In Spring 2026, 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) and Ao Sun (aos223 AT lehigh DOT edu).
Date: 2/4/2026
Speaker: Lingfu Zhang (Caltech)
Title: 2D random geometry from an i.i.d. field
Abstract: On Z^2, assign an i.i.d. nonnegative random weight to each edge. The distance between two points is defined as the minimum total weight over all paths connecting them, which induces a random metric on Z^2. A central problem in this area is the Strong KPZ Universality Conjecture, which predicts that this random metric converges to a universal random directed metric on R^2. I will discuss recent developments that reveal rich and surprising geometric properties of such random metrics (on Z^2 or R^2). These 2D geometries are also closely connected to a broad class of probabilistic models, from random growth to interacting particles. I will explain how geometric viewpoints can provide new insights into resolving open problems in these related models. No prior knowledge in this topic is assumed.
Date: 2/11/2026
Speaker: Davi Maximo (UPenn)
Title: Scalar Curvature, Minimal Surfaces, and the Shape of Manifolds
Abstract: Scalar curvature is a fundamental invariant in Riemannian geometry, capturing how volumes and distances deviate from those of flat space at infinitesimal scales. While comparison geometry has long used curvature bounds to derive strong geometric and topological consequences, scalar curvature has historically been more subtle and resistant to such techniques.
In this talk, I will give an introductory overview of how minimal surface methods can be used to extract global geometric and topological information from scalar curvature assumptions. After reviewing the basic ideas behind scalar curvature and minimal surfaces, I will explain how their interaction leads to restrictions on the topology and geometry of manifolds with positive scalar curvature.
The talk will emphasize ideas, examples, and geometric intuition rather than technical details, and is intended to offer non-experts a glimpse into the subject.
Date: 3/4/2026
Speaker: Hao Jia (University of Minnesota, Twin Cities)
Title: Recent progress in hydrodynamic stability at high Reynolds numbers
Abstract: Hydrodynamic stability of incompressible fluid flows is a classical topic, studied already by Rayleigh, Kelvin, Orr, Sommerfeld, Heisenberg, among many others. The focus then was on the possible existence of unstable eigenvalues for the linearized equation (the so-called Orr-Sommerfeld equations) around physically relevant flows such as laminar flows, in an attempt to explain the onset of turbulence. In recent years, tremendous progress has been made in the understanding of the asymptotic dynamics--not just spectrum--in the perturbative regime of shear flows and vortices, especially in the two dimensional case, for both the inviscid and slightly viscous fluid. It turns out that the question of stability in incompressible fluid flows is much more subtle than previously imagined. In this talk, we will introduce some recent results. In particular we will emphasize three fundamental stabilizing mechanisms in incompressible fluid flows: inviscid damping, enhanced dissipation and vorticity depletion, as well as the destabilizing "Orr mechanism", in the context of a recent proof on the sharp transition threshold for Kolmogorov flows on a non-square torus.
Date: 3/18/2026
Speaker: Shmuel Weinberger (University of Chicago)
Title: Manifolds of Postive Scalar Curvature
Abstract: The study of manifolds of positive scalar curvature has a long history and deep connections to general relativity and geometric topology. For several decades, through the work of Gromov and Lawson and Stolz, we know which simply connected closed manifolds have such metrics.
I will try to explain parts of this beautiful story, and then explain work with Manin, Xie and Yu that give information about how hard it is to find such metrics. Despite our focus on simply connected manifolds, geometric group theory, logic, C*-algebras and homotopy theory all play a role (although most of these will probably need to remain behind the curtains).
Date: 3/18/2026
Speaker: Shmuel Weinberger (University of Chicago)
Title: Manifolds of Postive Scalar Curvature
Abstract: The study of manifolds of positive scalar curvature has a long history and deep connections to general relativity and geometric topology. For several decades, through the work of Gromov and Lawson and Stolz, we know which simply connected closed manifolds have such metrics.
I will try to explain parts of this beautiful story, and then explain work with Manin, Xie and Yu that give information about how hard it is to find such metrics. Despite our focus on simply connected manifolds, geometric group theory, logic, C*-algebras and homotopy theory all play a role (although most of these will probably need to remain behind the curtains).
