Georgia Institute of Technology College of Sciences

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.

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.

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.

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.

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.

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.

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.

The permutation removal lemma was first proved by Klimosová and Král’, and later reproved by Fox and Wei in the context of permutation property testing. In this talk, we study a local version of the permutation removal problem. We show that for any permutation σ not equal to 12, 21, 132, 231, 213, or 312, there exists ε(σ) > 0 such that for any sufficiently large integer N, there is a permutation π of length N that is ε-far from being σ-free with respect to the ρ∞ distance, yet contains only a single copy of σ. Here, the ρ∞ distance is defined as an L∞-variant of the Earth Mover’s Distance between two permutations. We will also discuss our result on the local induced graph removal problem. This is joint work with Fan Wei.