Learning and The Price of Anarchy in Games

Eva Tardos

Cornell University

 We investigate repeated strategic interactions where participants use learning algorithms to guide their decisions. As machine learning increasingly powers online systems—from traffic and packet routing to ad auctions—it becomes essential to understand how strategic behavior affects performance, and how to design systems that ensure robust outcomes.Over the past two decades, researchers have developed powerful tools to quantify the inefficiency caused by selfish behavior, known as the Price of Anarchy. Foundational results show that when participants use learning algorithms satisfying the no-regret condition, the resulting inefficiency remains bounded—even in repeated games. However, these analyses typically assume that each round is independent, with no carryover effects from previous outcomes.In reality, many systems exhibit an evolving dynamic state. We explore such dynamic games, where outcomes in one round directly influence future interactions. We will highlight ongoing research studying this phenomenon in the context of a game modeling queuing system: routers compete for servers, and packets that fail to get served must be resent. This creates a feedback loop where the number of packets in each round depends on prior success, resulting in a highly dependent random process. We study how much excess server capacity is needed to guarantee system stability, even when participants behave selfishly and myopically.

The Shape of Math to Come

Alex Kontorovich

Rutgers University

We present an overview of how certain computational tools currently interact with mathematical practice, and reflect on the implications for research mathematics in the short to medium term, as the field navigates the emerging age of AI and formal verification systems.

Modern Machine Learning Methods: Large Step-Size Optimization, Implicit Bias, and Benign Overfitting

Peter Bartlett

University of California, Berkeley and Google DeepMind

The impressive performance of modern machine learning methods seems to arise through different mechanisms from those of classical statistical learning theory, mathematical statistics, and optimization theory. Simple gradient methods find excellent solutions to non-convex optimization problems, and without any explicit effort to control model complexity they exhibit excellent prediction performance in practice. This talk will describe recent progress in statistical learning theory and optimization theory that demonstrates the optimization benefits of step-sizes that are too large to allow gradient methods to be viewed as an accurate time discretization of a gradient flow differential equation, that characterizes the solutions that are favored by gradient optimization methods, and that illustrates when those solutions can overfit training data but still provide good predictive accuracy.

Optimization in Theory and Practice

Stephen Wright

University of Wisconsin-Madison

Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which  mathematical analysis of their convergence and complexity properties plays a central role. Besides their theoretical properties, optimization algorithms are interesting also for their practical usefulness as computational tools for solving real-world problems. There are often gaps between the practical performance of an algorithm and what can be proved about it. These two facets of the field - the theoretical and the practical - interact in fascinating ways, each driving innovation in the other.  This work focuses on the development of algorithms in two areas - linear programming and unconstrained minimization of smooth functions - outlining major algorithm classes in each area along with their theoretical properties and practical performance, and highlighting how advances in theory and practice have influenced each other in these areas. In discussing theory, we focus mainly on non-asymptotic complexity, which are upper bounds on the amount of computation required by a given algorithm to find an approximate solution of problems in a given class.

Geometric Concepts in Partial Differential Equations

Felix Otto

Max Planck Institute for Mathematics in the Sciences, Leipzig 

Being intrigued by the use of intuition and concepts from differential and in particular Riemannian geometry in the infinite-dimensional set-ting of field equations, I'd like to sketch a couple of examples:

1. The physics-informed interpretation of multi-phase flows in porous media – as gradient flows on the space of densities endowed with a metric from optimal transportation.

2. The renormalization of quasi-linear parabolic equations driven by noisy and thus rough right-hand sides – interpreted as the robust construction of canonical charts and transition maps for the solution manifold, with the help of derivatives with respect to the noise.

3. The connection between drift-diffusion equations with critical ensembles of divergence-free drifts in n-dimensional space – and the geometric Brownian motion on the Lie group Sl(n).

On New Challenges in Numerical Approximation of Partial Differential Equations

Annalisa Buffa

EPFL

Interactions with practitioners and researchers across a wide range of scientific disciplines continually pose new challenges for mathematics. Partial differential equations (PDEs) provide the fundamental language for modeling many physical phenomena, yet their mathematical analysis often remains incomplete and raises deep theoretical questions. Problems emerging from applications frequently require the development of new mathematical frameworks and, in numerical analysis, the design of novel computational methods capable of accurately approximating and simulating increasingly complex systems of PDEs.

In this lecture, I will take the audience on a journey through some of these interdisciplinary challenges and the mathematical ideas they inspire. While many of the motivating questions originate in applications, the numerical tools and theoretical frameworks that emerge often transcend their original context, opening new challenges in numerical mathematics.

Mathematics in the Age of AI

Terence Tao

University of California, Los Angeles

Current AI tools, when combined with formal verification and modern collaboration platforms, are enabling new ways to do mathematics at scale, with increasingly broad collaborations between professional mathematicians, other scientists, members of the public, and AI tools.  Yet these tools continue to have only a modest impact on more traditional mathematical objectives, such as making progress on individual highly difficult problems.  In this lecture we survey the recent achievements of these new paradigms, as well as their continuing limitations.

Between Numbers and People: The Art of Mathematical Communication

Talithia Williams

Harvey Mudd College

Mathematics shapes the modern world, from medicine and climate science to technology and public policy. Yet too often, it remains invisible or inaccessible to the very communities it serves. To meet the challenges of the future, mathematicians must not only discover new knowledge, but also learn to share it in ways that inspire curiosity, trust, and participation. In this talk, Talithia Williams will explore how storytelling can transform the way we communicate mathematics to the public. By connecting ideas to human experience, culture, and purpose, we can turn abstract concepts into meaningful narratives that invite people into the joy of discovery.

AI and Humanity's Long Conversation

Geordie Williamson

University of Sydney

Mathematics has been called humanity's long conversation. Observations of Euclid, Pythagoras, Euler and Poincaré still occupy the minds of mathematicians today. This conversation has experienced shocks and challenges, including the crisis of foundations in the early 20th century, and the first computer assisted proofs in the second half of the 20th century. We are currently in the midst of another shock, with the rise of formal proof and the first signs of AI systems helping to produce research-level mathematics. This raises questions of fundamental importance: How should mathematicians respond to AI? Will AI systems help (or hinder) our understanding of the mathematical world? Williamson will discuss some recent developments at the interface of mathematics and AI, with the aim of having a clearer picture of this unique point in the history of mathematics.