##### Description

**Organizing committee of Beijing-Moscow Mathematics Colloquium **

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

(2) Yi Liu (BICMR, Topology of
3-manifolds, hyperbolic geometry)

(3) Yue Yang (EC, computation
mathematics and mechanics)

(4) Ping Zhang (AMSS, P. D. E.: fluid
equation and semi-classical analysis)

(5) Ye Tian (UCAS, Number Theory,
Arithmetic Geometry, Iwasawa Theory)

(6) Jingsong Liu (AMSS, algebraic
geometry: singularity theory)

(7) Alexey Tuzhlin (MSU, geometry:
Riemannian and metric geometry)

(8) Alexander Zheglov (MSU, geometry:
algebraic geometry, integrable system)

(9) Sergey Gorchinskiy (SMI, algebra
and geometry: algebraic geometry, K-theory)

(10) Denis Osipov (SMI, algebraic
geometry, number theory, integrable system)

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The lecture announcements will be continually updated. The arrangement of the upcoming lectures is as follows:

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**Lecture Series 28 —— January 21, 2022**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:** https://us02web.zoom.us/j/84084958194?pwd=b2QxSTVGVzN6NVorc2NJSFQ2TUpCZz09

**Meeting ID:** 840 8495 8194

**Password:** 775348

**Lecture 1——Real algebraic and real pseudoholomorphic curves**

**Speaker:** Stepan Orevkov (Steklov Mathematical Institute of RAS; Paul Sabatier University)

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** According to Gromov's theory, smooth symplectic 2-surfaces in CP^2 share many properties with complex algebraic curves. The same phenomenon takes place in the real case. Namely, smooth symplectic surfaces invariant under the complex conjugation (we call them real pseudoholomorphic curves) have many common properties with plane projective real algebraic curves.

An open question (Symplectic Isotopy Problem): does each connected component of the space of symplectic surfaces contain an algebraic curve? The same question can be asked in the real case and an negative answer will be given in the talk. We shall prove certain inequalities for the complex orientations of plane real algebraic curves which are not satisfied by an infinite series of real pseudoholomorphic curves.

**Bio:** Stephan Orevkov, PhD (Phys&Math), is a senior researcher at Steklov Mathematical Institute, a lead researcher at MIPT, and also a researcher at Université Toulouse III - Paul Sabatier, France. His academic interests include topology of flat real algebraic curves and surfaces, the theory of braids, complex surface mapping (as applicable to the Jacobian hypothesis).

**Lecture 2——On Ahlfors currents**

**Speaker:** Song-Yan Xie

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We answer a basic question in Nevanlinna theory that Ahlfors currents associated to the same entire curve may be nonunique. Indeed, we will construct one exotic entire curve which produces infinitely many cohomologically different Ahlfors currents. Moreover, concerning Siu's decomposition, for an arbitrary positive integer k or k=infinity, some of the obtained Ahlfors currents have singular parts supported on k irreducible curves. In addition, they can have nonzero diffuse parts as well, which answers a question of Brunella. This is joint work with Dinh Tuan Huynh.

**Bio:** Song-Yan Xie got his Ph.D. from Paris-Sud (Orsay) University in 2016. In his thesis he proved an ampleness conjecture of Debarre—— the cotangent bundles of a large class of complete intersections are ample. He is currently an associate professor at the Academy of Mathematics and Systems Science. His research interest is complex geometry, especially complex hyperbolicity and Nevanlinna theory.

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**Lecture Series 27 —— December 17, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:** https://us02web.zoom.us/j/84960246303?pwd=N21VbzV5QWxyMVAyOFduTVI2bHJvdz09

**Meeting ID:** 849 6024 6303

**Password:** 329993

**Lecture 1——****Quantization and Index Theory**

**Speaker:** Si Li

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **In the first
part, we introduce some basic ideas and various recent mathematical
developments about quantization that arises from quantum field theory and
string theory. In the second part, we discuss several applications to geometry
and topology. In particular, we present an effective quantization theory for 2d
chiral field theories, and explain its connection with elliptic chiral homology
of chiral algebras and index theory.

**Bio: **Si Li got his Ph.D. in mathematics from Harvard University in 2011. He is
currently professor at Yau Mathematical Sciences Center (YMSC), Tsinghua
University. He works on algebraic and geometric aspects of quantum field theory
and string theory.

**Lecture 2——****Korevaar–Schoen's energy on strongly rectifiable
spaces**

**Speaker: **Alexander Tyulenev (Steklov Mathematical
Institute of RAS)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We extend
Korevaar-Schoen’s theory of metric valued Sobolev maps to cover the case of the
source space being an RCD-space. When the target space is CAT(0) we establish
that the corresponding energy functional is convex, lower semicontinuous and
admits a unique minimizer, in line with the smooth situation. The talk is based
on the joined work: Nicola Gigli, Alexander Tyulenev, “Korevaar–Schoen's energy
on strongly rectifiable spaces”, Calc. Var. Partial Differential Equations, 60 (2021),
235, 54 pp., arXiv: 2002.07440

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**Lecture Series 26 ****—****— ****December**** 03, 2021**

**Time**：16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:**** **__https://zoom.us/j/82239034166?pwd=cGdBNXpNVllMZ3RkcDBXSTlCWG9CQT09__

**Meeting ID: **822 3903 4166

**Password:** 001993

**Lecture 1——****Tikhonov's solution to a class of linear systems equivalent within perturbations**

**Speaker:** Eugene Tyrtyshnikov (INM RAS, Moscow)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **A standard approach to incorrect problems suggests that a problem of interest is reformulated with the knowledge of some additional a priori information. This can be done by several well-known regularization techniques. Many practical problems are successfully solved on this way. What does not still look as completely satisfactory is that the new reset problem seems to appear rather implicitly in the very process of its solution.

In 1980, A.N. Tikhonov proposed a reformulation that arises explicitly before the discussion of the solution methods. He suggested a notion of normal solution to a family of linear algebraic systems described by a given individual system and its vicinity comprising perturbed systems, under the assumption that there are compatible systems in the class notwithstanding the compatibility property of the given individual system. Tikhovov proved that the normal solution exists and unique. However, a natural queston about the correctness of the reset problem was not answered. In this talk we address a question of correctness of the reformulated incorrect problems that seems to have been missed in all previous considerations. The main result is the proof of correctness for Tikhonov's normal solution. Possible generalizations and diffculties will be also discussed.

**Bio: **Marchuk Institute of Numerical Mathematics of RAS,director, Academician of RAS.

**Lecture 2— Liouville Properties of the Incompressible Navier-Stokes Equations**

**Speaker: **Zhen Lei

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:**** **In this talk we will report recent results on the Liouville properties of bounded ancient solutions to the three-dimensional incompressible Navier-Stokes equations. These properties are crucial for excluding potential type I singularities or understanding the structure of possible singulairites of local smooth solutions of the corresponding Cauchy problem.

**Bio:** Dr. Zhen Lei is a distinguished professor of School of Mathematical Sciences at Fudan University. His honors include: Second-prize Winner of the National Prize of Natural Sciences; Winner of Shanghai Peony Prize of Natural Science; National Science Foundation for Distinguished Young Scholars; Changjiang Distinguished Professor; National Special Support Program for Leading Talents in Science and Technology Innovation. He is the Vice President of China Society for Industry and Applied Mathematics. Professor Lei's research is focused on the theory of PDEs arising from fluid mechanics and methematical physics. He introduced the concept of strong null condition and proved the global well-posedness of classical solutions to the incompressible elastodynamics in 2D. He has also made significant contributions to the well-posedness theory and Liouville properties of the incompressible Navier-Stokesequations. Professor Lei holds the position of associate editor-in-chief of Chinese Annals of Mathematics; associate editor-in-chief of Journal of Fudan University (Natural Science). He also serves at the editorial board for several academic journals, such as Communications in Mathematical Sciences, Communications on Pure and Applied Mathematics, Fundamental Research, etc.

