In Fall 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 Daren Chen (dac726 AT lehigh DOT edu) and Ao Sun (aos223 AT lehigh DOT edu).
Date: 9/2/2026
Speaker: Mark Skandera (Lehigh)
Title: Picturing a change-of-basis matrix
Abstract: Much work in algebraic combinatorics involves the comparison of two different bases of a vector space, and pictorial formulas for entries of the change-of-basis matrix that relates the two bases. Such formulas tell us how to compute a matrix entry by drawing certain pictures and playing a game with them. In one vector space known as the Hecke algebra, a certain basis defined by Kazhdan and Lusztig and another natural basis are related by a change-of-basis matrix having no known elementary formula. We will look at pictures and games which describe some entries of this matrix, and at the open problem of describing the remaining entries.
This is joint work with Tommy Parisi, Ben Spahiu, and Jiayuan Wang.
The talk will be accessible to students who have taken linear algebra.
Date: 9/9/2026
Speaker: Joe Kramer-Miller (Lehigh)
Title: The complexity of power series in positive characteristic
Abstract: An amazing theorem of Christol from the 1980s states that a power series over a finite field is algebraic if and only if its coefficient sequence is p-automatic. Informally, this means that the coefficients of an algebraic series can be computed by a "finite machine". This theorem has since become a cornerstone in the study of transcendence in positive characteristic. In this talk we prove analogues of Christol's theorem for series that are solutions to iterative differential equations with moderate singularities.
Date: 9/16/2026
Speaker: Luya Wang (Princeton/IAS)
Title: Interactions between smooth and symplectic topology
Abstract: Symplectic structures lie between smooth topology and complex geometry and exhibit both flexible and rigid phenomena. However, we still do not have a good understanding of what a general symplectic manifold should look like. In this talk, I will begin by surveying some classical results of Gromov and McDuff from the 1980s, which use pseudoholomorphic curves to classify symplectic manifolds containing certain distinguished submanifolds. I will then discuss a question of Donaldson in the 1990s, which relates smooth structures in dimension four to symplectic structures in dimension six. Finally, I will describe some recent refinements of the Gromov-McDuff techniques and their applications to classification problems in the compact setting.
Date: 9/23/2026
Speaker: Lei Wu (Lehigh)
Title: Diffusive Limit of Transport Equations in Bounded Domains
Abstract: In this talk, I will discuss the transition from microscopic particle models to macroscopic continuum equations, a subject at the intersection of PDE and asymptotic analysis. I will focus in particular on the diffusive limit of the neutron transport equation, which serves as a prototype for more general hydrodynamic limits of kinetic equations. It also provides a setting that captures nearly all of the fundamental difficulties associated with boundary effects. In this survey-style talk, I will review these difficulties and discuss various approaches to addressing them. I will present our work over the past decade on three types of domains: smooth convex domains, smooth non-convex domains, and, most recently, polygonal domains. After a brief introduction to the Hilbert expansion, I will focus on the formulation and well-posedness of boundary layer and wedge layer problems.
Date: 9/30/2026
Speaker: Lisa Traynor (Bryn Mawr)
Title: Geometry with a Twist: The Restricted World of Legendrian Links
Abstract: Contact manifolds are smooth manifolds that have an extra geometric structure — with roots in physics — given by a “twisting” hyperplane field. Legendrian submanifolds are the maximal dimensional integral submanifolds of the hyperplane field. For example, a contact structure on Euclidean 3-space is given by a field of 2-dimensional planes, and its Legendrian submanifolds are smooth knots and links that have restricted shapes and movements. A motivating question is to understand when two Legendrian submanifolds, which are equivalent as smooth submanifolds, are not equivalent as Legendrian submanifolds. Through a series of examples, I will introduce Legendrian submanifolds and Legendrian invariants of increasingly sophisticated depths.
