##### Description

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**Organizing committee of Beijing-Saint Petersburg Mathematics Colloquium **

(1) Huijun Fan (SMS PKU, symplectic geometry and mathematical physics, geometric analysis)

(2) Feimin Huang (CAS, institute of applied mathematics, partial differential equation, fluid mechanics)

(3) Sergey Kryzhevich (SPBU, differential and difference equations, dynamical systems, chaotic dynamics, structural stability, hyperbolic theory)

(4) Xiaonan Ma (Université de Paris, geometry, index theory)

(5) Maxim Vsemirnov (POMI RAS, SPBU, number theory, algebra, combinatorics)

(6) Wenyuan Yang (BICMR, geometric group theory and Kleinian groups)

(7) Dmitry Zaporozhets (POMI RAS, SPBU, Poisson-Voronoi tessellation zeros of random analytic functions and polynomials mixed volumes)

******

**Lecture Series 28 —— May 5, 2022（20:00-22:00 Beijing time / 15:00-17:00 St Petersburg time).**

**Recording: https://disk.pku.edu.cn:443/link/BBD59D040E4E91A35EFFC9E42CEF593A**

Valid Until: 2026-06-30 23:59

**Lecture 1——**Analysis on compressible Navier-Stokes equations with strong boundary layer

**Speaker:** Tong Yang (City University of Hong Kong)

**Time:** 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:** It is a classical problem in fluid dynamics about the stability and instability of different hydrodynamic patterns in various physical settings, in particular in the high Reynolds number limit of laminar flow with the boundary layer. However, there are very few mathematical results on the compressible fluid despite the extensive studies when the fluid is governed by the incompressible Navier-Stokes equations. In this talk, we will present a new approach to studying the compressible Navier-Stokes equations in the subsonic and high Reynolds number regime where a subtle quasi-compressible and Stokes iteration is developed. As a byproduct, we show the spectral instability of the subsonic boundary layer. This is a joint project with Zhu Zhang.

**Bio:** Tong Yang is a Chair Professor of Mathematics at the City University of Hong Kong. He was President of the Hong Kong Mathematical Society from 2016 to 2020. He has been working in the fields of partial differential equations and kinetic theory. He served as an Editor-in-Chief of Analysis and Applications from 2013 to 2017 and is one of the founding Editors-in-Chief of Kinetic and Related Models launched in 2008. He also serves on several international journals, including the Editorial Advisory Board of London Mathematical Society: Bulletin and Journal, and SIAM Journal on Mathematical Analysis. He is a member of the European Academy of Sciences, The World Academy of Sciences and the Hong Kong Academy of Sciences. The awards and distinctions that he has received include the 2nd class State Natural Science Award of China in 2012 and the Croucher Senior Research Fellowship in 2011.

**Lecture 2——Large bifurcation supports**

**Speaker:** Prof. Yulij Ilyashenko

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** The talk deals with a general problem of the bifurcation theory on the two-sphere. Consider a vector field with arbitrary degeneracies: complex singular points, multiple limit cycles and polycycles. What part of the phase portrait will bifurcate under a perturbation of this field, and what part will remain unchanged? Not only the degenerated parts of the phase portrait mentioned above will bifurcate; but some generic parts will also. A highly nontrivial question arises: WHO BIFURCATES? The answer will be given in the talk based on joint work with Natalya Goncharuk.

**Bio:** Prof. Yulij Ilyashenko is one of the most renowned Russian scientists working in Differential Equations, Dynamical Systems and Geometry. He obtained a big number of brilliant results in bifurcation theory and dynamics beyond uniform hyperbolicity. In particular, he proved the finiteness of the number of limit cycles for polynomial vector fields, thus giving a partial solution to Hilbert's 16th problem (the previous proof provided by Dulac contained a serious gap).

Yulij Sergeevich Ilyashenko is the Rector of the Independent University of Moscow, vice-president of Moscow Mathematical Society, and Professor at Moscow State University, Cornell University, Higher School of Economics, and Steklov Mathematical Institute. Besides, Prof. Ilyuashenko is a member of the editorial boards of several prestigious journals including Ergodic Theory and Dynamical Systems, Journal of Dynamics and control systems and Main Editor of Moscow Mathematical Journal.

******

**Lecture Series 27 —— April 7, 2022（20:00-22:00 Beijing time / 15:00-17:00 St Petersburg time).**

**To Join Zoom Meeting:**

**https://us02web.zoom.us/j/84735331476?pwd=Qk52clE5ZUhSbnpMR3FmNW9mNVMrZz09**

**Meeting ID:** 847 3533 1476

**Password:** 146996

**Lecture 1——Extending periodic maps on surfaces over the 4-sphere**

**Speaker:** Prof. Shicheng Wang, Peking University

**Time:** 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:** The topic indicated by the title has been addressed by Montesinos (1982) and Hirose (2002) using Rohlin intersection form, and by Ding-Liu-Wang-Yao (2012) using spin structures.

Based on the work above, we deduce a more computable criterion for extending periodic maps on surfaces over the 4-sphere.

Some applications are given. Results including:

(1) Let $F_g$ be the closed orientable surface of genus $g$ and $w_g$ be a periodic map of maximum order on $F_g$.

Then $w_g$ is extendable over $S^4$ for some smooth embedding $e: F_g\ to S^4$ if and only if $g=4k, 4k+3$.

(2) For infinitely many primes $p$, each periodic map of order $p$ on $F_g$ is extendable over $S^4$ for some smooth embedding $e: F_g\ to S^4$.

This is joint work with Zhongzi Wang.

**Bio:** Prof. Shicheng Wang is currently a Distinguished Professor at the Department of Mathematics, Peking University. He received a master's degree from Peking University in 1981 and a doctorate degree from UCLA in 1988. He has won the Shiing-Shen Chern Mathematics Award and the second prize of the National Natural Science Awards (China). Prof. Wang was an invited speaker at the ICM’2002 in Beijing. His research focuses on low-dimensional topology, involving geometric group theory, dynamical systems, and algebraic topology.

**Lecture 2——A Mozaic from Geometry, Dynamics, PDEs, and maybe more.**

**Speaker:** Prof. Dimitri Yu. Burago, Pennsylvania State University, State College PA

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** I am going to begin with some unsolved problems, which, in my opinion, deserve more attention that they receive. For some problems, there are partial solutions, which I would try to sketch or at least discuss. I plan then to go to problems where the progress is more substantial, as time permits. What I plan is indeed a mozaic. I am not sure how many topics I would be able to touch upon and which ones would be more of interest, my collaborators include D. Chen, S. Ivanov, B. Kleiner, Ya. Kurylev, M. Lassas, J. Lu, A. Novikov, L. Polterovich. Maybe, I would concentrate оn two or three topics, maybe many more short stories, depending on the reaction.

**Bio:** Dimitri Yurievich Burago is a worldwide renowned expert in Geometry, Topology and Dynamical Systems. He graduated from Saint Petersburg State University and has got his doctorate in 1994 under the supervision of Prof. Anatoly Vershik. Now he is working at Pennsylvania State University where he received the title of Distinguished Professor in Mathematics. Besides, Prof. Burago received many prestigious prizes including the Leroy P. Steele Prize by the American Mathematical Society. Also, he was an invited speaker of the ICM’1998 in Berlin.

******

**Lecture Series 26 —— March 24, 2022（20:00-22:00 Beijing time / 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——On Lyapunov exponents of quasi-periodic cocycles**

**Speaker: **Jiangong You (Chern Institute of Mathematics, Nankai University)

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **Lyapunov exponents are dynamical invariants. Quasi-periodic cocycles are a special important class of dynamical systems having strong background in quantum physics. Lyapunov exponents of a quasi-periodic cocycle depends sensitively on the smoothness of the cocycle, arithmetic property of the frequency and profile of the potential. So far the behind mechanism of the sensitivity has not been completely understood. In this talk, I will discuss the continuity, Holder regularity of the Lyapunov exponent as well as the quantitative version of Avila’s global theory. The talk is based on joint works with L. Ge, S. Jitomirskaya, Y. Wang, X. Zhao and Q. Zhou.

**Bio: **Jiangong You is a professor at Chern Institute of Mathematics of Nankai University. His research interests include Hamiltonian dynamics (especially KAM theory for Hamiltonian ODEs and PDEs), smooth ergodic theory (especially Lyapunov exponents) and mathematical physics (especially spectral theory of quasi-periodic Schrodinger operators).

**Lecture 2——Systems of isometries: dynamics and topology**

**Speaker:** Alexandra Skripchenko

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **Systems of partial isometries of the interval represent a simple combinatorial object which appears in topology in connection with measured foliations on a surface (orientable or non-orientable), in dynamics as a nice model to study billiards in rational polygons and in geometric group theory as a way to describe actions of free groups on R-trees. We will discuss several classes of systems of isometries (interval exchange transformations, interval exchange transformations with flips, interval translation mappings, band complexes) and compare their basic dynamical properties: minimality, ergodicity, invariant measures etc. The talk is mainly based on the joint works with Artur Avila and Pascal Hubert and with Serge Troubetzkoy.

