# Thematic programs 2021

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.

**Geometric and Mathematical Analysis, and Weak Geometric Structures**

#### March 23 – December 23, 2021

Organizers:

Roberta Musina, Università di Udine

Matteo Novaga, Università di Pisa

Eugene Stepanov, PDMI RAS

Dario Trevisan, Università di Pisa

This thematic program will be focused upon several areas of current mathematics research very closely interrelated between each other and having deep common roots in geometric measure theory (GMT):

- Shape optimization problems and geometric analysis: minimal surfaces (with respect to various notions of surface area/perimeter including nonlocal ones) and related problems, clusters of soap bubbles, problems with prescribed curvature, the Steiner problem and transportation networks, as well as applications of shape optimization in image analysis (e.g. Mumford-Shah problem and similar) and mechanics (e.g. eigenvalue or compliance optimization or ground states of Schrödinger equation).
- Weak geometric structures related to GMT, such as currents, as well as those arising in optimal mass transportation and the study of geometry of highly irregular metric measure spaces, in particular those without any differentiable structure, with applications to the analysis of PDEs with very low regularity (e.g. weak and/or or probabilistic notions of flows, or “differential equations” with “purely nondifferentiable”, for instance just Hölder continuous unknowns, like those in Rough paths theory) and to some modern harmonic analysis problems.
- Purely geometric problems involving weak geometric structures, like extensions of Frobenius theorem either to irregular differential forms or distributions of planes or to weaker notions of surfaces like De Rham currents; extensions of Chow-Rachevsky theorem to irregular vector fields or flows; applications to sub-Riemannian geometry, geometric control theory and dynamical systems.

The idea is to explore the deep connections between the above mentioned areas of research and to make them more comprehensible and attractive to both international and local mathematical

community.

The program includes the following minicourses:

- “Geometric flows of networks” by Matteo Novaga (Università di Pisa) and Alessandra Pluda (Università di Pisa), March 23 – April 1, 2021
- “Loops and Bubbles” by Roberta Musina (Università di Udine), April 13 – April 21, 2021
- “Gabor analysis for rational functions” by Yurii S. Belov (St. Petersburg State University), April 14 – May 26, 2021
- “One-dimensional optimization problems” by Emanuele Paolini (Università di Pisa), May 11 – May 19, 2021
- “Mathematics and applications of manifold learning: reconstructing hidden geometric structures in the data” by Serguei Barannikov (Skoltech and CNRS), Sergey Nechaev (CNRS), Vladimir Spokoiny (WIAS) and Dario Trevisan (Università di Pisa), dates TBA
- “Sobolev vector fields and their flows” by Elia Brué (IAS) and Dario Trevisan (Università di Pisa), dates TBA

The program also includes the conference “30th St.Petersburg Summer Meeting in Mathematical Analysis“, July 1 – 6, 2021.

**Moduli Spaces, Combinatorics and Poisson Geometry**

#### November 2021 – August 2022

Organizers:

Dmitry Korotkin, Concordia University and Centre de Recherches Mathématiques

Peter Zograf, PDMI RAS and St. Petersburg University

Moduli spaces have many non-trivial connections to other areas of mathematics: combinatorics, dynamics, integrable systems and Poisson geometry, to name a few. Among most celebrated results over the last 30 years one can mention several proofs of Witten’s conjecture about intersection numbers of ψ-classes (Kontsevich, Mirzakhani and others), computation of Euler’s characteristics of moduli spaces (Harer-Zagier), development of the higher Teichmüller theory (Fock, Goncharov) and its links with cluster algebras and associated Poisson structures (Fomin, Zelevinsky). The research problems central for the program are:

- Establishing a relationship between meandric systems (pairs of transversal multicurves) on higher genus surfaces and square-tiled surfaces.
- Study of the large genus asymptotics of the numbers of meandric systems of given topological type.
- Computation of Masur-Veech volumes of lower dimensional strata in the moduli space of quadratic differentials.
- Obtaining a relation of the distribution of geodesic multicurves to Masur-Veech volumes.
- Establishing a relation between Joyce’s structures by Bridgeland to Frobenius manifolds and topological recursion formalism.
- Description the complete WKB expansion of the generating function of monodromy symplectomorphism for second order differential equations on Riemann surfaces with second order poles

and establishing the link to topological recursion formalism. - Application of the WKB formalism to general isomonodromic tau-function and embed it into the topological recursion framework. Generalization to higher genus using the formalism of Krichever and Bertola-Malgrange.
- Construction of the dilogarithm line bundle over SL(2, R) cluster variety associated to the canonical symplectic form over the moduli spaces of bordered Riemann surfaces; description of the Bohr-Sommerfeld symplectic leaves and their quantization.

The following activities will be organized during the program:

- school “Moduli Spaces, Combinatorics and Integrable Systems“, November 15 – 26, 2021
- conference “Combinatorics of Moduli Spaces, Cluster Algebras and Topological Recursion“, June 13 – 17, 2022
- conference “Geometry and Dynamics of Moduli Spaces“, August 1 – 5, 2022
- a number of minicourses given by invited visitors.

**Spectral Theory and Mathematical Physics**

#### February 1 – December 31, 2021

#### rescheduled from 2020 due to COVID-19 pandemic

Organizers:

Alexander Fedotov, SPbU

Nikolai Filonov, PDMI

Alexander Its, IUPUI and SPbU

Alexander Sobolev, University College London

Tatiana Suslina, SPbU

Dmitri Yafaev, University of Rennes 1 and SPbU

The program is devoted to spectral theory and its applications. The spectral theory is one of the domains where the Saint Petersburg mathematical school is traditionally very strong, the names of L.D. Faddeev, M.S. Birman and V.S. Buslaev are among the names of the world-known leaders in the field.

The following activities will be organized during the program:

- warm-up online mini-courses, November 2 – 26, 2020
- Seminar in Spectral Theory and Related Topics, February – December, 2021
- Seminar in Mathematical Physics, February – December, 2021
- conference “Asymptotic Methods in Mathematical Physics”, June 20 – 22, 2021
- St. Petersburg Conference in Spectral Theory, June 23 – 26, 2021
- Summer School in Spectral Theory, June 27 – 30, 2021
- a number of minicourses given by invited visitors.

**New Trends in Mathematical Stochastics**

#### August 15 – December 31, 2021

#### rescheduled from 2020 due to COVID-19 pandemic

Organizers:

Ildar Ibragimov, PDMI

Yana Belopolskaya, SPbUACE

Andrei Borodin, PDMI

Yury Davydov, SPbU

Mikhail Lifshits, SPbU

Maria Platonova, PDMI and SPbU

Natalia Smorodina, PDMI and SPbU

Yuri Yakubovich, SPbU

Andrei Zaitsev, PDMI

Dmitry Zaporozhets, PDMI

The general idea of the program is to strengthen interactions between experts and young researchers working in the areas of probability theory, stochastic processes and their applications. Visits of leading scientists in these areas will serve to enrich the mathematical education in St. Petersburg and to develop possibilities for future collaboration.

The following activities will be organized during the program:

- introductory online school “Randomness online“, November 4 – 8, 2020
- conference “New Trends in Mathematical Stochastics”, August 30 – September 3, 2021
- workshop “St. Petersburg Youth Conference in Probability and Mathematical Physics“, December 20 – 23, 2021
- a number of minicourses given by invited visitors.