Date: 3/25/2026
Speaker: Michael Dougherty (Lafayette College)
Title: Noncrossing partitions, braid groups, and complex polynomials
Abstract: The n-strand braid group is one of the most important objects in geometric group theory, in part because it lies in the overlap of many different classes of groups, including Artin groups and mapping class groups. Topologically, this group can be viewed as the fundamental group for the space of monic complex polynomials with distinct roots, and algebraically it is often defined using a generating set of n-1 diagrams depicting half-twists of adjacent strands. Work of Birman-Ko-Lee, T. Brady, and Bessis around the turn of the century introduced a new generating set for these groups which used the lattice of noncrossing partitions, and the combinatorics of this lattice has led to breakthroughs in our understanding of braid groups and Artin groups more generally. In this talk, I will describe some new connections between noncrossing partitions, braid groups, and complex polynomials. This is joint work with Jon McCammond.
Date: 4/1/2026
Speaker: Sir John Ball (Heriot-Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh)
Title: Rank-one connections and microstructure in single crystals and polycrystals
Abstract: The talk will describe how nonlinear elasticity can be used to predict microstructure arising from solid phase transformations in alloys, and survey some recent results related by the common theme of finding rank-one connections between energy wells, these describing possible planar interfaces between different phases. The results concern (i) slip and twinning in Bravais lattices using the Ericksen energy-well picture, (ii) compatibility of martensitic microstructures across polycrystal grain boundaries and the Taylor set (joint with M. Galanopoulou), and (iii) remarkable martensitic microstructures observed in the alloy Ti76Nb22Al2, and their interpretation in terms of TN configurations (joint with T. Inamura & F. Della Porta).
Date: 4/1/2026
Speaker: Sir John Ball (Heriot-Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh)
Title: Rank-one connections and microstructure in single crystals and polycrystals
Abstract: The talk will describe how nonlinear elasticity can be used to predict microstructure arising from solid phase transformations in alloys, and survey some recent results related by the common theme of finding rank-one connections between energy wells, these describing possible planar interfaces between different phases. The results concern (i) slip and twinning in Bravais lattices using the Ericksen energy-well picture, (ii) compatibility of martensitic microstructures across polycrystal grain boundaries and the Taylor set (joint with M. Galanopoulou), and (iii) remarkable martensitic microstructures observed in the alloy Ti76Nb22Al2, and their interpretation in terms of TN configurations (joint with T. Inamura & F. Della Porta).
Date: 4/20/2026
Speaker: Sanat Sarkar (Temple University)
Title: On the False Discovery Rate of the Benjamini–Hochberg Procedure in Two-Sided Gaussian Mean Testing Under Dependence
Abstract: In this talk, we revisit the false discovery rate (FDR) of the Benjamini–Hochberg (BH) procedure for testing Gaussian means against two-sided alternatives in the presence of dependence. While BH is known to control the FDR under independence and certain positive dependence conditions, its behavior in correlated Gaussian settings remains theoretically unresolved, despite strong empirical evidence of its validity.
Date: 4/29/2026
Speaker: Elia Brué (Bocconi University)
Title: Nilpotent Structures under Curvature Bounds
Abstract: The study of the topological constraints imposed by curvature bounds is a classical and far-reaching theme in Riemannian geometry. Since Gromov’s pioneering work on almost flat manifolds, nilpotent structures have played a central role in this area. The goal of this colloquium is to illustrate the deep connections between nilpotency and curvature bounds through a survey of relevant examples, as well as classical and more recent results.
Date: 5/6/2026
Speaker: Yu Deng (UChicago)
Title: Recent progress on mathematical wave turbulence
Abstract: The theory of wave turbulence, which started in the 1920s as the wave analog of Boltzmann’s kinetic theory, has been an active field of physics in the last century, with substantial applications in science. In this talk I will review some recent works, joint with Zaher Hani, that establish the rigorous mathematical foundation of this subject. In particular, we present the justification of the wave kinetic equation up to arbitrarily large kinetic time, which is the first long time result ever obtained in any nonlinear kinetic limit.