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**Lecture Series 25 ****—— November**** ****19****, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:** https://zoom.us/j/82755522129?pwd=VkFSeDV6NmNPRldvWVhUckVrczVsQT09

**Meeting ID:** 82755522129

**Password: **648063

**Lecture 1——Propagation of quasi-particles on singular spaces. Relation to the behavior of geodesics and to certain problems of analytic number theory**

**Speaker:** Andrey Shafarevich

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **We study propagation of semi-classical localized solutions of Schroedinger or wave equations (Gaussian beams) on a certain class of singular spaces. These spaces are obtained by connecting of a number of smooth manifolds by several segments. Laplacians on such spaces are defined with the help of extension theory an depend on boundary conditions in the points of gluing. Statistics of a number of Gaussian packets is governed by the behavior of geodesics on manifolds and is connected with certain problems of analytic number theory - in particular, with the problem of distribution of abstract primes.

**Bio:** Prof. A.I. Shafarevich is currently the Dean of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University. He is also the Corresponding Member of the Russian Academy of Sciences.

The main scientific interests of A.I.Shafarevich lie in the field of mathematical physics, asymptotic and geometric theory of linear and nonlinear partial differential equations, quantum mechanics and hydrodynamics. He solved the problem posed by V.P. Maslov and widely discussed in the scientific literature on the multiphase asymptotics for the equations of hydrodynamics.

**Lecture 2——Toda equations and cyclic Higgs bundles over non-compact surfaces**

**Speaker: **Qiongling Li

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** For Higgs bundles over compact Kahler manifods, it is known by Hitchin and Simpson that the existence of Hermitian-Einstein metric is equivalent to the polystability of the Higgs bundle. There are some generalizations to non-compact cases. On a Riemann surface with a holomorphic r-differential, one can naturally define a Toda equation and a cyclic Higgs bundle with a grading. A solution of the Toda equation is equivalent to a Hermitian-Einstein metric of the Higgs bundle for which the grading is orthogonal. In this talk, we focus on a general non-compact Riemann surface with an r-differential which is not necessarily meromorphic at infinity. In particular, we discuss the Hermitian-Einstein metrics on the cyclic Higgs bundles determined by r-differentials. This is joint work with Takuro Mochizuki (Kyoto University).

**Bio: **Qiongling Li got her Ph.D. from Rice University in 2014. She is currently a research fellow at Chern Institute of Mathematics, Nankai University. Her main research fields are Higgs bundles, harmonic maps, and higher Teichmuller theory. Her recent works have been focused on understanding the non-abelian Hodge correspondence over Riemann surfaces.

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**Lecture Series 24 ——November 05, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:**

**https://zoom.us/j/89231753221?pwd=WDdZT0NBUVB6TVgxYSsUkVZcGREdz09**

**Meeting ID: **892 3175 3221

**Password: **758843

**Lecture 1——On isochronous dynamics.**

**Speaker:** Dmitry Treschev (Steklov Mathematical Institute of Russian Academy of Sciences).

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **We propose a systematic approach to the problem of isochronicity in Hamiltonian dynamics. In particular, we present an explicit neccesary and sufficient condition for isochronicity in the case of 1DOF in terms of the Taylor expansion of the Hamiltonian function.

**Bio:** Dmitry Treschev: Director of Steklov Mathematical Institute of RAS, Academician of RAS (2016), Invited speaker at ICM (2002), Lyapunov prize of RAS (2007), Russian Federation prize for young scintists (1995).

**Lecture 2——Smoothed particle hydrodynamics (SPH) for modeling fluid-structure interactions**

**Speaker: **Moubin Liu

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **SPH, as a truly Lagrangian and meshfree particle method, is very attractive in modeling fluid-structure interaction (FSI) problems. This talk reports some recent developments of SPH method in modeling FSI problems with rigid, elastic and flexible structures, with granular materials, and with extremely intensive loadings.

**Bio: **Moubin Liu is the Vice Dean and a Professor of the College of Engineering, Peking University. His areas of interest include computational fluid dynamics and computational fluid-structures interactions, using particle-based methods including SPH, and particle-grid coupling approaches including SPH-FEM, DEM-CFD and etc.

**Weblink: **http://www2.coe.pku.edu.cn/subpaget.asp?id=628

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**Lecture Series 23 ****—— October ****22, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:**

__https://us02web.zoom.us/j/82517283705?pwd=L2UvRjRuY1hPbGcwa0lmN1ZGRjVQQT09__

**Meeting ID:** 825 1728 3705

**Password:** 526064

**Lecture 1——What we know and what we do not know about the zeros of Riemann zeta-function**

**Speaker: **Prof. Maxim Korolev

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** In the talk, we will discuss from different points of view the connection between quite "transcendental" objects, that is, between zeros of the Riemann zeta-function, and purely arithmetic objects, that is, prime numbers.

**Bio: **Maxim Korolev is a Professor of the Russian Academy of Sciences. He has received the Vinogradov Prize of the Russian Academy of Sciences in 2019.

**Lecture 2——Finiteness and Duality for the Cohomology of Prismatic Crystals**

**Speaker:** Yichao Tian

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Prismatic site of a p-adic formal scheme was introduced in the recent pioneer work of Bhatt—Scholze. It provides a uniform framework for various p-adic cohomology theories. Prismatic crystals are natural analogues of classical crystalline crystals on prismatic sites. In this talk, after reviewing some basic definitions of the prismatic site, I will discuss some basic properties of the cohomology of prismatic crystals on smooth p-adic formal schemes. The key ingredient is an explicit local description of (reduced) prismatic crystals in terms of Higgs modules.

**Bio:**Yichao Tian got his Ph. D. from University Paris in 2008. He is currently professor in the Morningside Center of Mathematics at Chinese Academy of Science. His main research fields are Arithmetic algebraic geometry: p-adic Hodge theory, Geometry of Shimura varieties in characteristic p > 0, p-divisible groups, and p-adic modular forms.

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**Lecture Series 22 —— October 08, 2021**

**Time**：16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:**

https://us02web.zoom.us/j/84612239222?pwd=bXFtREVscWZGZ1BWV2xiK1Fyd0IxUT09

**Meeting ID:** 846 1223 9222

**Password: **012831

** Lecture 1——Weighted graphs, tetrahedron equation and loop quantum gravity**

**Speaker:** Prof. Dmitry Talalaev

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **Edge-weighted graphs are a combinatorial object parametrizing several archetypal models of statistical physics: the dimer model, the Ising model and the model of electrical networks. Each of these models describes some combinatorial problem, is associated with some solution of the tetrahedron equation, is associated with one of the versions of the completely positive Grassmannian. I will talk about these phenomena and how these problems are related to loop quantum gravity, namely spin foam evaluations.

**Bio: **Dmitry Talalaev Graduated from the Faculty of Mechanics and Mathematics of Moscow State University, received a doctorate in Physics and Mathematics at the Steklov Institute, specializes in classical and quantum integrable systems, including models of statistical physics, applications in low-dimensional topology and cluster algebras.