**Bio: **Prof. Alexandra Skripchenko is the Dean of Faculty of Mathematics at Higher School of Economics (HSE), Moscow. She is expert in Dynamical Systems, Low-dimensional Topology and Geometric Group Theory. In particular, she has got some outstanding results concerning interval exchange mappings and interval translation mappings. Having graduated from Moscow State University in 2009, Alexandra Skripchenko has got her PhD in 2012 and works at HSE since 2014.

******

**Lecture Series 25 —— March 8, 2022（20:00-22:00 Beijing time / 15:00-17:00 St Petersburg time).**

**Recording: https://disk.pku.edu.cn:443/link/EDA3BDF8D06B8765C1542BE7278C7D3B**

Valid Until: 2026-06-30 23:59

**Lecture 1——Sharp asymptotics for arm events in critical planar percolation**

**Speaker: **Xinyi Li

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **We consider critical planar site percolation on the triangular lattice and derive sharp estimates on asymptotics of the probability of half-plane $j$-arm events for $j\geq 1$ and whole-plane (polychromatic) $j$-arm events for $j>1$ under some specific boundary conditions. We also obtain up-to-constant estimates for other boundary conditions in the whole-plane case. These estimates greatly improve previous ones and solve a problem of Schramm (Proc. of ICM, 2006). In the course of proof, we also obtain a super-strong separation lemma, which confirms a conjecture by Garban, Pete and Schramm (J. Amer. Math. Soc., 2013) and is of independent interest. This is joint work in progress with Hang Du (PKU), Yifan Gao (PKU) and Zijie Zhuang (UPenn).

**Bio: **Dr. Xinyi Li is currently an assistant professor at Peking University. He received his Ph.D. in mathematics from ETH Zurich in 2016 under the supervision of Professor Alain-Sol Sznitman. His research focuses on Probability theory, and statistical physics.

**Lecture 2——On the number of level sets of smooth Gaussian fields**

**Speaker:** Dmitry Belyaev

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **The number of zeroes or, more generally, level crossings of a Gaussian process is a classical subject that goes back to the works of Kac and Rice who studied zeroes of random polynomials. The number of zeroes or level crossings has two natural generalizations in higher dimensions. One can either look at the size of the level set or the number of connected components. The surface area of a level set could be computed in a similar way using Kac-Rice formulas. On the other hand, the number of the connected components is a `non-local' quantity which is notoriously hard to work with. The law of large numbers has been established by Nazarov and Sodin about ten years ago. In this talk, we will briefly discuss their work and then discuss the recent progress in estimating the variance and deriving the central limit theorem. The talk is based on joint work with M. McAuley and S. Muirhead.

**Bio: **Dmitry Belyaev is a Professor of Mathematics and Tutorial Fellow at St . Anne’s College. Before coming to Oxford in 2011 he was an Assistant Professor at Princeton University (2008-2011) and Veblen Research Instructor at Princeton University and IAS (2005-2008). He received PhD in 2005 from the Royal Institute of Technology in Stockholm and B.Sc/M.Sc from St. Petersburg State University.

His main research interests are on the interface between analysis and probability including (but not limited to) Geometry of Gaussian fields, Growth models, Geometric function theory, Schramm-Loewner Evolution and critical lattice models.

** **

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**Lecture Series 24 —— December 16, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——**Random distances of Liouville quantum gravity: recent and very recent progresses

**Speaker: **Jian Ding

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **In this talk I will review some recent and very recent progress on random metric associated with Liouville quantum gravity with focus on the construction and the phase transition. The talk is based on works with Julien Dubédat, Alexander Dunlap, Hugo Falconet, Subhajit Goswami, Ewain Gwynne, Avelio Sepúlveda, Ofer Zeitouni and Fuxi Zhang in various combinations, and especially on a few very recent joint works with Ewain Gwynne.

**Bio: **Jian Ding is a Gilbert Helman Professor at University of Pennsylvania. His main research area is in probability theory, with focus on interactions with statistical physics and theoretical computer science. He also has a broad interest in probability questions that arise from "application-oriented" problems. Before joining Penn, he has been a postdoc at Stanford and a faculty at University of Chicago, after his Ph.D. at UC Berkeley in 2011.

**Lecture 2——**Random section and random simplex inequality

**Speaker:** Dmitry Zaporozhets

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **

**Bio: **Dmitry Zaporozhets is professor at the Department of Mathematics and Computer Science and Senior Researcher at Saint Petersburg Department of Steklov Mathematical Institute of Russian Academy of Science.

Prof. Zaporozhets is one of strongest researchers in Probability theory and Geometry, having major interests in Geometry of numbers, Stochastic Geometry, Convex Geometry, Poisson - Voronoi tessellation, zeros of random polynomials and random analytic functions.

Being a student of Acad. Il'dar Ibragimov, one of the most renowned experts in Probability and Statistics, Dmitry graduated from Saint Petersburg State University, defended his PhD thesis in 2005 and Doctoral thesis in 2015 (both thesis were devoted to Random Polynoms).

Later on, Prof. Zaporozhets has got the honorary degree of Professor of Russian Academy of Sciences.

******

**Lecture Ser****ies ****23**** —— ****December** **02****, ****2021**（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).

Valid Until：2026-04-30 23:59

**Lecture 1——Gabor analysis for rational functions**

**Speaker: **Prof. Yuri Belov

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:** Let $g$ be a function in $L^2(\mathbb{R})$. By $G_\Lambda$, $\Lambda\subset R^2$ we denote the system of time-frequency shifts of $g$, $G_\Lambda=\{e^{2\pi i \omega x}g(x-t)\}_{(t,\omega)\in\Lambda}$.

A typical model set $\Lambda$ is the rectangular lattice $\Lambda_{\alpha, \beta}:= \alpha\mathbb{Z}\times\beta\mathbb{Z}$ and one of the basic problems of the Gabor analysis is the description of the frame set of $g$ i.e., all pairs $\alpha, \beta$ such that $G_\Lambda_{\alpha,\beta}$ is a frame in $L^2(\mathbb{R})$. It follows from the general theory that $\alpha\beta \leq 1$ is a necessary condition (we assume $\alpha, \beta > 0$, of course). Do all such $\alpha, \beta $ belong to the frame set of $g$? Up to 2011 only few such functions $g$ (up to translation, modulation, dilation and Fourier transform) were known. In 2011 K. Grochenig and J. Stockler extended this class by including the totally positive functions of finite type(uncountable family yet depending on finite number of parameters) and later added the Gaussian finite type totally positive functions. We suggest another approach to the problem and prove that all Herglotz rational functions with imaginary poles also belong to this class. This approach also gives new results for general rational functions. In particular, we are able to confirm Daubechies conjecture for rational functions and irrational densities.

**Lecture 2—****—Planar Sobolev extension domains and quasiconformal mappings. **

**Speaker: **Yi Zhang

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** In 1979, Goldshtein et al. pointed out that a bounded simply connected planar domain is a W^{1,\,2}-extension domain if and only if it is a quasidisk. This result was later generalized by Jone to all dimensions, showing that uniform domains are W^{1,\,p}-extension domain for any 1\le p<\infty. However, there are simply connected W^{1,\,p}-extension domains in the plane which are not uniform when p\neq 2, pointed out by e.g. Maz'ya. In this talk I will present my joint work with Koskela and Rajala on this problem, and show its connection to the Ahlfors' quasiconformal reflection theorem.

**Bio: ** My main background is in complex analysis and geometric function theory. I have also done works on harmonic analysis, functional analysis, nonlinear elliptic partial differential equations (p-Laplacians and infinity Laplacian), free boundary problems, calculus of variations, eigenvalue problems of p-Laplacians and so on. Especially, I am interested in problems in the intersection of analysis and geometry.

B.S., Math & Applied Math, Beihang University, September 1, 2009 – June 30, 2013.

M.S., Mathematics, University of Jyväskylä (Supervisor: Pekka Koskela), August 1, 2013 – July 30, 2014.

Ph.D., Mathematics, University of Jyväskylä (Supervisor: Pekka Koskela), August 1, 2014 – July 30, 2017.

Doctoral Dissertation: Planar Sobolev extension domains. (Defended on May 5th, 2017)

******

**Lecture Series 22 —— November 18, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1**——Symmetry Results in Two-Dimensional Inequalities for Aharonov–Bohm Magnetic Fields.

**Speaker:** Prof. Ari Laptev,

**Time:** 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **We study functional and spectral properties of perturbations of a magnetic second order differential operator on a circle. This operator appears when considering the restriction to the unit circle of a two dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove some Hardy’s and sharp Keller-Lieb-Thirring inequalities.

