##### Description

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

To Join Zoom Meeting: https://zoom.com.cn/j/64207771930?pwd=ejlzYXcramtJU1RrbmdHUWZsRXZWUT09

Meeting ID: 642 0777 1930

Password: 992829

**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。

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

**To Join Zoom Meeting:**

https://zoom.com.cn/j/64856024353?pwd=MXEyZ3Q3OUJRUDhtc2U1a0hsZ2FEdz09

**Meeting ID: **648 5602 4353

**Password:** 479189

**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).

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

**To Join Zoom Meeting: **

https://zoom.com.cn/j/66506917258?pwd=NkNHZnFaaUlXTFlONXU3dGR2aWNEdz09

**Meeting ID: **665 0691 7258

**Password:** 841830

**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?”

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

**To Join Zoom Meeting:**

https://zoom.com.cn/j/64514151267?pwd=Q3F2NUZ2REdvemNaS3g1Snh4L2JHUT09

**Meeting ID：**645 1415 1267

**Password：**668479

**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).**

**To Join Zoom Meeting:https://zoom.com.cn/j/68067087375?pwd=MnlxYU1yV05BTnVrUzRQOHF6YUVBQT09**

**Meeting ID：**680 6708 7375

**Password：**313186

**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.

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**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**

**Video 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.

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**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**

**Video 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**

**Video 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**

**Video 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/Skorokhod_Geometric_Realisation_HomSets.pdf,

http://mishap.sdf.org/6a6ywke/6a6ywke.pdf,

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

**To Join Zoom Meeting**

https://zoom.com.cn/j/64194386353?pwd=UWNuaDdncWZSb0FtTzQ3KzRaK1ZvZz09

**Meeting ID：**641 9438 6353

**Password：**212480

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

**Speaker:** Prof. Zhang,Ping, 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.

Website: http://www.math.uni-sb.de/~ag-fuchs/ag-fuchs.html

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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)

**Abstract:(/upload/editor/file/20200711/11112603683.pdf)**

**Lecture 2 - Structure of Commutator Subgroups**

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

**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）__

**Video: **https://disk.pku.edu.cn:443/link/DDECD524A26C489F184BBB6F55C34388

Valid Until: 2025-08-31 23:59

**Lecture 1 - Dynamics in the space of metrics and new invariants in ergodic theor****y**
**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.

**Download **Slides:

__https://disk.pku.edu.cn:443/link/B42BC9D4300DF6D5FB9FB94E62D23105__

Valid Until: 2025-07-31 23:59

**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.

**Lecture 2 - The characteristic factors in dynamical sy****stems**
**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.

__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____）__

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

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

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

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**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).

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

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

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

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**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.