Georgia Institute of Technology College of Sciences

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.

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.

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.