In this talk, I will show how tools from VC-dimension theory can be used to address questions from incidence geometry, enumerating intersection graphs, and graph drawing.
Please Note: In order to derive his equation, Boltzmann used the assumption that the probability density function f(t,x1,v1, x2,v2, ..., xN,vN), describing the positions and velocities of a gas of N identical particles at time t, stays close to a product g(t,x1,v1) g(t,x2,v2)...g(t,xN,vN). Here g is the common 1-particle distribution. Mark Kac introduced a probabilistic model for N particles, for which Boltzmann's assumption is valid as N goes to infinity in a specific sense, provided that it is valid at an initial time. Kac's requirement concerning the N particle density functions at some initial time is nowadays known as chaos (or molecular chaos), and Kac's result is known as propagation of chaos. The aim of this talk is to retake the question, first asked and studied in [CCLLV] : For which density functionals g can we produce a family of symmetric densities {fN} supported on the constant energy sphere {v1^2+v2^2+ ... + vN^2 = N} which are chaotic to g? Using rescaled states, we show [CT] that the class of admissible g, obtained in [CCLLV] using other methods, can be expanded. We also mention some new ideas in this direction. This talk is introductory. References: [CCLLV] Carlen, Eric A., et al. "Entropy and chaos in the Kac model." Kinetic and Related Models 3.1 (2010): 85-122. [CT] Cortez, Roberto, and Hagop Tossounian. "Chaos for rescaled measures on Kac’s sphere." Electronic Journal of Probability 28 (2023): 1-29.
The bi-adjacency matrix of an Erdős–Rényi random bipartite graph with bounded aspect ratio is a rectangular random matrix with Bernoulli entries. Depending on the sparsity parameter $p$, its spectral behavior may either resemble that of a classical Wishart matrix or depart from this universal regime. In this talk, we study the extreme singular values at the critical density $np=c\log n$. We present the first quantitative characterization of the emergence of outlier singular values outside the Marčenko–Pastur law and determine their precise locations as functions of the largest and smallest degree vertices in the underlying random graph, which can be seen as an analogue of the Bai–Yin theorem in the sparse setting. These results uncover a clear mechanism by which combinatorial structures in sparse graphs generate spectral outliers. Joint work with Ioana Dumitriu, Haixiao Wang and Zhichao Wang.
Sperner's lemma is a simple combinatorial result that is surprisingly powerful and useful---bringing together ideas in combinatorics, geometry, and topology while attracting interest from economists and game theorists. I'll explain why, show some old and new proofs, and present some recent generalizations with diverse applications.
Dalton, Etnyre, and Traynor classified Legendrian cable links when the companion knot is both uniformly thick and Legendrian simple, and Etnyre, Min, and Chakraborty classified all cable knots of uniformly thick knots. Using convex surfaces, we build on these results to classify cable links of knots in $(S^3, \xi_\text{std})$ that are uniformly thick but not Legendrian simple, and address new questions that arise from their nonsimplicity. This is joint work with Rima Chatterjee, John Etnyre, and Hyunki Min.
Fluid models are used to make predictions about critical real-world systems arising in diverse fields including but not limited to meteorology, climate science, mechanical engineering, and geophysics. Simulations based on fluid models can, for example, be used to make predictions about the strength of a tornado or the stresses on an aircraft wing passing through turbulent air. The possibility that a mathematical model does not capture the full range of possible real-world scenarios is concerning if the predictions do not account for extreme events. It has been confirmed by computer assisted proof that the 3D Navier-Stokes equations possess non-unique solutions. The existence of such solutions can, in principle, pose a challenge to forecasters. This talk explores mathematical work aiming to quantify the rate at which non-unique solutions can separate.
In this talk, I will describe an excision construction for 3-manifolds and explain how (twisted) Heegaard Floer theory can be used to obstruct 3-manifolds from being related via such constructions. I will also discuss how the excision formula can be applied to compute twisted Heegaard Floer homology groups for specific 3-manifolds obtained by performing surgeries on certain links, including some 2-bridge links.
Classical solvers for large-scale scientific and data-driven problems often face limitations when uncertainty, multiscale effects, or ill-conditioning become dominant. In this talk, I will present hybrid algorithmic frameworks that unify ideas from numerical analysis, stochastic computation, and machine learning to address these challenges. In the first part, I will introduce Preconditioned Truncated Single-Sample (PTSS) estimators, a new class of stochastic Krylov methods that integrate preconditioning with truncated Lanczos iterations. PTSS provides unbiased, low-variance estimators for linear system solutions, log-determinants, and their derivatives, enabling scalable algorithms for inference and optimization. In the second part, I will discuss a data-driven approach to constructing approximate inverse preconditioners for partial differential equations (PDEs). By learning the Green’s function of the underlying operator through neural representations, this framework captures multiscale behavior and preserves essential spectral structure. The resulting solvers achieve near-linear complexity in both setup and application. Together, these developments illustrate how stochastic and learning-based mechanisms can be embedded into classical numerical frameworks to create adaptive and efficient computational methods for complex systems.
Please Note: There will be a pre-seminar 10:55-11:15 in Skiles 005.
At its core, numerical algebraic geometry is the business of solving zero-dimensional polynomial systems over the complex numbers. Thanks to incredibly fast state-of-the-art software implementations, the bottleneck in these algorithms has shifted from computation time to memory usage.
To address this, recent work has introduced iterator datatypes for solution sets. An iterator represents a list by storing a single element and providing a mechanism to obtain the next one, thereby reducing memory overhead.
In this talk, we present our design of 'homotopy iterators' and 'monodromy coordinates', two iterator datatypes based on the most widely used numerical methods for solving polynomial systems. We highlight the substantial benefits of this low-memory perspective through several iterator-friendly adaptations of existing algorithms, including parameter space searches, data compression, and certification.
This talk features joint work with subsets of Paul Breiding, Hannah Friedman, and David K. Johnson.