**Bio: **Prof. Ari Laptev is Professor at both KTH in Stockholm and Imperial College of London an Researcher at the Faculty of Mathematics and Computer Science at Saint Petersburg State University. Also, he is a member of the Council of Sirius Mathematical Center. He is a worldwide famous expert in Spectral Theory of Partial Differential Equations, in particular, Schrödinger operators; Polya conjecture; Lieb-Thirring inequalities; Trace formulae; Inverse problems; Global solutions of Wave Equations; Generalized Szegö Problems, Pseudodifferential Operators. Ari Laptev graduated from Leningrad State University and obtained in 1978 his PhD under the supervision of Michael Solomyak. In 2001-2003 Prof. Laptev was the President of the Swedish Mathematical Society. In 2007 he received the Royal Society Wolfson Research Merit Award. From 2007 to 2011 Prof Laptev served as the President of European Mathematical Society. He was the Director of Institut Mittag-Leffler between 2011 and 2018. He is Editor-in-Chief of Acta Mathematica, Editor-in-Chief of Arkiv för Matematik, Deputy Chief Editor of the Journal of Spectral Theory, Editor of the Bulletin of Mathematical Sciences, Editor of Problems in Mathematical Analysis, and Editor of Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika.

**Lecture 2**——Microscopic conservation laws for the derivative nonlinear Schrodinger equation

**Speaker:** Guixiang Xu (Beijing Normal University)

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** Using the perturbation determinant introduced by B. Simon, we show the microscopic conservation laws for the Schwarz solutions of the derivative nonlinear Schrodinger equation (DNLS) with small mass, which can be used to show uniformly boundedness and local smoothing estimate of DNLS in the lower regularity space. This is joint work with Xingdong Tang. Lastly, we will show some progresses about DNLS.

**Bio:** Guixiang Xu got Ph. D from the Institute of Applied Physics and Computational Mathematics in China on 2006, and now is a full professor at School of Mathematical Sciences of Beijing Normal Universiity. His research fields are harmonic analysis and partial differential equations, in particular, global well-posedness and scattering theory, stability theory and blow-up dynamics of nonlinear dispersive equations.

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**Lecture Series 21 —— November 04, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——The main cubioid and the modulus of renormalization**

**Speaker: **Vladlen Timorin

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **The main cubioid (CU) is a central part in the parameter space of cubic polynomials of one complex variable viewed as dynamical systems. It plays a similar role to that of the main cardioid in the (quadratic) Mandelbrot set, hence the title. However, in contrast to the main cardioid, topology and combinatorics of the CU are rather involved. We discuss a dynamical characterization of the CU, a proof of which has been recently completed. The last (most recent) ingredient is an upper bound on the moduli of quadratic-like restrictions. This is a joint project with A. Blokh and L. Oversteegen.

**Bio：**Vladlen Timorin is a Professor at the Department of Mathematics of the Higher School of Economics, Moscow. His major research interests include Geometry (convex polytopes, toric varieties, projective differential geometry, classical geometric structures), dynamics (rational functions, surgery, invariant laminations), and quadratic forms. Graduated from the Independent Moscow University (NMU) in 1999 and the Faculty of Mechanics and Mathematics of Moscow State University in 2000 Vladlen defended his Ph.D thesis at the Steklov Mathematical Institute in 2003 and University of Toronto in 2004. In 2012 he defended his doctoral dissertation "Dynamics and geometry of quadratic rational mappings" at the Institute for Information Transmission Problems. Kharkevich RAS. Vladlen Timorin has experience of working at Stony Brook University, Jacobs University in Bremen and Max Plank Institute. Since 2009 he has been working at the Faculty of Mathematics of the National Research University Higher School of Economics. In particular, Prof. Timorin has been Dean of the Department of Mathematics of HSE for several years. He is member of the Editorial Board of Journal of Dynamical and Control Systems; Managing Editor of Arnold Mathematical Journal; member of the Moscow Mathematical Society; permanent professor and board member of the Independent Moscow University.

**Lecture 2——Kahler manifolds with quasi-negative k-Ricci curvature**

**Speaker: **Jianchun Chu

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **In this talk, I will show that a compact Kahler manifold with quasi-negative k-Ricci curvature is projective with canonical line bundle big and nef. This is a joint work with Man-Chun Lee and Luen-Fai Tam.

**Bio: **Dr. Jianchun Chu is currently an assistant professor at Peking University. He received his Ph.D. in mathematics from Peking University in 2017 under the supervision of Professor Gang Tian. His research focuses on differential geometry and partial differential equations.

*****

**Lecture Series 20 — October 21, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——Multiplication Between (Local) Hardy Spaces and Their Dual Spaces**

**Speaker: **Yang Dachun

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **It is well known that bilinear decompositions of products of (local) Hardy spaces and their dual spaces play an important role in the study on various problems from analysis. In this talk, we present some recent progresses on such bilinear decompositions of products of (local) Hardy spaces and their dual spaces. Some open questions are also mentioned in this talk.

**Bio: **Yang Dachun, Professor, School of Mathematical Sciences, Beijing Normal University. Focus on the research of the real-variable theory of function spaces and its applications. Rewarded by National Science Foundation for Distinguished Young Scholars of China in 2005.

**Lecture 2 —— Elementwise sub-representations of groups, and their automorphic analogue.**

**Speaker: **Dipendra Prasad, Prof.

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **Suppose V and W are two finite dimensional representations of a group G such that for each element g of G, the eigenvalues of g (counted with multiplicity) on V are contained in the eigenvalues of g on W. What can we say about V and W? Such a question appears naturally in Automorphic representations which we will briefly discuss, but then focus on the abstract question on group representations.

**Bio: **Professor Dipendra Prasad works at Indian Institute of Technology Bombay (Mumbai, India) having the second position at Saint - Petersburg State University where he heads the Laboratory ‘Modern Algebra and Applications’ at the Department of Mathematics and Computer Science. He obtained his bachelor’s degree from the St. Xavier College, Mumbai in 1978 before moving to the Indian Institute of Technology Kanpur which he completed in 1980. From 1980–1985, Prasad worked as a research scholar at the Tata Institute of Fundamental Research, Mumbai (TIFR Mumbai). He then completed his PhD under the supervision of Benedict Gross at Harvard, in 1989.

Prof. Prasad is a worldwide known expert in Number Theory, Group Theory and Representation Theory (mainly, on Automorphic Representations and the Gan-Gross-Prasad conjecture).

He has got a big number of prestigious awards on national and international levels. In 2018, he was Invited Speaker of the International Congress of Mathematics in Rio de Janeiro.

*****

**Lecture Series 19 — October 7, 2021（19:00-21:00 Beijing time or 14:00-16:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——Norm-resolvent convergence to zero-range models with internal structure in models with strong inhomogeneities**

**Speaker:** Alexander V. Kiselev

**Time: **19:00-20:00 Beijing time (14:00-15:00 St Petersburg time)

**Abstract:**

**Bio: **Dr. Alexander Kiselev, an outstanding expert in Mathematical Physics, whose interested include but not are limited to:

- Functional model for non-selfadjoint linear operators in Hilbert spaces

- Functional analysis;

- Stability Theory and Similarity Theory for non-selfadjoint linear operators;

- Theory of perturbations.

Having graduated from Saint Petersburg State University in 1996 (Faculty of Physics), Dr. Kiselev obtained his PhD degree in 2001. After that he was working in a big number of top-level universities including Dublin Institute of Technology, Universidad Nacional Autónoma de México, University College London, University of Bath, and Lomonosov Moscow State University,

The number of Alexander Kiselev’s collaborators and coauthors is very impressive as well as the number and scientific diversity of his research projects and students.

Now Dr. Kiselev is working at the Department of Mathematics and Computer Science of Saint Petersburg State University.

**Lecture 2 —— Elementwise sub-representations of groups, and their automorphic analogue.**

**Speaker: **Mingying Zhong

**Time:** 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov-Poisson-Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one of the Boltzmann equation. Based on this, we give the optimal time decay rates of solutions to the equilibrium.

**Bio: **Zhong Mingying, Professor, College of Mathematics and Information Science, Guangxi University. Focus on the spectral analysis, Green's function and fluid dynamical limits of the kinetic equations with external force, such as the Vlasov-Poisson(Maxwell)-Boltzmann systems, and so on. Rewarded by National Natural Science Foundation--Outstanding Youth Foundation, 2019.

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**Lecture Series 18 — June 3, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——Nevanlinna factorization in classes of analytic functions smooth up to the boundary.**

**Speaker:** N.A. Shirokov

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **The definition of all objects used in this announcement will be given in the talk. Let an analytical function f belong to the Hardy class Hp in the unit disc D. Then f may be represented as the product, f=IOf, where I is the so-called inner function, it means that |I(z)|<1 for z belonging D, and |I(z)|=1 for almost every z on the unit circle, and the so-called outer function Of is defined by values of |f(z)| on the unit circle. One of those two factors may be absent. The cited statement is the classical result, the factors I and Of are in a sense independent for the functions f from Hp. Let us consider a class X which is contained in H1 and consists of functions f continuous in the closed disc D--. Then f=IOf ,the inner function I is in general discontinuous in D-- what implies that the outer function Of is to compensate the points of discontinuity of the inner function I. The talk is devoted to the concrete way of this compensation and to the specific properties of outer functions belonging to the analytical Holder classes and to the classes of functions of variable smoothness. The consequences about the half-smoothness of an analytical function in comparison with the smoothness of its modulus on the boundary will be given too.

