Georgia Institute of Technology College of Sciences

Given a finite set of points $\Gamma$ in $\mathbb{P}^n$, we say that $\Gamma$ satisfies the Cayley-Bacharach condition with respect to degree r polynomials, or is CB(r), if any degree r homogeneous polynomial F vanishing on all but one point of $\Gamma$ must vanish at the last point. In recent literature, the condition has played an important role in computing a birational invariant called the degree of irrationality of complex projective varieties. However, the condition itself has not been studied extensively, and surprisingly little is known about the geometric properties of CB(r) points. 

In this talk, I will discuss a new combinatorial approach to the study of the CB(r) condition, using matroid theory, and present some examples of how matroid theory can shed light on the underlying geometry of such sets.
 

Leveraging large-scale data and systems of computing accelerators, statistical learning has led to  significant paradigm shifts in many scientific disciplines. Grand challenges in science have been tackled with exciting synergy between disciplinary science, physics-based simulations via high-performance computing, and powerful learning methods.

In this talk, I will describe several vignettes of our research in the theme of modeling complex dynamical systems characterized by partial differential equations with turbulent solutions. I will also demonstrate how machine learning technologies, especially advances in generative AI technology,  are effectively applied to address the computational and modeling challenges in such systems, exemplified by their successful applications to  weather forecast and climate projection. I will also discuss what new challenges and opportunities have been brought into future machine learning research.

The research work presented in this talk is based on joint and interdisciplinary research work of several teams at Google Research, ETH and Caltech.


Bio: Dr. Fei Sha is currently a research scientist at Google Research, where he leads a team of scientists and engineers working on scientific machine learning with a specific application focus towards AI for Weather and Climate. He was a full professor and the Zohrab A. Kaprielian Fellow in Engineering at the Department of Computer Science, University of Southern California. His primary research interests are machine learning and its application to various AI problems: speech and language processing, computer vision, robotics and recently scientific computing, dynamical systems, weather forecast and climate modeling.  Dr. Sha was selected as a Alfred P. Sloan Research Fellow in 2013, and also won an Army Research Office Young Investigator Award in 2012. He has a Ph.D from Computer and Information Science from U. of Pennsylvania and B.Sc and M.Sc from Southeast University (Nanjing, China). More information about Dr. Sha's scholastic activities can be found at his microsite at http://feisha.org.

In a recent note Francesco Lin showed that if a rational homology sphere Y admits a taut foliation then the Heegaard Floer module HF^-(Y) contains a copy of F[U]/U as a summand. This implies that either the L-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. In a recent paper in collaboration with Fraser Binns we verified that Lin's geography restriction holds for a wide class of rational homology spheres. I shall discuss our argument, and advance the hypothesis that the Heegaard Floer module HF^-(Y) may satisfy a stronger geography restriction than the one suggested by Lin’s theorem.

The hydrostatic Euler equations, also known as the inviscid primitive equations, are derived from the Euler equations by taking the hydrostatic limit. They are commonly used when the aspect ratio of the domain is small, such as the ocean and atmosphere in the planetary scale. In this talk, I will first present the stability of finite-time blowup of smooth solutions to this model, then discuss the effect of fast rotation (from Coriolis force) in prolonging the lifespan of solutions. Finally, I will talk about the regularization effects that arise when the model is driven by certain random noise.

In this talk, we address the challenge of bivariate probability density estimation under low-rank constraints for both discrete and continuous distributions. For discrete distributions, we model the target as a low-rank probability matrix. In the continuous case, we assume the density function is Lipschitz continuous over an unknown compact rectangular support and can be decomposed into a sum of K separable components, each represented as a product of two one-dimensional functions. We introduce an estimator that leverages these low-rank constraints, achieving significantly improved convergence rates. Specifically, for continuous distributions, our estimator converges in total variation at the one-dimensional rate of (K/n)^{1/3} up to logarithmic factors, while adapting to both the unknown support and the unknown number of separable components. We also derive lower bounds for both discrete and continuous cases, demonstrating that our estimators achieve minimax optimal convergence rates within logarithmic factors. Furthermore, we introduce efficient algorithms for the practical computation of these estimators.

The Hales--Jewett theorem, one of the fundamental results of Ramsey theory, guarantees that when an $n$-dimensional $t \times t \times \dots \times t$ grid is colored with $r$ colors, if $n$ is sufficiently large depending on $r$ and $t$, then the grid contains a line of $t$ collinear points of the same color (possibly with some further restrictions on the line). If you know a second fact about the Hales--Jewett theorem, it is probably that the upper bounds on $n$ grow incredibly quickly (even after tremendous improvement from Shelah in 1988).

In this talk, we will survey the general upper bounds on the Hales--Jewett numbers and move on to results for specific values of $r$ and $t$. We show an upper bound of ``only'' about $10^{11}$ on $n$ when $r=2$ and $t=4$, and discuss the challenges and open questions in extending this to larger cases.

Motivated by question in the dynamics of phase fields, we study the Allen-Cahn equation in dimension 2 with white noise initial datum. We prove the appearance of a universal initial condition for mean curvature flow in a small noise scaling. We also obtain a weak coupling limit when the noise is not tuned down: the effective variance that appears can be described as the solution to an ODE. I will discuss ongoing applications in the perturbative study of other critical SPDEs. Joint works with Simon Gabriel, Martin Hairer, Khoa Lê and Nikos Zygouras.

The Georgia Scientific Computing Symposium (GSCS) is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. Established in 2009, this annual symposium welcomes participants from the broader research community. The event features a day-long program of invited talks, lightning presentations and ample opportunities for networking and collaboration.  Please check this year's information at https://wliao60.math.gatech.edu/2025GSCS.html