**Lecture 2——An invitation to categorical enumerative invariants.**

**Speaker: **Junwu Tu

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **In this talk, we shall sketch the definition of categorical enumerative invariants following Costello (2004/2005) and Caldararu-T. (2020). Then we survey known calculations of these categorical invariants. In the end, we also discuss some research problems in this direction.

**Bio:** Junwu Tu got his Ph.D. from University of Wisconsin-Madison in 2011. He is currently an associate professor in the Institute of Mathematical Sciences (IMS) at ShanghaiTech University. His main research fields are homological algebras related to mirror symmetry. His recent works have been focused on understanding categorical enumerative invariants defined by Costello back in 2004.

**Lecture Series 21 —— September 17, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:**

https://zoom.com.cn/j/86714579603?pwd=L0h4azBrNmZodi9zTkxOQ3FGWEZlZz09

**Meeting ID:** 867 1457 9603
**Password: **042703

**Lecture 1——Liouville theorem for a class of semilinear elliptic equation on Heisenberg group**
**Speaker: **Prof. Xinan Ma, University of Science and Technology of China.
**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **We obtain a Liouville type theorem for a class of semilinear subcritical elliptic equation on Heisenberg group. The proof is based on a'priori integral estimate from a generalized differential identity found by Jerison and Lee in 1988. We also get a point-wise estimate near the isolated singularity. This is a joint work with Qianzhong Ou.

**Bio: **Xinan Ma got Ph. D from Hangzhou University in 1996. He is now working at the School of Mathematical Sciences, University of Science and Technology of China. His main research fields are nonlinear elliptic equation and geometry analysis.

**Lecture 2 ——Symplectic and Contact Geometry of Monge–Ampère equation: Introduction and application.**

**Speaker: **Vladimir Rubtsov (Université d'Angers)

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: ** I am going to present an introduction into the geometric approach to Monge–Ampère operators and equations based on contact and symplectic structures of cotangent and the 1st jet bundles of a smooth manifold. This approach was developed by V. Lychagin and goes back to the ideas of E.Cartan and his successor T. Lepage. I shall try to make my talk self-contained. I also plan to discuss various applications and links with important geometric structures.

**Bio: **Vladimir (Volodya) Rubtsov graduated in Mathematics from the Moscow State University in 1974 with a MSc in Differential Geometry and Applications. He has a PhD in Higher Geometry and Topology (1983) and held research and teaching positions at various Mathematics and Applied Mathematics Laboratories in the former Soviet Union. Presently he is Professor at the Department of Mathematics, Université d'Angers, and a member of LAREMA (Anjou Research Mathematical Laboratory) of CNRS (France). Since 1993 he is a Senior Researcher at the Theory Division in the Alikhanov Institute for Theoretical and Experimental Physics (ITEP) in Moscow. He held visiting positions at Ecole Polytechnique (Palaiseau, France), Universities of Lyon, Lille and Strasbourg (France), University of Uppsala (Sweden), SISSA (Italy) and others. He was invited member at IHES, MPIM (Bonn), the Newton Institute (Cambridge, UK) and others. His research is in the area of Poisson geometry, quantum Groups, integrable systems, symplectic and contact geometric methods in non-linear differential equations and applications in hydrodynamics.

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**Lecture Series 20 —— June 4, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting: ****https://zoom.com.cn/j/83122858822?****pwd=clgwMkhjNGNxYkRFZEFtQTY4ZStyQT09**

**Meeting ID:** 831 2285 8822

**Password: **338416

**Lecture 1——Integrable systems with elliptic dependence on momenta and related topics**

**Speaker:** Andrei Zotov (Steklov Mathematical Institute of Russian Academy of Sciences)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **We discuss a family of integrable many-body systems of classical (and quantum) mechanics. Some interrelations (dualities) predict existence of integrable many-body systems with elliptic dependence on particles momenta – the most general representative of this family. We describe some recent results on this topic. Next, we discuss relations of the many-body systems to other families of integrable models including integrable tops and spin chains. Finally, some interesting open problems are formulated.

**Bio: **Andrei Zotov is a leading researcher at Steklov Mathematical Institute. Also, associative professor at Moscow Institute of Physics and Technology and researcher at ITEP and HSE – Skoltech International Laboratory of Representation Theory and Mathematical Physics. Main field of research is mathematical physics and integrable systems.

**Lecture 2 ——Off-diagonal Bethe ansatz approach to quantum integrable models **

**Speaker:** Wenli Yang

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: ** Applying the recent developed method-the off-diagonal Bethe ansatz method, we construct the exact solutions of the Heisenberg spin chain with various boundary conditions. The results allows us to calculate the boundary energy of the system in the thermo dynamic limit. The method used here can be generalized to study the thermodynamic properties and boundary energy of other high rank models with non-diagonal boundary fields.

**Bio: **Wenli Yang is a professor and PhD supervisor in the School of Physics at North-western University. He is currently the executive director of the Chinese Physical Society and a member of the National Committee on Condensed Matter Theory and Statistical Physics. He received his bachelor's degree from Xi'an Jiaotong University in 1990 and his Ph.D. degree from North-western University in 1996, and has worked at the University of Bonn, Germany, Kyoto University, Japan, and the University of Queensland, Australia. In 2009, he was selected as one of the first "Hundred Talents Plan" in Shaanxi Province, and in 2014, he was awarded the National Outstanding Youth Fund, and in 2015, he was selected as one of the Changjiang Distinguished Professors of the Ministry of Education. His achievements were awarded the Second Prize of Natural Science of the Ministry of Education in 2010 and the First Prize of Science and Technology of Shaanxi Province in 2012.

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**Lecture Series 19 —— May 21, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom
Meeting: **

https://zoom.com.cn/j/85486883613?pwd=djM5eUlJMHdYRmhYOE9ObFRPd0RQUT09

**Meeting ID: **854 8688 3613

**Password: **303680

**Lecture 1 ****—— On the application of the Ważewski method to the problem of global stabilization**

**Speaker:** Ivan Polekhin (Steklov Mathematical Institute of RAS).

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **In 2000, S.P. Bhat and D.S. Bernstein proved that if the configuration space of an autonomous control mechanical system is closed (compact without boundary), then the system cannot have a globally asymptotically stable equilibrium [1]. We will present a similar result for non-autonomous control systems defined on manifolds with non-empty boundaries. The talk is based on the paper [2].

[1] Bhat S.P., Bernstein D.S. A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon Systems Control Lett., 39 (1) (2000), pp. 63-70

[2] I. Polekhin, “On the application of the Ważewski method to the problem of global stabilization”, Systems & Control Letters, 153 (2021) Share Link: https://authors.elsevier.com/a/1d2Qoc8EXim67

**Bio: **Russian Academy of Sciences prize for young Russian scientists (2020)

**Lecture 2 ——High Fidelity Simulations of High-Pressure Turbines Cascades for Data-Driven Model Development**

**Speaker: **Yaomin Zhao College of Engineering, Peking University

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Gas turbines (GT) are, and will continue to be, the backbone of aircraft propulsion, as well as power generation and mechanical drive. A small GT performance improvement is expected to have a fuel-spend advantage of the billion-dollar order, together with a significant CO2 emission benefit. Part of the possible performance improvements can be enabled by continuously advancing the understanding of the GT flow physics and thus further reducing the inaccuracy of current design tools based on computational fluid dynamics (CFD). By exploiting the capability of our high-fidelity CFD solver on leadership GPU-accelerated supercomputers, we have been able to perform state-of-the-art high fidelity simulations of turbomachinery flows. The generated data, therefore, can shed light on the detailed fundamental flow physics, in particularly the behavior of transitional and turbulent boundary layers affected by large-scale violent freestream turbulence, under strong pressure gradient and curvature. Furthermore, machine learning methods are applied to the high-fidelity data to develop low order models readily applicable to GT designs.

