北京雁栖湖应用数学研究院

Mathematics has been the foundational language of the Universe, providing the tools to describe the law of physics. Physics also has inspired new mathematics through its history. In this talk, I review recent developments in theoretical physics, especially in quantum field theory and string theory, and new mathematical insights arising from them.

The Fields Prize is considered as the Nobel Prize of mathematics and is awarded every four years at the International Congress of Mathematicians. I will discuss the topics recently studied in probability theory by looking at the work of Fields medalists. In fact, they are all related to statistical physics. These include percolation theory by W. Werner (2006) and S. Smirnov (2010); singular stochastic partial differential equations by M. Hairer (2014); the triviality of four-dimensional Euclidean field theory by H. Duminil-Copin (2022); the limit shape of three-dimensional random Young tableaux by A. Okounkov (2006).

The continued fractions is a tool providing an algorithm of finding  best possible rational approximations to irrational numbers. But it is also a beautiful example of a dynamical system with non-trivial ergodic behaviour. In this talk I will explain the basic facts of the theory of continued fractions. I will then use the example of continued fractions to  introduce important concepts of ergodic theory. Ergodic theory has many applications to different areas of modern mathematics, in particular to number theory. I plan to introduce two conjectures formulated about 100 years ago: the Oppenheim conjecture and the Littlewood conjecture. The first one was solved in 1986 by Grigory Margulis. The second one remains open despite the progress made in the last 25 years.

Algebraic geometry stands at the heart of modern mathematics — a field where the elegance of geometry meets the precision of algebra, revealing deep structures that connect seemingly distant domains. In this talk, we will journey through the revolutionary ideas of Grothendieck, whose reformulation of geometry through schemes, functors, and categories reshaped the very language of mathematics. We will explore how this conceptual framework became the foundation for modern developments such as Gromov–Witten and Donaldson–Thomas theories in enumerative geometry, and the rich interplay between geometry and quantum field theory — the key ingredients in the mathematical formulation of topological string theory. In the final part of the talk, we will turn to a more unexpected frontier: the influence of algebraic geometry on today’s most advanced technologies in artificial intelligence. From algebraic varieties underlying data manifolds to geometric deep learning and the categorical structures emerging in AI theory, we will reflect on how ideas born from Grothendieck’s vision continue to shape the conceptual foundations of computation, learning, and abstraction itself. This talk aims to inspire a sense of wonder — to show how the “magic” of algebraic geometry transcends its classical boundaries and continues to illuminate both the deepest corners of mathematics and the most cutting-edge frontiers of technology.

Number Theory is famous for its numerous long-standing open problems, as well as for its broad connections with other areas of Science. In particular, Number Theory strongly motivated creation and intensive development of Representation Theory and Harmonic Analysis. On the other hand, Representation Theory is known to have deep and tight links with Quantum Physics. Therefore, one might pose a question about interconnections between Number Theory and Quantum Physics. My talk is devoted to one such interconnection between automorphic $L$-functions and quantum integrable systems, that I discovered jointly with A. Gerasimov and D. Lebedev, and that we have been developing over the last 15 years. First, I review properties of irreducible (principal series) representations of the general linear group $GL(n,\mathbb{R})$ and the associated local $L$-functions. Then I discuss the example of $GL(2)$ (non-holomorphic) Eisenstein series and elucidate how the associated global $\zeta$-functions arise via action of the so-called Hecke-Baxter operator.