**Bio:** Professor Nikolai A. Shirokov is a worldwide renowned expert in harmonic and complex analysis which includes the theory of approximations, boundary properties of holomorphic functions and the theory of singular integrals. Having graduated from Saint Petersburg State University in 1971, he has got his Ph.D.in 1973, became Doctor of Science (Saint Petersburg department of Steklov Institute) in 1985 and got his Professor degree in 1988. Professor Shirokov is the author of 82 research publications (Scopus) and a supervisor of a big number of students (both graduate and undergraduate). In 2014 he was officially recognized as ‘the best teacher’ of Higher School of Economics which is one of the most prestigious Russian Universities. N.A. Shirokov heads the department of Math.Analysis of Saint Petersburg State University since 2005 and the Department of Applied Mathematics at Higher School of Economics (Saint Petersburg branch) since 2013.

**Lecture 2 —— Factoring Quasiconformal & quasisymmetric mappings**

**Speaker: **Jinsong Liu (Institute of Mathematics, Chinese Academy of Sciences)

**Time: **21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** It follows from the Measurable Riemann Mapping Theorem that we can always present a 2-dimensional quasi-conformal mapping as a composition of quasi-conformal mappings with smaller dilatation. In this talk we will construct n (≥3)-dimensional quasi-conformal homeomorphism between Euclidean spaces which admit no minimal factorization in linear, inner, or outer dilatation. If time permits, I will discuss the composition of quasi-symmetric mappings between metric spaces.

**Bio: **Jinsong Liu is a professor of institute of mathematics, Chinese Academy of Sciences. His research field is complex analysis and Teichmuller theory.

*****

**Lecture Series XVII — May 20, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——Fractional Calculus for m-accretive Operators**

**Speaker: **Dr. Maxim Kukushkin, Moscow State University for Civil Engineering; Institute for Applied Mathematics and Automatization

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **In this report we aim to construct an abstract model of a differential operator with a fractional integro-differential operator composition in final terms, where modeling is understood as an interpretation of concrete differential operators in terms of the infinitesimal generator of a corresponding semigroup. We study such operators as a Kipriyanov operator, Riesz potential, difference operator. Along with this, we consider transforms of m-accretive operators as a generalization, introduce a special operator class and provide a description of its spectral properties.

**Bio: **Maksim Kukushkin is a specialist in boundary problems, fractional derivatives and operator theory being author of a wide range of publications in all these areas. He has got his PhD degree in 2016 and, after the worked at Moscow State University of Civil Engineering, Kabardino-Balkarian Scientific Center and Saint Petersburg State Transport University.

**Lecture 2 —— Cohn-Vossen inequality on certain noncompact Kahler manifolds**

**Speaker: **Prof. Gang Liu, East China Normal University

**Time: **21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **We generalize the Cohn-Vossen inequality to certain noncompact Kahler manifolds. This is related to a conjecture of Yau.

**Bio:**Gang Liu graduated from the University of Minnesota. He was a postdoctoral fellow at the University of California, Berkeley, and an assistant professor at Northwestern University. Now he is a professor at the East China Normal University.

*****

**Lecture Series XVI — May 13, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——Newton’s aerodynamic problem: an overview of recent results and open questions**

**Speaker:** Alexander Plakhov University of Aveiro (Portugal) and Institute for Information Transmission Problems RAS (Moscow)

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:** Newton (1687) posed the problem of finding a convex axisymmetric bodies of the smallest aerodynamic drag. So, we are looking for the optimal curve which is the generatrix of the body. In 1993, a similar problem was formulated in a wider class of convex (not necessarily symmetric) bodies. This task proved to be much more difficult: it is about finding the optimal surface. The talk will provide an overview of recent results and methods used, and besides, we formulate some open questions. А special attention will be paid to the following statement. If all points of an open (in the relative topology) subset of the boundary of some optimal body are regular (that is, C ^ 1-smoothness holds), then this set does not contain any extreme points of the body. This statement is a strengthening of a similar result (Brock, Ferone, Kawohl, 1996), formulated for C ^ 2-smooth subsets.

**Lecture 2 ——Distributed and Secure Algorithm for Dominant SVD**

**Speaker: **Xin Liu

**Time: **21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **We propose and study a distributed and secure algorithm for computing dominant (or truncated) singular value decompositions (SVD) of large and distributed data matrices. We consider the scenario where each node privately holds a subset of columns and only exchanges “safe” information with other nodes in a collaborative effort to calculate a dominant SVD for the whole matrix. In the framework of alternating direction methods of multipliers (ADMM), we propose a novel formulation for building consensus by equalizing subspaces spanned by splitting variables instead of equalizing themselves. This technique greatly relaxes feasibility restrictions and accelerates convergence significantly, while at the same time yielding simple subproblems. We design several algorithmic features, including a low-rank multiplier formula and mechanisms for controlling subproblem solution accuracies, to increase the algorithm's computational efficiency and reduce its communication overhead. More importantly, unlike many existing distributed or parallelized algorithms, our algorithm preserves the privacy of locally-held data; that is, none of the nodes can recover the data stored in another node through information exchanged during communications. We present convergence analysis results, including a worst-case complexity estimate, and extensive experimental results indicating that the proposed algorithm, while safely guarding data privacy, has a strong potential to deliver a cutting-edge performance, especially when communication costs are high.

**Bio:** Dr. Xin Liu, professor of the Academy of Mathematics and Systems Science (AMSS), Chinese Academy Sciences (CAS). He got his bachelor degree from the School of Mathematical Sciences, Peking University in 2004, and PhD from the University of Chinese Academy of Sciences in 2009, under the supervision of Professor Ya-xiang Yuan. His research interests include the optimization problems over the Stiefel manifold, linear and nonlinear eigenvalue problems, nonlinear least squares and distributed optimization. Dr. Xin Liu is the principal investigator of four NSFC (National Science Foundation of China) grants including the Excellent Youth Grant. He was granted the Jingrun Chen Future Star Program from AMSS in 2014, the Science and Technology Award for Youth from The Operations Research Society of China (ORSC) in 2016, and the Fifth Chinese Society for Industrial and Applied Mathematics (CSIAM) Young Scholar Prize in 2020. He serves as an associate editor of “Mathematical Programming Computation”, “Asia-Pacific Journal of Operational Research”, “Journal of Computational Mathematics” and “Operations Research Transactions”.

*****

**Lecture Series XV——April 22, 2021（15:00-17:00 Beijing time or 10:00-12:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1—— POINT PROCESSES AND INTERPOLATION**

**Speaker: **Alexander BUFETOV (CNRS Steklov IITP RAS)

**Time:** 2021-04-22 15:00-16:00 Beijing time or 10:00-11:00 St Petersburg time

**Abstract: **The Kotelnikov theorem recovers a Paley-Wiener function from its restriction onto an arithmetic progression. A Paley-Wiener function can also be recovered from its restriction onto a realization of the sine-process with one particle removed. If no particles are removed, then the possibility of such interpolation for the sine-process is due to Ghosh, for general determinantal point processes governed by orthogonal projections, to Qiu, Shamov and the speaker. If two particles are removed, then there exists a nonzero Paley-Wiener function vanishing at all the remaining particles.

How explicitly to interpolate a function belonging to Hilbert space that admits a reproducing kernel, given the restriction of our function onto a realization of the determinantal pont process governed by the kernel? In the case of the zero set of the Gaussian analytic function, or, in other words, the determinantal point process governed by the Bergman kernel, in joint work with Qiu, the Patterson-Sullivan construction is used for uniform interpolation in dense subspaces of the Bergman space. The invariance of our point process under Lobachevskian isometries plays a key rôle.

For the sine-process, the Ginibre process, the determinantal point process with the Bessel kernel and the determinantal point process with the Airy kernel, A.A. Borichev, A.V. Klimenko and the speaker proved that if the function decays as a sufficiently high negative power of the distance to the origin, then the answer is given by the Lagrange interpolation formula.

**Bio: **Alexander Bufetov is an expert in Ergodic Theory, Probability, Dynamical Systems and Statistics.

He graduated from Moscow State University being student of Acad. Ya. Sinai, one of the worldwide greatest experts in Ergodic Theory. Later on, he has got his PhD from Princeton University. In 2011 he became Doctor of Sciences. In 2015 he won the prestigious Sofia Kovalevskaya price and, thereafter, has got a honorary degree of ‘Professor of Russian Academy of Science’. He is an author of an impressive number of outstanding results https://scholar.google.com/citations?user=nAkXSowAAAAJ&hl=ru.

