Applied Math Colloquium

Abstract: Differential privacy (DP) has emerged as a powerful framework for designing algorithms that protect sensitive data. In this talk, I will present our work at the intersection of differential privacy and linear algebra, introducing efficient DP algorithms for fundamental algebraic tasks: solving systems of linear equations over arbitrary fields, solving linear inequalities over the reals, and computing affine spans and convex hulls.

Our algorithms for equalities are strongly polynomial, while those for inequalities are only weakly polynomial—and this gap is provably inherent.

As applications, we obtain the first efficient DP algorithms for learning halfspaces and affine subspaces.

The talk will not assume prior familiarity with differential privacy; I will begin with a review of the definition.

Abstract: We discuss a novel class of swarm-based gradient descent (SBGD) methods for non-convex optimization. The swarm consists of agents, each is identified with position, $x$, and mass, $m$. There are two key ingredients in the SBGD dynamics:
(i) persistent transition of mass from agents at high to lower ground; and
(ii) time stepping protocol which decreases with $m$.

The interplay between positions and masses leads to dynamic distinction between ‘leaders' and 'explorers': heavier agents lead the swarm near local minima with small time steps; lighter agents use larger time steps to explore the landscape in search of improved global minimum, by reducing the overall 'loss' of the swarm. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer.

Abstract: The 2024 Nobel Prizes in Physics and Chemistry were both awarded in the field of Artificial Intelligence (AI). John J. Hopfield and Geoffrey E. Hinton pioneered most of the machine learning methods that we nowadays take for granted, and established “the physicist’s way of thinking” in developing and understanding early neural networks. David Baker, Demis Hassabis and John M. Jumper used advanced computational tools, including contemporary supervised and unsupervised learning with built-in chemical principles, to model and design complex protein structures: one of the most vexing problems in the life sciences. Were the back-to-back awards a coincidence or planned?

While of course I cannot claim to know what the committees were thinking, in this talk I will argue that the double award strengthens the urgency to reorient engineering research in a new direction: while large language models and their implications are occupying the everyperson’s involvement with AI at present, the future belongs to the interaction of intelligent machines with physical systems, including the human body and its environs. My own work presents some interesting cases where endowing optical physics with data-driven approximations to complex phenomena, e.g. strong volumetric scattering, led us to discover new fundamental principles in imaging theory. We then put these in service to practical applications, such as metrology and inspection for the semiconductor and pharmaceutical industries, with the added benefit of explainability through adherence to the underlying physics.

At the concluding part of the presentation, I would like to invite everyone to imagine a future where humans and intelligent machines work seamlessly together to accomplish demanding, delicate and sensitive physical tasks: child and elderly care at the home and hospital; food production, quality assurance and preparation; transportation, travel and hospitality; urban management and sanitation; and so on. From our positions as leading academics and entrepreneurs we are presented with rich opportunity and, at the same time, grave responsibility to maximize the benign potential of the new technologies, while also educating the general public, legislators and regulators toward balanced, ethical and safe societal adoption.

Abstract: Machine learning is transforming mathematical discovery, enabling advances on longstanding open problems. This talk explores two complementary approaches illustrating different paradigms for AI and mathematics.

On the one hand, I will present a systematic discovery of unstable singularities in multi-dimensional partial differential equations. While most numerical methods historically found only stable singularities, we discover families of unstable singularities requiring infinitely precise initial conditions. Combining curated machine learning architectures with high-precision optimization and mathematical analysis, we achieve in some cases near machine precision, meeting requirements for rigorous computer-assisted proofs.

On the other, I will discuss AlphaEvolve, a general-purpose evolutionary coding agent that uses large language models to autonomously discover old and new mathematical constructions and potentially go beyond them. AlphaEvolve tackles a wide variety of problems across analysis, geometry, combinatorics, and number theory. This illustrates how general-purpose AI systems can systematically successfully explore broad mathematical landscapes at an unprecedented speed, leading us to do mathematics at scale.

Together, these examples reveal complementary roles for machine learning and mathematics in the future: deep, precision-focused small models for specific problems versus broad, systematic exploration across domains via large models.

Abstract:Fast Direct Solvers (FDS) address the problem of solving a system of linear equations $A x = b$ arising from the discretization of either an elliptic PDE or of an associated integral equation. The matrix $A$ will be sparse when the PDE is discretized directly, and dense when an integral equation formulation is used. For decades, industry practice for large scale problems has been to use iterative solvers such as multigrid, GMRES, or conjugate gradients. In contrast, a direct solver builds an approximation to the inverse of $A$ or an easily invertible factorization of $A$ (e.g.\ LU or Cholesky). A major development in numerical analysis in the last couple of decades has been the emergence of algorithms for constructing such factorizations or performing such inversions in linear or close to linear time. Such methods must necessarily exploit that the inverse of $A$ is ``data-sparse,'' e.g.\ that it can be tessellated into blocks that have low numerical rank. This talk will cover the development of FDS's for both sparse and dense matrices, recent developments in the field, as well as future challenges and opportunities.