**Bio: **Dr. Yaomin Zhao is an Assistant Professor at Peking University, China. He obtained his B.Sc. in 2011 and then Ph.D. in 2017 both from Peking University. The Ph.D. thesis, Lagrangian investigation on transitional wall-bounded flows, was awarded the Outstanding PhD Thesis by Chinese Society of Theoretical and Applied Mechanics in 2018. From 2017 to late 2020, he was a post-doctocal research fellow at the University of Melboune, Australia. His research interests include high-fidelity simulations of turbomachinery flows, turbulence model development with machine learning methods, and boundary layer transition, etc.

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**Lecture Series XVIII —— May 14, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting:** https://zoom.com.cn/j/64938888630?pwd=ODBQcFpJSHJ2ejJ2Q29DVVBuK3NQZz09

**Meeting ID:** 649 3888 8630

**Password：**774874

**Lecture 1 —— Lorentz geometry and contact topology**

**Speaker:** Stefan Nemirovski (MIAN).

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** Roger Penrose observed four decades ago that the space of light rays of a reasonable spacetime carries a natural contact structure and raised the problem of describing the causality relation of the spacetime in its terms. The talk will survey the progress made in this direction from the seminal work of Robert Low to the more recent applications of global contact rigidity.

**Bio: **Corresponding member of the Russian Academy of Sciences, winner of the European Mathematical Society prize (2000).

**Lecture 2 ——Gromov-Hausdorff limit of manifolds and some applications**

**Speaker:** Wenshuai Jiang(Zhejiang University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **Gromov-Hausdorff distance is a distance between two metric spaces, which was introduced by Gromov 1981. From Gromov’s compactness theorem, we knew that any sequence of manifolds with uniform lower Ricci curvature bounds has a converging subsequence in Gromov-Hausdorff topology to a limit metric space. The limit metric space in general may not be a manifold. The structure of such limit metric space has been studied by Cheeger, Colding, Tian, Naber and many others since 1990. It turns out that such theory has powerful application in geometry. In fact, the resolution of Yau-Tian-Donaldson conjecture was largely relied on the development of the study of the limit metric space.

In the first part of the talk, we will discuss some recent progress of the Gromov-Hausdorff limit of a sequence of manifolds with Ricci curvature bounds. In the second part, we will discuss some applications based on the study of Gromov-Hausdorff limits.

**Bio: **Wenshuai Jiang studied in the Department of mathematics of Nanjing University from 2007 to 2011 and obtained his bachelor's degree. From 2011 to 2016, he studied in school of Mathematical Sciences of Peking University and obtained a doctorate under the guidance of Professor Gang Tian. He has been working in Zhejiang University since 2016, and is currently an associate professor of Zhejiang University. His major research interest is geometric analysis.

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**Lecture Series XVII —— April 23, 2021**

Time：16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting: **

https://zoom.com.cn/j/64563211865?pwd=Njd2M3NzSFcvWmlONTlJOUd5bmRjdz09

Meeting ID: 645 6321 1865

Password：341825

**Lecture 1 —— VALUES OF PERMANENT AND POSITIVE SOLUTION OF WANG-KRÄUTER PROBLEM**

**Speaker: **A.E. GUTERMAN (RUSSIA, MSU)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:**

**Lecture 2 —— From Sphere Packings to Post-Quantum Cryptography **

**Speaker: **Chuanming Zong (Tianjin University)

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **

**Bio: **Chuanming Zong obtained his PhD from Vienna University of Technology in 1993. He was a professor at the Chinese Academy of Sciences and Peking University. Currently, he is a distinguished professor at Tianjin University. He mainly works in number theory. He has made important contribution in Hilbert’s 18th problem and tiling theory. He has been awarded a Conant Prize by Amer Math Soc in 2015, a National Science Prize by the Chinese government in 2009, and a S. S. Chern Prize by the Chinese Math Soc in 2007. He was a plenary speaker at Asiacrypt2012, a author of two solicited papers in Bull AMS and three books at Springer and Cambridge University Press.

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**Lecture Series XVII —— April 9, 2021**

**Time**：16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting: **

https://zoom.com.cn/j/63363611209?pwd=VHdkbHF5KzNBVG1nTklxeWpZSExUZz09

**Meeting ID**:633 6361 1209

**Password：**127853

**Lecture 1 —— Mathematical methods of quantum key distribution.**

**Speaker: **Anton Trushechkin (MIAN).

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **Quantum key distribution and, more generally, quantum cryptography is a modern branch of science where methods of secure communication based on principles of quantum mechanics are studied. The rigorous proof of the security of quantum key distribution gave rise to a complex and beautiful mathematical theory, which is based on methods of quantum information theory, namely, quantum entropic measures and entropic uncertainty relations. In particular, to estimate secret key rate, one needs to minimize the quantum relative entropy (a convex function) subject to linear constraints. The problem is, in general, infinite-dimensional, but symmetry properties of the problem reduces the dimensionality and allows one to solve this problem analytically. However, currently, an important task is to prove the security of quantum key distribution with imperfect (i.e., practical) devices. Imperfections introduce asymmetries and thus make the problem more complicated. In the talk, estimations for the secret key rate in the case of detection-efficiency mismatch will be presented. Using entropic uncertainty relations, an infinite-dimensional problem is reduced to a one-dimensional one.

**Bio：**Results of Anton Trushechkin in quantum cryptography were nominated as one of most important mathematical achievements of the Russian Academy of Sciences in 2020.

**Lecture 2 ——A quantum leap in security.**

**Speaker:** Prof. Feihu Xu, University of Science and Technology of China.

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** Quantum cryptography or quantum key distribution (QKD) offers information-theoretic security based on the laws of physics. This is the technology at the basis of the quantum satellite “Mozi”, put in orbit by the Chinese Academy of Sciences in 2016. In practice, however, the imperfections of realistic devices might introduce deviations from the idealized models used in the security proofs of QKD. Can quantum code breakers successfully hack real systems by exploiting the side channels? Can quantum code makers design innovative countermeasures to foil quantum code breakers? In this talk, I will talk about the theoretical and experimental progress in the practical security aspects of quantum code making and quantum code breaking. After numerous attempts over the past decades, researchers now thoroughly understand and are able to manage the practical imperfections. Recent advances, such as the decoy-state, measurement-device-independent (MDI) and twin-field (TF) protocols, have closed critical side channels in the physical implementations in a rigorous and practical manner. Further readings in [Xu et al., Rev. Mod. Phys. 92, 025002 (2020)].

**Bio：**Feihu Xu has been a Professor at USTC since Oct. 2017. Before joining USTC, he was a Postdoctoral Associate at MIT in 2015-2017. He received an M.A.Sc and Ph.D from University of Toronto in 2011 and 2015. He works on quantum information science and has co-authored more than 70 journal papers. As the first/corresponding author, he has published more than 40 journal papers in Rev. Mod. Phys. (1), Nat. Photon. (4), Nat. Phys. (1), etc. He is the recipient of Early Career Award by NJP in 2020, 35 Innovators Under 35 of China (by MIT Technology Review) in 2019, Outstanding Dissertation Award (by OCPA) in 2015, and Best Paper Award of QCrypt in 2014.