Prof. Bufetov had a wide range of prestigious positions: at Mathematical Institute of Russian Academy of Sciences, Higher School of Economics, Rice Univeristy, Chebyshev Laboratory at Saint Petersburg State University etc. Now he is working at University Aix-Marseille at CNRS Director position.

**Lecture 2—— Differential Network Analysis via Lasso Penalized D-Trace Loss**

**Speaker:** Ruibin Xi (Peking University)

**Time:** 2021-04-22 16:00-17:00 Beijing time or 11:00-12:00 St Petersburg time

**Abstract: **Biological networks often change under different environmental and genetic conditions. In this paper, we model the network change as the difference of two precision matrices and propose a novel loss function called the D-trace loss, which allows us to directly estimate the precision matrix difference without attempting to estimate precision matrices. Under a new irrepresentability condition, we show that the D-trace loss function with the lasso penalty can give consistent estimators in high-dimensional settings if the difference network is sparse. A very efficient algorithm is developed based on the alternating direction method of multipliers to minimize the penalized loss function. Simulation studies and a real data analysis show that the proposed method outperforms other methods.

**Bio: **

Dr. Ruibin Xi is an associate professor at School of Mathematical Sciences, Peking University. He obtained his PhD from Washington University in St. Louis and received post doc training at Harvard Medical School. His main research interests include statistical analysis of big biological data, cancer genomics, network and graphical models, Bayesian analysis and high-dimensional statistics.

*****

**Lecture Series XIV——April 8, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1——Transcendence Theory over Function Fields on Quotients of Bounded Symmetric Domains**

**Speaker: **Ngaiming Mok (The University of Hong Kong)

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **Finite-volume quotients of bounded symmetric domains Ω, which are naturally quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic Geometry, Arithmetic Geometry and Number Theory, and an important topic revolves around functional transcendence in relation to universal covering maps of such varieties. While a lot has already been achieved in the case of Shimura varieties by means of methods of Model Theory, Hodge Theory and Complex Differential Geometry, techniques for the general case of not necessarily arithmetic quotients Ω/Γ =: XΓ have just begun to be developed. For instance, Ax-type problems for subvarieties of products of arbitrary compact Riemann surfaces of genus ≥ 2 have hitherto been intractable by existing methods. We will explain how uniformization theorems for bi-algebraic varieties can be proven by analytic methods involving the Poincar´e-Lelong equation in the cocompact case (joint work with S.-T. Chan), generalizing in the absence of the emisimplicity theorem of Andr´e-Deligne for monodromy groups (proven for arithmetic lattices). Klingler-Ullmo-Yafaev (2016) resolved the hyperbolic Ax-Lindemann Conjecture for Shimura varieties in the affirmative ascertaining that the Zariski closure of the image π(S) of an algebraic subset S ⊂ Ω under the universal covering map π : Ω → XΓ is totally geodesic. I will explain how the arithmeticity condition can be dropped in the cocompact case by a completely different proof using foliation theory, Chow schemes, partial Cayley transforms and K¨ahler geometry.

**Bio: **莫毅明教授为香港大学明德教授与讲座教授，自1999年始兼任数学研究所所长。莫毅明1980年在斯坦福大学获得数学系哲学博士学位，旋即在普林斯顿大学开展其职业生涯，历任美国哥伦比亚大学正教授与法国巴黎大学（奥赛）正教授，1994年回香港任职香港大学数学系讲座教授。1984年莫毅明获美国斯隆研究基金，1985年获美国总统年青研究人员奖，1998年获香港裘槎优秀科研者奖，2007年获国家自然科学奖二等奖， 2009年获美国数学会伯格曼奖(Bergman Prize).莫毅明为美国数学会会士。莫毅明自1992年起担任“数学年鉴(Mathematische Annalen)”编辑委员，并于2002至2014年期间担任 “数学发明(Inventiones Mathematicae)”编辑委员。莫毅明1994年获邀在苏黎世于国际数学大会(ICM)做学术演讲，并获委任为ICM 2010(海得拉巴)的菲尔兹奖选委。2015莫毅明获选中国科学院院士与香港科学院院士。

**Lecture 2 ——A New Life of the Old Sieve**

**Speaker: **Yuri Matiyasevich

**Time: **21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **Prime numbers are one of the most important objects of study in Number Theory. Greek mathematician Eratosthenes (276--194 B.C.) invented a method (known nowadays under his name) for revealing primes among all natural numbers. A more modern and powerful for tool for studying prime numbers is Riemann's zeta function. In recent years the speaker performed intensive computer calculations in the search for new properties of this function. Unexpectedly, the Sieve of Eratosthenes naturally appeared in one calculation with the non-trivial zeros of the zeta function. This form of the sieve demonstrates a rich fractal structure lacking in the original sieve. Later another calculation revealed a different kind of a sieve which is dual to the classical Sieve of Eratosthenes. So far no theoretical explanation was found for the observed phenomena. Calculation were not an easy computational task. They required calculations of several thousands of initial non-trivial zeros of the zeta function with several thousands decimal digits, and solving systems consisting of several thousands linear equations.

**Bio: **马蒂亚舍维奇教授是当代俄罗斯最著名的数学家之一，圣彼得堡数学学会主席。1966年，19岁的他在逻辑学方面取得了一些突破性的成果，应邀在当年莫斯科举行的国际数学家大会上作邀请报告。1970年，尤里-马蒂亚舍维奇解决了希尔伯特第十问题。到目前为止，他在其他经典领域，如四种颜色问题和黎曼问题上取得了广泛的突破性成果。 他获得了许多著名的奖项，包括洪堡奖、苏联科学院的Markov奖等。他曾获得奥弗涅大学克莱蒙费朗分校和皮埃尔和玛丽居里大学的荣誉学位。Matiyasevich教授是AMS、巴伐利亚科学院和许多其他科学团体的成员。Yuri Vladimirivich Matiyasevich自1970年起在圣彼得堡的斯捷科洛夫研究所工作，1995年成为圣彼得堡国立大学教授。 2002年起，他担任圣彼得堡市数学奥林匹克竞赛负责人。2008年，Matiyasevich教授当选为俄罗斯科学院正式院士。Matiyasevich院士培养了一大批优秀的学生，其中有多位教授和院士。

*****

**Lecture Series XIII——March 25, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

**To Join Zoom Meeting:**

Valid Until：2026-04-30 23:59

**Lecture 1—— Polynomial complexity and Sarnak conjecture**

**Speaker: **Prof. Wen Huang, University of Science and Technology of China(USTC)

**Time:** 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:**We will review some recent progresses in Sarnak conjecture related to complexity. In particular, we will investigate the relation between {0,1}- sequences with polynomial mean complexity and the Logarithmic Sarnak conjecture.

**Bio:** Huang Wen received his Ph.D. from the Department of Mathematics, USTC in 2003. His research field is topological dynamical system and ergodic theory, related to the entropy and chaos theory, multipleergodic Theorem, Sarnak’s conjecture and so on. He and his collaborators proved the following results: 1) positive entropy implies weak horseshoes. 2) pointwise multiple ergodic theorem holds for distal measure-preserving systems. 3) Sarnak’s conjecture holds for dynamical systems with sub-polynomial measure complexity. He achieved many rewards: 2012 China National Science Funds for Distinguished Young Scientists; 2018 National Ten Thousand Talent Program Leading Scientists, P.R.China; 2018 Second Class National Natural Science Award, P.R.China（rank 2）.

**Lecture 2 ——Dynamical models of some sociological problems.**

**Speaker: **Sergei Yu. Pilyugin, Faculty of Mathematics and Computer Science,St. Petersburg State University.

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** We study a dynamical system modeling an iterative process of choice in a group of agents between two possible results. The studied model is based on the principle of bounded confidence introduced by Hegselmann and Krause. According to this principle, at each step of the process, any agent changes his/her opinion being influenced by agents with close opinions. The resulting dynamical system is nonlinear and discontinuous.

We study both cases of finite and infinite groups of agents. We are mostly interested in the structure and stability of fixed points of the system and in conditions under which any positive trajectory tends to a fixed point.

**Bio: **Sergei Yu. Pilyugin is Former vice President of Saint Petersburg Mathematical Society, he is a professor at Faculty of Mathematics and Computer Science, St. Petersburg State University, Russia. His research interests are in dynamical systems (especially theory of attractors and shadowing theory) and in applications (including sociological models).

*****

**Lecture Series Ⅻ —— March 11, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

**To Join Zoom Meeting: **

Valid Until：2026-04-30 23:59

**Lecture 1—— The J-equation and the deformed Hermitian-Yang-Mills equation.**

**Speaker: **Prof. Gao Chen, University of Science and Technology of China.

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **The deformed Hermitian-Yang-Mills (dHYM) equation is the mirror equation for the special Lagrangian equation. The "small radius limit" of the dHYM equation is the J-equation, which is closely related to the constant scalar curvature K\"ahler (cscK) metrics. In this talk, I will explain my recent result that the solvability of the J-equation is equivalent to a notion of stability. I will also explain my similar result on the supercritical dHYM equation.

