# Thematic programs 2022

Each calendar year Leonhard Euler International Mathematical Institute hosts thematic programs gathering recognized experts in the respective fields as well as early career researchers and postdocs to carry out investigations with a special emphasis on collaborative activities. Every program features a summer school as well as one or two conferences or workshops. The stays of the visitors and speakers are fully or partially financially covered by the Institute.

The themes of the programs are open to suggestion via the application process. The selection criteria for accepting a proposal are its scientific strength and the degree to which the program would benefit mathematical research and post-graduate training in the Russian Federation.

**New Trends in Topology**

### February 1 – June 30, 2022

Organizers:

Evgeny Fominykh, St. Petersburg University

Ilia Itenberg, Sorbonne University

Viatcheslav Kharlamov, Université de Strasbourg

Vladimir Turaev, Indiana University

Oleg Viro, Stony Brook University

There were several far-reaching conceptual developments in topology at the turn of the century. In the late 1980’s – early 1990’s, a spectacular progress in the theory of knots and 3-dimensional manifolds was made by the Fields medalists E. Witten and V. Jones followed by the work of N. Reshetikhin, V. Turaev, O. Viro and others who related this area of topology to the theory of quantum groups. As a result, a new mathematical field was born, the topological quantum field theory.

In parallel to the appearance of the topological quantum field theory, there emerged a theory of Gromov-Witten invariants, under the influence of implantation of pseudo-holomorphic curve technique into symplectic geometry by M. Gromov and the holomorphic curve counting into quantum 2-dimensional gravity by E. Witten. This had led, in particular, to the Kontsevich-Manin theory of quantum cohomology and a breakthrough in enumerative geometry.

These new research areas are not only linked by the time of their appearance and the string theory as a common root, but also by deep relations between the techniques and underlying algebraic structures. The very recent developments in Gromov-Witten theory brought to light direct bridges between enumerative geometry of open strings and the theory of knot invariants. The proposed program is devoted to a number of the most active areas of these research fields.

The following activities will be organised during the program:

- introductory school “Classical and Quantum Topology in dimension three”, April 11 – 15,
- introductory school “New Methods in Enumerative Geometry”, April 25 – 29,
- conference “Low-Dimensional Topology”, June 6 – 10,
- conference “On the Crossroad of Topology and Enumerative Geometry”, June 13 – 17,
- a weekly research seminar.

**Algebraic Methods in Complexity Theory**

### August 1 – November 30, 2022

Organizers:

V. Arvind, Institute of Mathematical Sciences Chennai

Alexander S. Kulikov, PDMI

Rahul Santhanam, University of Oxford

Jacobo Toran, University of Ulm

The broad goal of Computational Complexity is the study of computational resources required by algorithms to detect properties of combinatorial objects and structures. An approach that has proven successful in the past is, broadly, via connections to algebraic settings. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. These include most of the methods for obtaining circuit lower bounds, the PCP characterization of NP, or the Agrawal-Kayal-Saxena polynomial-time primality testing, just to name some well known results.

While these examples are now classical results, algebraic methods continue to play a central role in exciting recent progress in computational complexity. These areas include:

- Algebraic Complexity Theory: this subfield is rooted in algebraic questions, including algebraic circuit lower bounds, and limits to the methods, algorithms for polynomial identity testing and

related problems, and connections to randomness in computation. - Algorithms based on algebra: traditionally strong fields here are algorithmic coding theory and constraint satisfaction problems. There are new surprising connections with algebraic methods

like the work of Ryan Williams on all-pair shortest path algorithms based on the polynomial method, and the more recent fast exponential-time algorithm for solving a system of polynomial equations based on the Razborov-Smolensky method. To this area belongs also the study of algorithms on algebraic structures like groups, matrices, or polynomials. - Many other areas of complexity theory, like proof complexity, communication complexity, quantum computation or algorithmic game theory abound with algebraic techniques and methods.

The program is planned with activities around these three broad themes. We aim to bring together researchers using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics, and this program can play an important role in educating the complexity theory community about the latest new techniques, bringing together experts using algebraic methods for different problem domains to exchange ideas spurring further progress.

The following activities will be organized during the program:

- student school “Bootcamp in Complexity Theory”,
- conference “Complexity Interactions”,
- mini-courses for advanced students,
- weekly seminars.