******

**Lecture Series XVI ——March 26, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time).

**To Join Zoom Meeting: **

https://zoom.com.cn/j/67379810561?pwd=ekM5NmVBV0pYaTl2RllNV2hKdVpCQT09

**Meeting ID:** 673 7981 0561

**Password**：263867

**Lecture 1 —— Transference principle in Diophantine approximation**

**Speaker:** Oleg German (MSU)

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The talk will be devoted to one of the fundamental principles in Diophantine approximation called transference principle. It reflects the relation of duality between certain problems. This principle is usually formulated in terms of Diophantine exponents - they generalise to the multidimensional case the measure of irrationality of a real number. We plan to give an account on the existing relations Diophantine exponents satisfy and try to reveal the geometric nature of those relations. After having described some basic geometric constructions, we shall look from this perspective at the famous linear independence criterion that belongs to Nesterenko. It appears that our approach provides an alternative proof of this criterion, which bases on rather simple geometric considerations.

**Bio：**Oleg German graduated from Moscow State University in 2001, defended the Candidate thesis in 2005 at MSU and the Doctorate thesis in 2013 at Steklov Mathematical Institute. He works at the Department of Number Theory, Faculty of Mechanics and Mathematics, MSU. His research interests include geometry of numbers, Diophantine approximation, multidimensional continued fractions.

**Lecture 2 ——Introduction to p-adic Langlands program for GL_2**

**Speaker: **Hu Yongquan

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **The p-adic and mod p Langlands program is an avatar of the classical Langlands program and has been first initiated by C. Breuil. In this colloquium talk, I will give a brief introduction to the program and survey some recent progress in the case of GL_2.

**Bio:**Yongquan Hu received PhD degree from University Paris-Sud in 2010. After that, he has worked at University of Rennes 1 (France) as a Maître de Conférence. Starting from 2015, he is a Professor at Morningside Center of Mathematics, Academy of Mathematics and Systems Science. His research interest lies in p-adic and mod p Langlands program.

******

**Lecture Series XV —— March 12, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time), March 12, 2021.

**To Join Zoom Meeting: **

https://zoom.com.cn/j/64623316558?pwd=ODJUSkpOekV0ZlR3RHpUN000ZW8zUT09

**Meeting ID: **646 2331 6558

**Password**：574262

**Lecture 1 ——Mathematical problems in the theory of topological insulators**

**Speaker: **Armen Sergeev (Steklov Mathematical Institute, Moscow).

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:** The talk is devoted to the theory of topological insulators - a new and actively developing direction in solid state physics. To find a new topological object one have to look for the appropiate topological invariants and systems for which these invariants are non-trivial. The topological insulators are characterized by having wide energy gap stable for small deformations. A nice example is given by the quantum Hall spin insulator. It is a two-dimensional insulator invariant under the time reversal. It is characterized by the non-trivial topological Z_2-invariant introduced by Kane and Mele.

In our talk we consider the topological insulators invariant under time reversal. In the first part we present the physical basics of their theory while the second part deals with the mathematical aspects. These aspects are closely related to K-theory and non-commutative geometry.

**Bio：**Prof. Armen Glebovich Sergeev is a leading scientific researcher of the department of complex analysis in Steklov Mathematical Institute and a professor in Mechanical and Mathematical department of Moscow State University. He Got Ph. D in Moscow State University in 1975 and Doctor of Sciences in Steklov Mathematical Institute at Moscow in 1989. He is a foreign member of Armenian Academy of Sciences, member of the board of Moscow Mathematical Society and a member of Executive Committee of European Mathematical Society. He is the Chief-editor of many mathematical journals and published 106 papers and is the author of 10 books. His Principal fields of research include Pseudoconvex polyhedral, Invariant domains of holomorphy, Geometric quantization, Twistor quantization, Seiberg-Witten equations, Pseudoholomorphic curves and Vortex equations. He has been the scientific advisors of many doctors. Together with Prof. Xiangyu Zhou, they have organized a series of Sino-Russia Joint mathematical conferences for many years which has promoted greatly the mathematical cooperation between two countries.

**Lecture 2 ——Some recent applications of the strong openness property.**

**Speaker:** Qi'an Guan

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: ** The multiplier ideal sheaf plays an important role in several complex variables, complex geometry and algebraic geometry. The strong openness property for multiplier ideal sheaves was conjectured by Demailly and proved by Guan-Zhou. In this talk, we will recall some recent applications of the strong openness property on the restriction formula and subadditivity property related to multiplier ideal sheaves. This is joint work with Professor Xiangyu Zhou.

**Bio: **Qi'an Guan graduated from the Institute of mathematics and systems science, Chinese Academy of Sciences in 2011 as a Ph.D, and his advisor is Professor Xiangyu Zhou.

After graduation, he worked as a postdoctoral researcher in Beijing International Center for Mathematics Research for two years, and his co-advisor is Professor Xiaobo Liu.

In 2013, he joined the School of Mathematical Sciences of Peking University and is now a professor.

Qi'an Guan is mainly engaged in the study of several complex variables.

Qi'an Guan has won the "outstanding postdoctoral Award" (2013) of Peking University, the "Young Teacher Award" (2016) of Huo Yingdong education foundation, and the "Chang Jiang Scholars Program - Young Scholars" (2016) and the "Science Research Famous Achievement

Award in Higher Institution – Youth Science Award" (2017) of the Ministry of Education, the "Qiu Shi Outstanding Young Scholars Award " (2016), the " National Award for Youth in Science and Technology--Special Prize" (2019) of the Chinese Association for science and technology, “The Tan Kah Kee Young Scientist Award in Mathematics & Physics” (2020).

Qi'an Guan was supported by the "Excellent Young Scientists Fund"(2015) and "National Science Fund for Distinguished Young Scholars" (2018) of the National Natural Science Foundation of China.

******

**Lecture Series XIV —— January 29, 2021**

**Time：16:00-18:00 Beijing time (11:00-13:00 Moscow time), January 29, 2021.**

To Join Zoom Meeting:

https://zoom.com.cn/j/63780031309?pwd=cTVIeldxZms4cXo0MzRsS0ZBVXQ2dz09

**Meeting ID：**637 8003 1309

**Password：**037788

**Lecture 1 ——The Aerothermal Performance of Tip Leakage Flow in High Pressure Turbines.**

**Speaker: **Prof. Chao Zhou, College of Engineering, Peking University, Beijing, China

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **Future aero engines are expected to achieve higher efficiencies and lower emissions, which brings new challenges for turbine designers. In a high-pressure turbine, tip clearances exist between the turbine rotor blade tip and the casing to prevent rubbing. The pressure difference between the blade pressure side and the suction side drives the gas across this tip clearance gap. The high-temperature gas results in excessively high metal temperatures on the blade tip, which lead to thermal erosion and oxidation. Obtaining good aerothermal performance is the key to maintain the performance of the high pressure turbines. The current talk will present a combined experimental and numerical study, which aims to understand the performance of the tip leakage flow and to develop new tip configurations for higher engine efficiency. First, the aerodynamic and heat transfer performance of squealer tips will be discussed. The effects of the squealer height and thickness will be investigated. Then, the tips with

coolant injection are investigated to understand the effects of the cooling air on the loss mechanism and tip heat transfer. Finally, winglet configurations are used on blade tips to reduce the tip leakage loss. The results showed that by using the winglet tip developed in the current study, the turbine stage efficiency increases.