**Bio:**Chen, Gao got Bachelor's degree in Univeristy of Science and Techonology of China in 2012 and Ph.D. in Stony Brook University in 2017. Since then he has worked in IAS and University of Wisconsin-Madison. From January of 2021 he became a tenure-track professor in USTC.

**Lecture 2 ——Various types of spectra and spectral measures on Cayley graphs of finitely generated groups and their actions.**

**Speaker: **Tatiana Nagnibeda (University of Geneva)

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** We will be interested in the Laplacian on graphs associated with infinite finitely generated groups: Cayley graphs and more generally, Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [0,2], but not much more can be said about it in general. Little is known about the associated spectral measures either. We will discuss various spectral problems arising in this context and stemming from the famous question “Can one hear the shape of a drum?”

*****

**Lecture Series Ⅺ—— January 28, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1—— On local combinatorial formulas for Euler class of spherical fiber bundle. **

**Speaker: **Dr. Nikolai Mnev, Senior Research Fellow of Chebyshev Laboratory at Saint-Petersburg State University and Saint Petersburg Department of Steklov Mathematical Institute

**Time:** 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:** We will discuss the classical problem on local combinatorial formulas of characteristic classes on example of Euler class. Suppose we have a PL spherical fiber bundle with a fiber S^n triangulated over the base simplicial complex. The bundle determines n+1 dimensional Euler characteristic class in the base. Local combinatorial formula for the Euler class is a universal combinatorial function of elementary triangulated S^n-bundles over n+1 simplices universally representing Euler cocycle of the bundle in simplicial cohomology of the base. Such functions exist for rational coefficients in cohomology. They can be constructed as "twisting cochains" -- explicit local chain-level formulas for Gysin homomorphism in the Gysin sequence of the bundle. To get an access to local chain combinatorics of spectral sequence of the bundle we may use Guy Hirsh homology model of the bundle as a local system and then applying homology perturbation theory obtain local formulas as certain measure of twisting in combinatorial Hodge structure of the elementary bundle. The answer can be interpreted and evaluated statistically as certain combinatorial counting using Catanzaro-Chernyak-Klein higher Kirchhoff theorems. The formulas are resulting in a combinatorial form of Gauss-Bonne theorem. For example we easily obtain otherwise difficult to access statement: One can triangulate only trivial and Hopf circle bundles over a 2-dimensional sphere if the base sphere is triangulated as the boundary of 3-simplex.

**Bio: **Dr. Nikiolay Mnev graduated from the faculty of Mathematics and Mechanics of Saint Petersburg State University in 1980. In 1986 he defended his PhD thesis under supervision of Prof. Anatoly Vershik. Since that time he works at Steklov Institute and, besides, he has got a position at Chebyshev Laboratory starting from its foundation in 2011. Additionally, Dr. Mnev curates the so-called Fizmatklub, the organization intended to boost the level of teaching maths in Saint Petersburg Universities by organizing additional lecture courses, delivered by leading specialists.

Dr. Nikolai Mnev’s research interests cover Algebraic Topology, Geometric Topology and Combinatorial Geometry. In 1991, he received Delbert Ray Fulkerson prize for outstanding papers in the area of discrete mathematics.

**Lecture 2 —— Recent developments in exact Lagrangian fillings.**

**Speaker:** Honghao Gao, Michigan State University

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** Legendrian knots and their exact Lagrangian fillings are important geometric objects to study in low dimensional contact and symplectic topology. It was not known whether there exists a Legendrian knot that admits infinitely many exact Lagrangian fillings. In 2020, this statement was proven affirmatively, and infinitely many Lagrangian fillings have been constructed for all torus links of infinite type (with Casals), a family of positive braid links (Casals-Zaslow), all positive braid links of infinite type (with Shen-Weng), two examples that are not positive braid links (Casals-Ng). In this talk, I will review these results, and explain the construction for the torus (3,6) link appeared in the work with R. Casals.

**Bio: **Honghao Gao received his PhD degree from Northwestern University in 2017, under the supervision of Eric Zaslow. He was previously a postdoc at Institut Fourier, and is currently a postdoc at Michigan State University. His research interest is in contact and symplectic topology.

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**Lecture Series Ⅹ—— January 14, 2021（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Valid Until：2026-04-30 23:59

**Lecture 1—— The Problem of Changing The Dimension in Tasks of Constructing Optimal Designs**

**Speaker:**Dr. Petr. Shpilev

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:** The lecture is based on a joint talk with Prof. Vyacheslav Melas. Within the framework of the report, a brief overview of the history of the development of the theory of optimal experiment designs will be given, the basic concepts, definitions, and some key results of this theory are considered.The problem of changing the dimension of the initial optimization task and some approaches to the solution of this problem will be considered on the examples of tasks of constructing specific optimal designs studied by the author of the report.

**Bio: **Dr. Shpilev graduate from the Faculty of Mathematics and Mechanics in 2004 and got his PhD degree in 2007. His research interests include: Optimal Experimental Design Theory, Regression Analysis, Statistics, Mathematical Simulation, Data Analysis, Computer Science, and Approximation Theory.

**Lecture 2 —— Exploring Stochastic Methods For Deep Learning and Reinforcement Learning**

**Speaker: **Zaiwen Wen, Beijing International Center for Mathematical Research.

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** Stochastic methods are widely used in machine learning. In this talk, we present a structured stochastic quasi-Newton method and a sketchy empirical natural gradient method for deep learning. We also introduce a stochastic quadratic penalty algorithm for reinforcement learning.

**Bio:** Wen's research interests include large-scale computational optimization and their applications in data sciences. Together with his coauthors, he has developed both deterministic and stochastic semi-smooth Newton algorithms for composite convex program and Newton type algorithms for Riemannian optimization, as well as academic software packages such as SSNSDP, ARNT, Arrabit, LMSVD and LMAFIT, etc. He was awarded the Science and Technology Award for Chinese Youth in 2016, and the Beijing Science and Technology Prize-Outstanding Youth Scholar Zhongguan Village Prize in 2020. He is an associate editor of Journal of the Operations Research Society of China, Journal of Computational Mathematics and a technical editor of Mathematical Programming Computation.

*****

**Lecture Series Ⅸ—— December 17th, 2020（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Video: https://disk.pku.edu.cn:443/link/A59A44F4FDF41825DCBF8AE9578C8D2B

Valid Until: 2025-01-01

**Lecture 1——Hierarchical Behavior of Solutions to the Maryland Equation in the Semiclassical Approximation.**

**Speaker:** Prof. Alexander Fedotov, Department of Mathematics and Mathematical Physics of the Physics Faculty of Saint Petersburg State University.

**Time: **20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **We describe a multiscale selfsimilar struture of solutions to one of the most popular models of the almost periodic operator theory, the difference Schroedinger equation with a potential of the form a ctg(b n+c), where a, b and c are constants, and n is an integer variable. The talk is based on a joint work with F.Klopp.

**Research interests:** Asymptotic methods of mathematical physics (quasi-classical, short-wave and adiabatic asymptotics); Spectral theory of ergodic Schr\»odinger operators; Analytic theory of difference equations on the complex plane.

**Bio：**Professor Fedotov was an invited speaker at several prestigious conferences including XII International Congress on mathematical Physics (Brisbane, Australy) and 20th European Math. Congress, Minisimposium “Almost periodic equations”(Amsterdam, Holland). He was invited to several famous universities and math centers in France, Germany, Sweden, Canada, Austria and other countries.

**Lecture 2 ——Vanishing dissipation limit of planar wave patterns to the multi-dimensional compressible Navier-Stokes equations.**

**Speaker: **Prof. Yi Wang, Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences of Chinese Academy of Sciences, Beijing

**Time: **21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **The talk is concerned with our recent results on the vanishing viscosities limit of planar rarefaction wave to both 2D compressible isentropic Navier-Stokes equations and 3D full compressible Navier-Stokes equations and the vanishing dissipation limit of planar contact discontinuity to 3D full compressible Navier-Stokes equations. Remark that the planar shock wave is non-unique and the planar rarefaction wave is unique in the class of entropic solutions to 3D compressible Euler equations and whether the planar contact discontinuity is unique or not for entropic weak solutions is still open to 3D compressible Euler equations. And our vanishing dissipation limit for planar contact discontinuity, in particular, impies the positive answer to the uniqueness of a planar contact discontinuity for 3D compressible Euler equations in the class of zero dissipation limit of full compressible Navier-Stokes equations.

**Research areas:** PDEs of fluid mechanics and from other applied sciences, including the compressible Navier-Stokes and Euler equations, the mathematical theory of viscous/inviscid systems of conservation laws, kinetic equations, and other related fluid mechanic equations.

**Honors: ** won the National Science Fund for Excellent Young Scholars in 2013 and the national youth talent support program in 2015.

