. Algebraic K-theory provides a higher-homotopical framework for studying objects up to decomposition and reassembly. In this talk, I will give an overview of algebraic K-theory and explain how different variants—such as the classical K-theory of spaces and the more recent scissors congruence K-theory—can be applied to classification problems in the study of manifolds.
Date: 10/15/2025
Speaker: Paul Feehan (Rutgers)
Title: Gauge theory and the Bogomolov-Miyaoka-Yau inequality for symplectic 4-manifolds
Abstract: The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we shall describe our program to prove the conjecture that symplectic 4-manifolds with b^+ > 1 obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles. The method aims to use the fact that there is at least one non-vanishing Seiberg-Witten invariant to produce a solution to the anti-self-dual Yang-Mills equation on a vector bundle with suitable topology over the symplectic 4-manifold. The talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789 (to appear in AMS Memoirs) and https://arxiv.org/abs/2206.14710.
Date: 10/22/2025
Speaker: Francesco Lin (Columbia)
Title: Geometry and topology of vector fields in three-dimensions
Abstract: Understanding the locus at which a vector field on a manifold vanishes (or more in general k-vector fields are linearly dependent) is a fundamental problem in geometry and topology, dating back at least to the famous Hairy Ball theorem. In this talk, I will discuss some of the history of the problem, with a focus on three dimensions, and discuss a new proof using the Dirac equation of the following theorem of Gromov (and in fact a Riemannian refinement of it): on a closed three-manifold equipped with a volume form, there exist three-vector fields which are volume-preserving and are linearly independent at every point.