**Bio:** Chao Zhou is a tenured associate professor and the director of turbomachinery laboratory at the college of Engineering in Peking University, China. Before join Peking University, he obtained his doctorate degree at the Whittle Laboratory of Cambridge University in 2010. He is also educated in Nanjing University of Aeronautics and Astronautics, China, and received his Master degree and Bachelor in 2006 and 2003 respectively. His main research area is the aerodynamic and heat transfer of turbomachinery, including aerothermal performance of high pressure turbines, high-lift low pressure turbines; inter-turbine duct flows, unsteady flows and loss mechanism of turbomachinery, advanced cooling methods and highly loaded compressors. He has published 9 papers on the Journal of Turbomachinery, which is the top Journal in the research area. Dr. Zhou is a member of ASME IGTI Tubomachinery committee. He serves as the review co-chair of 2020 GPPS (Global Power and Propulsion Society) conference.

**Lecture 2 —— Mathematical Models of Mediums in Continuum Mechanics.**

**Speaker:** Dmitrii Georgievskii

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **Elements of the theory of constitutive relations. The tangent modulus and the tangent compliance. Physical nonlinearity, tensor linearity (quasilinearity), and nonlinearity. The material functions. Rheonomic and scleronomic media. Homogeneous and inhomogeneous media. Composites. Elastic bodies. Viscous liquids. Media with memory. Non-local media. Tensor functions and their invariants in the theory of constitutive relations. Potential media and conditions of potentiality. Incompressible materials (liquids).Nonlinear elastic-viscoplastic models. Classification of incompressible media (quasilinear models, Bingham bodies, perfectly plastic media, Newtonian viscous fluids). Statement of the linearized boundary value problem of flow stability with respect to small perturbations of the initial data.

**Bio: **Dmitrii Georgievskii received his PhD degree at MSU, 1989, and DSc degree at MSU, 1996. He is a Professor of Russian Academy of Sciences since 2015and is a Chair of Lab. of Elasticity and Plasticity in Institute of Mechanics (MSU) since 2020. His research interests include the Theory of constitutive relations in continuum mechanics, Phenomenological description of stress-strain state by multiscale simulation, Asymptotic methods in theory of thin solids, see also web-page http://mech.math.msu.su/~georgiev/first_e.htm for more details.

******

**Lecture Series XIII —— January 15, 2021**

**Time：**16:00-18:00 Beijing time (11:00-13:00 Moscow time), January 15, 2021.

**To Join Zoom Meeting:** https://zoom.com.cn/j/64523635960?pwd=dVo4eWdQeW90MThXd0NWZk1HdVBoUT09

**Meeting ID：**645 2363 5960

**Password：**714716

**Lecture 1 ——Linear stability of pipe Poiseuille flow at high Reynolds number regime**

**Speaker:** Zhifei Zhang, School of Mathematical Sciences, Peking University

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **The linear stability of pipe Poiseuille flow is a long standing problem since Reynolds experiment in 1883. Joint with Qi Chen and Dongyi Wei, we solve this problem at high Reynolds regime. We first introduce a new formulation for the linearized 3-D Navier-Stokes equations around this flow. Then we establish the resolvent estimates of this new system under favorable artificial boundary conditions. Finally, we solve the original system by constructing a boundary layer corrector.

**Bio:** Zhifei Zhang received his PhD from Zhejiang university in 2003. Then he spent 2 years in Mathematics Institute of AMSS as a Postdoc. He joined Peking university in 2005. His research interest is in the mathematical problems in the fluid mechanics such as the well-posedness of the Navier-Stokes equations, free boundary problem, hydrodynamic stability.

**Lecture 2 —— Partial spectral flow and the Aharonov–Bohm effect in graphene.**

**Speaker:** Vladimir E. Nazaikinskii Ishlinsky Institute for Problems in Mechanics RAS

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract:** We study the Aharonov–Bohm effect in an open-ended tube made of a graphene sheet whose dimensions are much larger than the interatomic distance in graphene. An external magnetic field vanishes on and in the vicinity of the graphene sheet, and its flux through the tube is adiabatically switched on. It is shown that, in the process, the energy levels of the tight-binding Hamiltonian of π-electrons unavoidably cross the Fermi level, which results in the creation of electron–hole pairs. The number of pairs is proven to be equal to the number of magnetic flux quanta of the external field. The proof is based on the new notion of partial spectral flow, which generalizes the ordinary spectral flow introduced by Atiyah, Patodi, and Singer and already having well-known applications (such as the Kopnin forces in superconductors and superfluids) in condensed matter physics.

**Bio:** Vladimir Nazaikinskii received PhD degree from Moscow Institute of Electronic Engineering in 1981 and DSc degree from Steklov Mathematical Institute of RAS in 2014 and was elected Corresponding Member of RAS in 2016. He works at Ishlinsky Institue for Problems in Mechanics of RAS as a principal researcher. His research interests include asymptotic methods in the theory of differential equations and mathematical physics; asymptotic methods in the statistics of many-particle systems and relations to number theory; C*-algebras and noncommutative geometry; elliptic theory and index theory.

******

**Lecture Series Ⅻ —— December 18th, 2020**

**Video:https://disk.pku.edu.cn:443/link/BFD8999366FC0522C130630CFF7B5304**

**Video Until: 2025-01-01**

**Lecture 1 —— Polynomial structures in higher genus enumerative geometry**

**Time:** 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Speaker: **Shuai Guo, School of Mathematical Sciences, Peking University).

**Abstract: **It is important to calculate the enumerative invariants from various moduli theories in mirror symmetry. The polynomial structure is often appeared in those quantum theories, including the Calabi-Yau type and the Fano type theories. Such conjectural structure is also called the finite generation conjecture in the literature. For each genus, it is conjectured that the computation of infinite many enumerative invariants can be converted to a finite computation problem. The original motivation of studying such structures will also be mentioned. This talk is based on the joint work with Janda-Ruan, Chang-Li-Li, Bousseau-Fan-Wu and Zhang respectively.

**Bio: **Shuai Guo got Ph. D in Tsinghua University, 2011 and now is an associate professor in SMS of Peking University.

**Research interests: **Higher genus enumerative geometry and mirror symmetry.

**Honors: **2019 "QiuShi" Outstanding Youth Award (2019), Selected as the national youth talent support program of China (2019).

**Lecture 2 ——Smooth compactifications of differential graded categories**

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Speaker: **Prof. Alexander Efimov, Steklov Mathematical Institute of Russian Academy of Sciences.

**Abstract:** We will give an overview of results on smooth categorical compactifications, the questions of theire existence and their construction. The notion of a smooth categorical compactification is closely related with the notion of homotopy finiteness of DG categories.

First, we will explain the result on the existence of smooth compactifications of derived categories of coherent sheaves on separated schemes of finite type over a field of characteristic zero. Namely, such a derived category can be represented as a quotient of the derived category of a smooth projective variety, by a triangulated subcategory generated by a single object. Then we will give an example of a homotopically finite DG category which does not have a smooth compactification: a counterexample to one of the Kontsevich's conjectures on the generalized Hodge to de Rham degeneration.