*****

**Lecture Series Ⅷ ——December 3rd, 2020（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Video: https://disk.pku.edu.cn:443/link/51D4B1FC28EA321B4C4BC25FEF15C28F

Valid Until: 2025-01-01

**Lecture 1——Graph-walking automata.**

**Speaker:Prof. Alexander Okhotin.**

**Time: 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time) .**

**Abstract:** Graph-walking automata are a model studied in theoretical computer science: they traverse an undirected graph by following its edges, and use a memory of constant size to plan their movements. Graph-walking automata can be regarded both as a model of a robot navigating an unknown environment, and as a generic model of computation, with the graph modelling its memory. This paper presents the known results on these automata, ranging from their limitations in traversing graphs, studied already in the 1970s, to the later work on the logical reversibility of their computations, including the most recent lower bounds on their size.

**Bio:**Alexander Okhotin (Ph.D. 2004, Queen's University) is a professor of theoretical computer science at St. Petersburg State University. His main research subjects are formal grammars, finite automata and their complexity.

**Lecture 2 ——Modeling and Verification of Concurrent and Distributed Systems: From Reo to Mediator.**

**Speaker: Prof. Meng Sun.**

**Time: **21:00-22:00 Beijing time (16:00-17:00 St Petersburg time) .

**Abstract:** In this talk I will introduce the channel-based coordination language Reo and the component-based modeling language Mediator, and show how to formalize and verify component-based concurrent and distributed system models in them. Reo provides a channel-based model which focuses on complex interactions among system components. Mediator supports a two-step hierarchical modeling approach: Automata, which provide an interface of ports, are the basic behavior units; Systems declare components or connectors through automata, and glue them together. With the help of Reo and Mediator, both components and systems can be modeled separately and precisely.

**Bio:**Meng Sun received his BSc and PhD degrees in applied mathematics from Peking University, in 1999 and 2005, respectively. He then spent one year as a postdoctoral researcher in National University of Singapore. From 2006 to 2010, he worked as a scientific staff member at CWI, the Netherlands. He has been a faculty member at Peking University since 2010 and became a full professor in 2017. Currently, his research interests mainly lie in software theory and formal methods. His recent work includes coordination models and languages, coalgebra theory, model checking, theorem proving, software testing, cyber-physical systems, service-oriented and cloud computing, modeling and verification of blockchain and smart contracts, big data analysis, theoretical foundations of machine learning, deep learning and their application in formal verification.

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**Lecture Series Ⅶ ——November 19th, 2020（20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Video: https://disk.pku.edu.cn:443/link/B06B1459AA07B80DD51E17F8406655C2

Valid Until: 2025-01-01

**Lecture 1——Deep CT Imaging by Unrolled Dynamics**

**Speaker:** Bin Dong, Beijing International Center for Mathematical Research, Peking University

**Time:** 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **In this talk, I will start with a brief review of the dynamics and optimal control perspective on deep learning (including supervised learning, reinforcement learning, and meta-learning), especially the so-called unrolled dynamics approach and its applications in medical imaging. Then, I will present some of our recent studies on how this new approach may help us to advance CT imaging. Specifically, I will focus on our thoughts on how to combine the wisdom from mathematical modeling with ideas from deep learning. Such combination leads to new data-driven image reconstruction models and new data-driven scanning strategies for CT imaging, and with a potential to be generalized to other imaging modalities.

**Bio:**Bin Dong received his B.S. from Peking University in 2003, M.Sc from the National University of Singapore in 2005, and Ph.D from the University of California Los Angeles (UCLA) in 2009. Then he spent 2 years in the University of California San Diego (UCSD) as a visiting assistant professor. He was a tenure-track assistant professor at the University of Arizona since 2011 and joined Peking University as an associate professor in 2014. His research interest is in mathematical modeling and computations in imaging and data science. A special feature of his research is blending different branches in mathematics which include: bridging wavelet frame theory, variational techniques, and nonlinear PDEs; bridging differential equations and optimal control with deep learning.

**Lecture 2 —— Gaussian random fields in machine learning**

**Speaker:** Viacheslav Borovitskiy, Department of Mathematics and Mechanics，Saint Petersburg State University.

**Time:** 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **I will introduce the Gaussian process regression algorithm, a widely used approach to probabilistic modeling in machine learning, and talk about its applications and challenges associated with its use. In the end, I will mention our own recent research in this topic that was presented on International Conference on Machine Learning and will be presented on this year's conference on Neural Information Processing Systems.

**Bio:** He has graduated from Saint-Petersburg State University, Faculty of Mathematics and Mechanics. His main interests cover PDEs, Banach spaces, Satistics and Informatics.

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**Lecture Series VI - November 5th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Video: https://disk.pku.edu.cn:443/link/E68335CAFDAD6A71A943A348EF8FC42A

Valid Until: 2025-01-01

**Lecture 1——Heights and separation of characters of finite groups**

**Speaker:** Prof. Yanjun Liu, Jiangxi Normal University

**Time: **2020-11-05 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **The question of which prime powers can occur as divisors of irreducible character degrees of finite groups has a long history. One related conjecture is given by Geoffrey Robinson in 1996, who conjectured that the p-part of character degrees in a p-block of a finite group can be bounded in terms of the center of a defect group of the block. I will mention recent progress on Robinson's conjecture for odd primes,and then turn to the question of when a p-block of a finite group G is also a q-block of G. A series related work have been done by Navarro-Willems,Bessenrodt-Mall-Olsson, Navarro-Turull-Wolf and etc. The block separation property is studied by Bessenrodt-Zhang and they proved that the nilpotency, p-nilpotency of a finite group can be characterized by intersections of principal blocks of some (quotient) groups. Thus it is natural to ask if the solvability or p-solvability of a finite group can also be characterized in this way. By introducing the so-called block graph of a finite group this problem was solved affirmatively. Finally, I will talk about a conjecture relating the trivial intersection of principal blocks to the existence of nilpotent Hall subgroups.

**Bio:** Dr. Yanjun Liu got Ph. D. from Peking University and is now working in Jiangxi Normal University. Recently He got Humboldt Research Fellowship based on his excellent work in some canonical conjecture of representation theory.

**Lecture 2——Basic topology in terms of simplicial sets with a notion of smallness.**

**Speaker:** Prof. Mikhail Gavrilovich, Saint Petersburg Department of Higher School of Economics

**Time: **2020-11-05 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** We consider simplicial sets equipped with a notion of smallness, and observe that this slight “topological” extension of the “algebraic” simplicial language allows a concise reformulation of a number of classical notions in topology, e.g. continuity, limit of a map or a sequence along a filter, various notions of equicontinuity and uniformconvergence of a sequence of functions; completeness and compactness; in algebraic topology, locally trivial bundles as a direct product after base-change and geometric realisation as a space of discontinuous paths. These reformulations are elementary and can perhaps be used in teaching to give motivated examples of elementary concepts in category theory. Surprisingly, this category is not well-studied and thus these observations raise many easy but open problems, which we like to think are in line with goals of tame topology put by Grothendieck. In the talk, we will work through of a couple of example, briefly mention some others, and indicate a number of open problems, who we like to think are in line with the goals of tame topology put by Grothendieck.

We will preceed the main part of the talk by explaning a category-theoretic characterisaiton of finite solvable and nilpotent groups in terms of the Quillen lifting property, which is also used in reformulations of the notions of limit, compactness, and completeness, and others.

This talk is based on preliminary notes

http://mishap.sdf.org/by:gavrilovich/treplo-groups.pdf.

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**Lecture Series V - October 22th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time).**

Video：https://disk.pku.edu.cn:443/link/FD9BF2C2F871B7FE56EFD891777A6E7E

Valid Until：2025-01-01 00：00

**Lecture 1—— Global existence and decay of solutions to Prandtl system with small analytic data**

**Speaker:** Prof. Ping Zhang, Mathematics Institute of Academy of Mathematics and Systems Science of Chinese Academy of Sciences, Beijing

**Time：**2020-10-22 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstarct:** In this talk, we shall prove the global existence and the large time decay estimate of solutions to Prandtl system with small initial data, which is analytical in the tangential variable. The key ingredient used in the proof is to derive sufficiently fast decay-in-time estimate of some weighted analytic energy estimate to a quantity, which consists of a linear combination of the tangential velocity with its primitive one, and which basically controls the evolution of the analytical radius to the solutions. Our result can be viewed as a global-in-time Cauchy-Kowalevsakya result for Prandtl system with small analytical data, which in particular improves the previous result in \cite{IV16} concerning the almost global well-posedness of two-dimensional Prandtl system. In the last part, I shall also mention our recent result on the global well-posedness of this system with optimal Gevrey data. (This is partially joint work with Marius Paicu, Chao Wang and Yuxi Wang).

**Bio：**Director of Mathematics Institute of Academy of Mathematics and Systems Science of Chinese Academy of Sciences; B. S., Department of Mathematics of Nanjing University, China, 1991；Ph. D. ，Nanjing University, China, 1997. Research interests: Navier-Stokes equations and semi-classical analysis.

