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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.