Finally, we will formulate a K-theoretic criterion for existence of a smooth categorical compactification, using DG categorical analogue of Wall's finiteness obstruction from topology.

**Research interests: **algebraic geometry, mirror symmetry, non-commutative geometry.

**Honors:** European Mathematical Society Prize (2020), Russian Academy of Sciences Medal with the Prize for Young Scientists (2017), Moscow Mathematical Society award (2016).

******

**Lecture Series Ⅺ ——December 4th, 2020**

**Video:https://disk.pku.edu.cn:443/link/F4C07EE03EBDA810BD5935193DD8D832**

**Video until: 2025-01-01**

**Lecture 1 —— On homology of Torelli groups**

**Time: 1**6:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Speaker: **Prof. Alexander Gaifullin, Steklov Mathematical Institute & Skolkovo Institute of Science and Technology, Russia.

**Abstract: **The mapping class groups of oriented surfaces are important examples of groups whose properties are closely related to geometry and topology of moduli spaces, topology of 3-manifolds, automorphisms of free groups. The mapping class group of a closed oriented surface contains two important subgroups, the Torelli group, which consists of all mapping classes that act trivially on the homology of the surface, and the Johnson kernel, which is the subgroup generated by all Dehn twists about separating curves. The talk will be devoted to results on homology of these two subgroups. Namely, we will show that the k-dimensional homology group of the genus g Torelli group is not finitely generated, provided that k is in range from 2g-3 and 3g-5 (the case 3g-5 was previously known by a result of Bestvina, Bux, and Margalit), and the (2g-3)-dimensional homology group the genus g Johnson kernel is also not finitely generated. The proof is based on a detailed study of the spectral sequences associated with the actions of these groups on the complex of cycles constructed by Bestvina, Bux, and Margalit.

**Bio: **Prof. Alexander Gaifullin is the Correspondent member of the Russian Academy of Sciences (since 2016). He got the following honours: Prize of the President of the Russian Federation in the field os science and innovations for young scientists (2016), Prize of the Moscow Mathematical Society (2012). He is the invited speaker at the 5th European Congress of Mathematics (Krakow, 2012); plenary speaker at the 6th European Congress of Mathematics (Berlin, 2016)

**Lecture 2 ——Stable homotopy groups of spheres**

**Time:** 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Speaker:** Prof. Guozhen Wang, Shanghai Center for Mathematical Sciences, Fudan University.

**Abstract:** We will discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.

**Bio:** Guozhen Wang received PhD degree from MIT in 2015. From 2016, he is working at Shanghai Center for Mathematical Sciences, Fudan University. His research field is algebraic topology, including stable and unstable homotopy groups, applications of computers in homotopy theory, motivic homotopy theory and topological cyclic homology.

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**Lecture Series X —— November 20th, 2020**

**video：https://disk.pku.edu.cn:443/link/20B14B526417D526F2945DE3AE5BC4A0**

**Video until：2025-01-01**

**Lecture 1 —— Right-angled polytopes, hyperbolic manifolds and torus actions**

**Speaker：**Taras Panov, Moscow State University, Russia

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract: **A combinatorial 3-dimensional polytope P can be realized in Lobachevsky 3-space with right dihedral angles if and only if it is simple, flag and does not have 4-belts of facets. This criterion was proved in the works of A.Pogorelov and E.Andreev of the 1960s. We refer to combinatorial 3 polytopes admitting a right-angled realisation in Lobachevsky 3-space as Pogorelov polytopes. The Pogorelov class contains all fullerenes, i.e. simple 3-polytopes with only 5-gonal and 6-gonal facets. There are two families of smooth manifolds associated with Pogorelov polytopes. The first family consists of 3-dimensional small covers (in the sense of M.Davis and T.Januszkiewicz) of Pogorelov polytopes P, also known as hyperbolic3-manifolds of Loebell type. These are aspherical 3-manifolds whose fundamental groups are certain extensions of abelian 2-groups by hyperbolic right-angled reflection groups in the facets of P. The second family consists of 6-dimensional quasi toric manifolds over Pogorelov polytopes. These are simply connected 6-manifolds with a 3-dimensional torus action and orbit space P. Our main result is that both families are cohomologically rigid, i.e. two manifolds M and M' from either family are diffeomorphic if and only if their cohomology rings are isomorphic. We also prove that a cohomology ring isomorphism implies an equivalence of characteristic pairs; in particular, the corresponding polytopes P and P' are combinatorially equivalent. This leads to a positive solution of a problem of A.Vesnin (1991) on hyperbolic Loebell manifolds, and implies their full classification. Our results are intertwined with classical subjects of geometry and topology such as combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology.

This is a joint work with V. Buchstaber, N. Erokhovets, M. Masuda and S.Park.

**Bio:** Higher geometry and topology chair, Professor. Research interests: Algebraic and differential topology, cobordism theories, toric topology. Honors: I. I. Shuvalov Prize, 1st degree, Moscow State University (2013), Moscow Mathematical Society award (2004).

**Lecture 2 —— Finite covers of 3-manifolds**

**Speaker：**Yi Liu, Beijing International Center for Mathematical Research.

**Time: **17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Abstract: **In this talk, I will discuss some developments in 3-manifold topology of this century regarding finite covering spaces. These developments led to the resolution of Thurston’s virtual Haken conjecture and other related conjectures around 2012. Since then, people have been seeking for new applications of those techniques and their combination with other branches of mathematics.

**Bio：**Yi Liu is a professor at Beijing International Center for Mathematical Research (BICMR) in Peking University. His research interest lies primarily in 3-manifold topology and hyperbolic geometry. He received his Ph.D. degree in 2012 in University of California at Berkeley. In 2017, he received the Qiushi Outstanding Young Scholar Award. He has been a principal investigator of the NSFC Outstanding Young Scholar since 2019. Below are some selected research works of Yi Liu: (1) proving J. Simon’s conjecture about knot groups (joint with I. Agol, 2012); (2) resolving fundamental properties of the L2 Alexander torsion for 3-manifolds, (2017); (3) proving C. T. McMullen’s conjecture about virtual homological spectral radii of surface automorphisms (2020).

**Lecture Series Ⅸ ——November 6th, 2020**

**Video：https://disk.pku.edu.cn:443/link/7CF5DA289E3EE66336FED3FD391E2527**

**Valid Until：** 2025-01--01 00：00

**Lecture 1 —— Spectrum rigidity and integrability for Anosov diffeomorphisms.**

**Speaker：**Assistant Prof. Yi Shi, School of Mathematical Sciences, Peking University

**Time: **16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract:**

**Bio: **Yi Shi obtained PhD from Peking University and Universite de Bourgogne in 2014, and then did postdoc in IMPA. He is now an assistant professor in School of Mathematical Sciences at Peking University. His research field is differentiable dynamical systems, including partially hyperbolic dynamics and singular star vector fields.

**Lecture 2 —Аn application of algebraic topology and graph theory in microeconomics**

**Speaker：**Lev Lokutsievskiy (Steklov Mathematical Institute of RAS)

**Time：**17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**
**
**Abstract：**One of the important questions in mechanism design is the implementability of allocation rules. An allocation rule is called implementable if for any agent, benefit from revealing its true type is better than benefit from lying. I’ll show some illustrative examples.

Obviously, some allocation rules are not implementable. Rochet’s theorem states that an allocation rule is implementable iff it is cyclically monotone. During the talk, I’ll present a new convenient topological condition that guarantees that cyclic monotonicity is equivalent to ordinary monotonicity. The last one is easy to check (in contrary to cyclic one). Graph theory and algebraic topology appear to be very useful here.