Date: 10/07/2026
Speaker: Gaoyong Zhang (NYU)
Title: Reconstruction of Convex Geometric Shapes and Monge-Ampère Equations
Abstract: Can a convex geometric shape be fully reconstructed from its local measurements? This fundamental question drives a number of geometric problems known as Minkowski-type problems. For example, a convex polytope can be recovered from its facet areas and unit normals, and a smooth closed convex hypersurface can be reconstructed from its Gauss curvature.
Solving these shape recovering problems relies on both analytical and geometrical tools. In the smooth setting, they translate into Monge-Ampère-type partial differential equations. When dealing with general, non-smooth shapes, they lead to singular Monge-Ampère and convex geometric measure equations.
This talk will introduce several of the Minkowski-type problems through accessible examples. We will talk about their analytical equations, recent developments, and open problems in the field.
Date: 10/14/2026
Speaker: Marco Zaninelli (U Penn)
Title: Sums of squares in hyperelliptic function fields
Abstract: Building on work by Becher and Van Geel, we study sums of squares in function fields of hyperelliptic curves, mostly over hereditarily pythagorean fields. A refinement of their techniques allows us to obtain upper bound 2 for the Pythagoras number of certain such fields. As a consequence, we give a negative answer to Becher and Van Geel’s question whether Pythagoras number 2 for the function field of the elliptic curve X(X^2+1) implies that the base field K is infinity-hereditarily pythagorean or -1 belongs to K^2.
Date: 10/21/2026
Speaker: Aaron Naber (IAS)
Title: Nonlinear Harmonic Maps and the Energy Identity
Abstract: We begin this talk with an introduction to linear and nonlinear harmonic maps between Riemannian manifolds, with a first goal of understanding some basic examples and uses in geometry and analysis. Unlike linear harmonic maps, which are always smooth and well behaved, nonlinear harmonic maps may exhibit singularities of various sorts. One type of singularity which appears is in the form of blow up not for a single solution, but for a sequences of solutions. We will give examples and discuss how they relate to the general theory, attempting to end at an explanation of the energy identity of nonlinear harmonic maps. The talk should have a minimal background requirement, with most discussions being in the form of examples. Work discussed is joint with Daniele Valtorta.
Date: 10/28/2026
Speaker: Hongkai Zhao (Duke)
Title: Iterative Truncated SVD for Inverse Scattering Problems
Abstract: We propose an iterative truncated SVD (iTSVD) algorithm for Newton's method to solve inverse scattering problems. The key idea is to estimate the effective rank of the corresponding linearized operator (or the Jacobian matrix after discretization), which defines the dimension of the signal/information space of the associated forward operator. Once the effective rank is estimated, TSVD reconstruction in the signal space can be computed efficiently based on fast random SVD (RSVD). A key point is that random SVD only needs the application of the matrix to a set of random vectors whose cardinality is comparable to the effective rank. Moreover, these matrix vector multiplications have a matrix free formulation which only requires solving the same forward problem with different forcing terms.
Our analysis and numerical experiments show that iTSVD can markedly improve computational efficiency by reducing the number of iterations, and enhance stability by effective rank estimate. Moreover, the effective rank estimate is naturally model- and data-based. Extensive experiments, including far- or near-field, limited aperture, and random measurement data without or with noise, are presented.
Date: 11/4/2026
Pitcher's lecture, see https://math.cas.lehigh.edu/events/2026-everett-pitcher-lecture-series-dr-gerhard-huisken for more information.
Speaker: Gerhard Huisken (University of Tübingen/Oberwolfach)
Title: Geometric Evolution Equations
Abstract: The lecture describes the deformation of hypersurfaces in normal direction with a speed depending on the curvature of the surface at the given point and and time. This leads to non-linear systems of partial differential equations of parabolic type that have a smoothing effect similar to the heat equation for a short time while allowing the development of singularities for larger times that in some cases can be overcome by surgery methods. We describe the geometric structure of solutions and the analytical techniques needed for their study.
Date: 11/9/2026 (Note the special time)
3:15-4:15pm in CU 218
Speaker: Paulo Lima-Filho (Texas A&M University)
Title: TBD
Abstract: TBD
Date: 11/25/2026
Thanksgiving