Georgia Institute of Technology College of Sciences

The relationship between analytic and algebraic geometry (GAGA) is a rich area of study. For example, Grothendieck's existence theorem states that if $X$ is proper over a complete local Noetherian ring $A$, then a compatible system of coherent sheaves on the thickenings $X_n$ of the special fiber is algebraizable. Such GAGA-type results are now standard tools for studying varieties and their families. In this talk, we answer a question Grothendieck posed in the 1960s: can a Brauer class on $X$ be determined from a compatible system of classes on the $X_n$'s? This is joint work with Andrew Kresch.

This is joint work with Jen Hom and Tye Lidman. Attaching a Casson handle to a slice disk complement gives a smooth manifold homeomorphic to R^4. In the 90's De Michelis and Freedman asked how these choices affect the smooth type of the resulting manifold. This problem has seen some progress since then but is still not well understood. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic R^4's made with the simplest Casson handle are distinct. This gives a countably infinite family of exotic R^4's made with different slice disk complements. We then produce exotic R^4's with various phenomena, and re-prove a theorem of Bizaca-Etnyre on smoothings of product manifolds Y x R. Our main tool is Gadgil's end Floer homology, which we show how to compute effectively by analyzing a certain cobordism map. Time permitting, I'll discuss an upcoming result on exotic planes in R^4 and branched covers, and plans to study more noncompact exotic phenomena.

A simple algorithm for computing the diameter of an unweighted $n$-vertex graph is to run a BFS from every vertex of the graph. This leads to quadratic time algorithms for computing diameter in sparse graphs and geometric intersection graphs. There are matching fine-grained lower bounds which show that in many cases, it is not possible to get a truly subquadratic time algorithm for diameter computation.

To contrast, we give the first truly subquadratic time algorithm for computing the diameter of an $n$-vertex unit-disk graph. The algorithm runs in with $O^*(n^{2-1/18})$ time. The result is obtained as an instance of a general framework, applicable for distance problems in any graph with bounded distance VC-dimension. To obtain these results, we exploit bounded VC-dimension of neighborhood balls,  low-diameter decompositions, and geometric data structures.

Based on paper in FOCS 2025, joint work with Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, and Hung Le. Arxiv: https://arxiv.org/abs/2510.16346

Exploring Parametrized Systems of Real Polynomial Equations with Neural Networks

We investigate the use of machine learning techniques to assist in the solution of parametrized systems of real polynomial equations. In particular, we discuss the problem of jointly predicting the number of real solutions to a problem and approximating the paramter-to-solution map. Throughout, we emphasize interactions between the algebraic geometry of the underlying problem and considerations in deep learning.
This is based on ongoing work with Julianne Barnhart, Jonathan Hauenstein, Ikenna Nometa, Margaret Regan, Trong-Thuc Trang, and Charles Wampler.

The work is devoted to the investigation of the solvability of an integro-differential equation in the case of the double scale anomalous diffusion with a sum of two negative Laplacians in different fractional powers in $R^{3}$. The proof of the existence of solutions relies on a fixed point technique. Solvability conditions for the elliptic operators without the Fredholm property in unbounded domains are used.

 Each year, millions complete brackets to predict the outcomes of the NCAA men’s and women’s basketball tournaments—an activity centered on a fundamental question in sports analytics: Who is number one? Ranking algorithms provide mathematical frameworks for addressing this question and are widely used in postseason selection and predictive modeling.

This talk examines two influential rating systems—the Colley Method and the Massey Method—both of which compute team rankings by solving systems of linear equations based on game outcomes. We discuss extensions that incorporate factors such as late-season momentum and home-field advantage, and we evaluate their impact on predictive performance.

Applications across sports, including basketball and soccer, will be presented, with particular attention to NCAA tournament bracket construction. Research-driven implementations of these methods have produced brackets that outperformed over 90% of millions of ESPN submissions. The talk concludes with open questions and broader applications of ranking methodology.

Bio: 

Bio: Dr. Tim Chartier is the Joseph R. Morton Professor of Mathematics and Computer Science at Davidson College, where he specializes in data analytics. He has consulted with ESPN, The New York Times, the U.S. Olympic & Paralympic Committee, and teams in the NBA, NFL, MLB, and NASCAR. He founded and grew a sports analytics group to nearly 100 student researchers annually. The group, now student-run, provides analytics for Davidson College athletic teams.

His scholarship and leadership have been recognized nationally through service in the Mathematical Association of America (MAA) and with multiple honors, including an Alfred P. Sloan Research Fellowship, the MAA Southeastern Section Distinguished Teaching Award, and the MAA’s Euler Book Prize. He has also collaborated with educational initiatives at Google and Pixar and served as the 2022–23 Distinguished Visiting Professor at the National Museum of Mathematics.

We study a broad class of space–time continuous directed polymers in a Gaussian environment that is white in time and spatially correlated (Itô sense). Under Dalang’s condition, we prove key properties of the partition function—positivity, stationarity, scaling, homogeneity, and a Chapman–Kolmogorov relation—and establish pathwise regularity of the polymer (Hölder continuity and quadratic variation). We give a sharp singularity criterion: the polymer measure is singular w.r.t. Wiener measure iff the spectral measure has infinite total mass. Finally, for d≥3, we prove diffusive large-time behavior in the high-temperature regime, providing a unified framework for polymers driven by singular Gaussian noise.

Joint work with Cheng Ouyang (UIC), Samy Tindel (Purdue), and Panqiu Xia (Cardiff).

I'll review a few known results about quantum graphs that maximize or minimize eigenvalues or combinations of eigenvalues, and will then concentrate on ratios of eigevalues under topological constraints on the graph.  In particular a new discovery with James Kennedy and Gabriel Ramos is that the largest ratio of the first two eigenvalues of the Laplacian on a finite tree graph with Dirichlet conditions at the ends is achieved by equilateral stars. Some related, Weyl-sharp estimates of arbitrary eigenvalue ratios can be obtained using similar ideas.  If time permits, I will also describe some optimal results about differences and other combinations of eigenvalues.

An important class of problems at the intersection of harmonic analysis and geometric measure theory asks how large the Hausdorff dimension of a set must be to ensure that it contains certain types of geometric point configurations. We apply these tools to study configurations associated to the problem of bounding the VC-dimension of a naturally arising class of indicator functions on fractal sets.

The degeneracy of a graph is a measure of sparseness that appears in many contexts throughout graph theory. In extremal graph theory, it is known that graphs of bounded degeneracy have Ramsey number which is linear in their number of vertices (Lee, 2017). Also, the degeneracy gives good bounds on the Turán exponent of bipartite graphs (Alon--Krivelevich--Sudokav, 2003). Extending these results to hypergraphs presents a challenge, as it is known that the naïve generalization of these results -- using the standard notion of hypergraph degeneracy -- are not true (Kostochka--Rödl 2006). We define a new measure of sparseness for hypergraphs called skeletal degeneracy and show that it gives information on both the Ramsey- and Turán-type properties of hypergraphs.

Based on joint work with Jacob Fox, Maya Sankar, Michael Simkin, and Yunkun Zhou