**Lecture 2 —— When Hopf’s lemma remains valid?**

**Speaker: **Apushkinskaya Darya, St. Petersburg State University; Peoples’ Friendship University of Russia, Moscow; Saarland University, Saarbruecken

**Time：**2020-10-22 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:** The Hopf lemma, known also as the“boundary point principle”, is one of the important tools in qualitative analysis of partial differential equations. This lemma states that a supersolution of a partial differential equation with a minimum value at a boundary point, must increase linearly away from its boundary minimum provided the boundary is smooth enougn. For general operators of non-divergence type with bounded measurable coefficients this result was established in elliptic case independently by E. Hopf and O. Oleinik (1952) and in parabolic case by L. Nirenberg (1953). The first result for elliptic equations with divergence structure was proved by R. Finn and D. Gilbarg (1957). Later the efforts of many mathematicians were aimed at the extension of the classes of admissible opeartors and at the reduction of the boundary smoothness. We present several versions of the Hopf lemma for general elliptic and parabolic equations in divergence and non-divergence forms under the sharp requirements on the coefficients of equations and on the boundary of a domain. Also we provide a new sharp counterexample. The talk is based on results obtained in collaboration with Alexander Nazarov.

**Bio:** Graduated from Faculty of Mathematics and Mechanics of Leningrad State University (LGU) in 1990 (department of mathematical physics). Ph. D. thesis (advisor prof. N. N. Uraltseva) was defended in St. Petersburg State University in 1993. Research interests: Differential equations, dynamical systems, and optimal control.

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__Lecture Series IV ____- July 14th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）__

Video： Part I：https://disk.pku.edu.cn:443/link/ADEA1AC5CB6C898F8809E6FFBC93B54C

Part II：https://disk.pku.edu.cn:443/link/8395E85B327FAC1E80993F79F2CACCF2

Valid Until：2025-01-01 00：00

**Lecture 1 - WORD MAPS ON SIMPLE ALGEBRAIC GROUPS AND RELATED TOPICS**

**Speaker:** Prof. Nikolai Gordeev, Saint Petersburg State Pedagogical University

**Time：**2020-07-14 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

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**Lecture 2 - Structure of Commutator Subgroups**

**Speaker:** Prof. Zuhong Zhang， Beijing Institute of Technology

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**Time：**2020-07-14 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **

This talk is based on the joint works with R. Hazrat, A. Stepanov and N. Vavilov in the past decade. In his seminal paper, more than half a century ago, Hyman Bass initialed the study of commutator subgroups and commutator formulas over rings. Since then, it attracted great attend of many leading experts including A. Bak, A.A. Suslin, L.N. Vaserstein, etc.. Various commutator formulas have been obtained in stable and non-stable settings and for a range of classical and algebraic like-groups.

In this talk, we will describe some recent results on the study (higher/birelative) commutators in general linear groups $GL(n,A)$ as well as their elementary generators. we will also discuss some further related research and applications.

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__Lecture Series III ____- June 30th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）__

Valid Until: 2025-08-31 23:59

**Lecture 1 - Dynamics in the space of metrics and new invariants in ergodic theory**

**Speaker: **Prof. Anatoly Vershik, St. Petersburg Department of Steklov Institute of Mathematics, Russian Academy of Sciences

**Time：**2020-06-30 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract:**

1. Admissible metrics on the measure space: new trend in the theory of mm spaces.

2. Classification of mm-spaces. Matrix distributions.

3.Thе actions of measure preserving groups in the space of admissible metrics in measure space. ($\ca M \subset L^1(X\otimes X, \mu \otimes \mu).

4.Average metrics, ergodic limit and asymptotic invariants,

5.Sclaing entropy. Scaling entropy function. Examples.

6.Theorem. Bounded scaling entropy and discrete spectra. Sequential entropy by Kushnirenko. All possible scalings.

7.New geometrical problems.

**Bio:**

Prof. Vershik is the Head of the Laboratory of Representation Theory and Dynamical Systems at Saint Petersburg Department of Steklov Mathematical Institute and professor at Saint Petersburg State University. He was President of the Saint Petersburg Society from 1998 to 2008. He is a member of European Academy of Sciences (since 2015), he was a member of Executive Committee of European Mathematical Society (1996-2000), Laureate of Humboldt Research Award - 2007 and an ICM invited speaker (1974 and 1994), Miller-professor (Berkeley, 1995) and Simons-professor (MSRI, 2008). His research interests include but are not limited to

- infinite-dimensional groups and their Asymptotic Representation Theory;

- Lie groups;

- New methods of Representation Theory for finite symmetric groups;

- Combinatorial Probability Theory and limit forms for configurations (the Vershik-Kerov theorem on limit forms of Young diagrams, Bratelli-Vershik diagrams, etc) ;

- Universal objects in Combinatorics Geometry and Dynamical Systems;

- Dynamical Systems and Ergodic Theory;

- Non-holonomic Geometry and Mechanics;

- Random processes, random walks and random matrices;

- Optimization.

**Download **Slides:

Valid Until: 2025-07-31 23:59

**Lecture 2 - The characteristic factors in dynamical systems**

**Speaker**: Prof. Ye Xiangdong, University of Science and Technology of China

**Time：**2020-06-30 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract: **

It is an ideal situation if we can reduce a problem P stated for general ergodic or minimal systems to the same problem in their "simple factors". We will explain how one did so for the problem on the convergence of multiple ergodic averages in ergodic theory. Moreover, we will present a recent work by Glasner-Huang -Shao-Weiss-Ye on the similar problem in the topological setup.

On the way to do so, we will address the parallels between topological dynamics and ergodic theory, and their applications to combinatoric number theory.

**Bio:**

Prof. Ye got his Ph. D. in Mechanics and Mathematics Department of Moscow State University in 1991, and then did postdoc work in ICTP during 1991-1993. He was a faculty of Mathematics School in USTC since 1993. He was selected as a member of the Chinese Academy Sinica in 2019. His research interests include topological dynamics, ergodic theory and combinatoric number theory.

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__Lecture Series II ____- June 16th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）__

Video：Part I：https://disk.pku.edu.cn:443/link/F3034E87D1FC25E824BBDB14302AB24E

Part II：https://disk.pku.edu.cn:443/link/E55DFDEE3CA697F5C15F978B04ABC727

Valid Until：2025-01-01 00：00

**Lecture 1 - Multiple structures for quasilinear equations by the variational method**

**Speaker: **Alexander Nazarov, PDMI RAS and Math&Mech Faculty, St. Petersburg State University

**Time：**2020-06-16 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **

We study entire bounded solutions to the equations of variational nature. The model example here is $\Delta u - u + u^3 = 0$ in $\mathbb R^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in an unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.),both positive and sign-changing. It is also applicable for more general equations in any dimension.

The talk is based on the joint paper Lerman L.M., Naryshkin P.E., Nazarov A.I., Abundance of entire solutions to nonlinear elliptic equations by the variational method, Nonlinear Analysis -- TMA. 190 (2020), DOI 10.1016/j.na.2019.111590, 1-21.

**Lecture 2 - Periodic and quasi-periodic solutions of 1-d Q-curvature equation. **

**Speaker:** Jiang Meiyue, School of Mathematical Sciences, Peking University

**Time：**2020-06-16 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract（/upload/editor/file/20200609/09145940157.pdf）: **

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__Lecture Series I - June 2nd, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）__

**Lecture 1 - Spectral synthesis for systems of exponentials and reproducing kernels **

**Speaker：**Anton Baranov (Saint Petersburg State University)

**Time：**2020-06-02 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)

**Abstract: **

**Let $x_n$ be a complete and minimal system of vectors in a Hilbert space $H$. We say that this system is hereditarily complete or admits spectral synthesis if any vector in $H$ can be approximated in the norm by linear combinations of partial sums of the Fourier series with respect to $x_n$. It was a long-standing problem whether any complete and minimal system of exponentials in $L^2(-a,a)$ admits spectral synthesis. Several years ago Yu. Belov, A. Borichev and myself gave a negative answer to this question which implies, in particular, that there exist non-harmonic Fourier series which do not admit a linear summation method. At the same time we showed that any exponential system admits the synthesis up to a one-dimensional defect. In the talk we will also discuss related problems for systems of reproducing kernels in Hilbert spaces of entire functions (such as Paley-Wiener or Fock).**

**Lecture 2 - On gauged linear sigma model and related problems **

**Speaker: **Prof. Huijun Fan (Peking University)

**Time：**2020-06-02 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)

**Abstract:**

**Gauged linear sigma model was proposed by Witten in the early of 90's to explain the mirror symmetry phenomenon and the CY/LG correspondence conjecture. In this lecture, I will firstly formulate the mathematical framework of the GLSM, and then describe an algebraic way to construct the quantum invariants of GLSM in narrow case (for general gauge group) via quasimaps. **

**This was a joint work with Jarvis and Ruan. Finally I will report the recent progress in this field and related problems. **