**Bio: **Lokutsievskiy L.V. is a specialist in geometric optimal control theory. He proved his habilitation thesis in 2015. Starting from 2016 he works at Steklov Mathematical Institute as a leading researcher.

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Previous Lectures:

**Lecture Series Ⅷ ——October 23th, 2020**

Video： https://disk.pku.edu.cn:443/link/8219AEEF731697BBD8C480F050327EBF

Vaild Until： 2025-01-01 00：00

**Lecture 1 - Topological cyclic homology for p-adic local fields.**

**Speaker:** Prof. Ruochuan Liu, School of Mathematical Sciences, Peking University

**Time:** 18:15-19:15 Beijing time (13:15-14:15 Moscow time)

**Abstract:** We introduce a new approach to compute topological cyclic homology using the descent spectral sequence and the algebraic Tate spectral sequence. We carry out computations in the case of a p-adic local field with coefficient Fp. Joint work with Guozhen Wang.

**Bio: **Ruochuan Liu is working on p-adic aspects of arithmetic geometry and number theory, especially p-adic Hodge theory, p-adic automophic forms and p-adic Langlands program. He got his PhD from MIT at 2008. After several postdoc experience at Paris 7, McGill, IAS and Michigan, he joined the Beijing International Center for Mathematical Research at 2012. Starting from this year, he holds professorship at the School of Mathematical Sciences of Peking University.

**Lecture 2 — Additive divisor problem and Applications**

**Speaker: **Dimitry Frolenkov, Steklov Mathematical Institute (Moscow)

**Time:** 19:15-20:15 Beijing time (14:15-15:15 Moscow time)

**Abstract:** Additive Divisor Problem (ADP) is concerned with finding an asymptotic formula for the sum $\sum_{n<X}d(n)d(n+a)$, where $d(n)=\sum_{d|n}1$ is the divisor function. Surprisingly, the ADP arises naturally in quite different problems of number theory. For example, it is related to the investigation of the 4th moment of the Riemann zeta-function, the second moment of automorphic $L$-functions and the mean values of the length of continued fractions. In the talk, I will describe the ADP and its applications.

**Bio: **Dmitry Frolenkov received his PhD degree from Steklov Mathematical Institute in 2013. Starting from 2014 he works at Steklov Mathematical Institute as a senior researcher. Besides he got the RAS award for young scientists of Russia. His research interests are centered around an analytic number theory with a special emphasis on the theory of L-functions associated to automorphic forms.

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**Second activity: Online conference Algebraic geometry and arithmetic **

The conference is on the occassion of the 70-th anniversary of our friend and colleague Vyacheslav Valentinovich Nikulin, to celebrate his huge contributions to the theory of K3 surfaces and other areas of geometry and arithmetic including reflections groups, automorphic forms and infinite-dimensional Lie algebras. The topics covered at the conference reflect the mathematical interests of V.V. Nikulin.

**Zoom meeting** ：948 1049 3176, **password ：**HWEG2Q

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**Lecture Series Ⅶ - October 16th, 2020**

Video：https://disk.pku.edu.cn:443/link/50D48681952109CBB56EC62458B46065

Valid Until：2025-01-01 00：00

**Lecture 1**** - Restriction of unitary representations of Spin(N,1) to parabolic subgroups**

**Speaker**: Prof. Yu Jun, Beijing International Center for Mathematical Research

**Time**：17：30-18：30 Beijing time （12：30-13：30 Moscow time）

**Abstract:** The orbit method predicts a relation between restrictions of irreducible unitary representations and projections of corresponding coadjoint orbits. In this talk we will discuss branching laws for unitary representations of Spin(N,1) restricted to parabolic subgroups and the corresponding orbit geometry. In particular, we confirm Duflo's conjecture in this setting. This is a joint work with Gang Liu (Lorraine) and Yoshiki Oshima (Osaka).

**Bio:** Jun Yu obtained PhD from ETH Zurich in 2012, and then did postdoc in IAS Princeton and MIT. He is now an assistant professor in Beijing international center for mathematical research at Peking University. His research field is representation theory and Langlands program, including the branching rule problem, the orbit method philosophy, and the beyond endoscopy program.

**Lecture 2: Characterizing homogeneous rational projective varieties with Picard number 1 by their varieties of minimal rational tangents.**

**Speaker:** Prof. Dmitry Timashev, Moscow State University

**Time**：18：30-19：30 Beijing time （13：30-14:30 Moscow Time)

**Abstract:** It is well known that rational algebraic curves play a key role in the geometry of complex projective varieties, especially of Fano manifolds. In particular, on Fano manifolds of Picard number (= the 2nd Betti number) one, which are sometimes called "unipolar", one may consider rational curves of minimal degree passing through general points. Tangent directions of minimal rational curves through a general point $x$ in a unipolar Fano manifold $X$ form a projective subvariety $\mathcal{C}_{x,X}$ in the projectivized tangent space $\mathbb{P}(T_xX)$, called the variety of minimal rational tangents (VMRT).

In 90-s J.-M. Hwang and N. Mok developed a philosophy declaring that the geometry of a unipolar Fano manifold is governed by the geometry of its VMRT at a general point, as an embedded projective variety. In support of this thesis, they proposed a program of characterizing unipolar flag manifolds in the class of all unipolar Fano manifolds by their VMRT. In the following decades a number of partial results were obtained by Mok, Hwang, and their collaborators.

Recently the program was successfully completed (J.-M. Hwang, Q. Li, and the speaker). The main result states that a unipolar Fano manifold $X$ whose VMRT at a general point is isomorphic to the one of a unipolar flag manifold $Y$ is itself isomorphic to $Y$. Interestingly, the proof of the main result involves a bunch of ideas and techniques from "pure" algebraic geometry, differential geometry, structure and representation theory of simple Lie groups and algebras, and theory of spherical varieties (which extends the theory of toric varieties).

**Bio**: Dmitry Timashev recieved PhD degree from Moscow State University in 1997. From 1997, he is working at the Department of Higher Algebra in the Faculty of Mathematics and Mechanics, Moscow State University, currently at the position of associate professor. His research interests include Lie groups and Lie algebras, algebraic transformation groups and equivariant algebraic geometry, representation theory and invariant theory.

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**Lecture Series VI - September 25th, 2020**

Video：Part I：https://disk.pku.edu.cn:443/link/6814AE00FAEED6A3E3D347F3DBAD7BF5

Part II： https://disk.pku.edu.cn:443/link/95EC3FD5EBDA27A4EE6FEE7A1A13C90A

Valid Until：2025-01-01 00：00

**Lecture 1** - ** Geometric description of the Hochschild cohomology of Group Algebras**

**Speaker**: A. S. Mishchenko (Lomonosov Moscow State University)

**Time**: 2020-09-25 16:00-17:00 Beijing time (11:00-12:00 Moscow time)

**Abstract**: /upload/editor/file/20200921/21143204927.pdf

**Bio**:

Professor A.S. Mischenko graduated from Moscow State University in 1965. He became a Professor of the Department of Higher Geometry and Topology, Faculty of Mechanics and Mathematics of this University in 1979. He also holds a position of Leading researcher at the Mathematical Steklov Institute. He is a Honored Professor of Moscow University since 2006. _{}^{}

His research interests include geometry and topology and their applications. The main direction of his work is related to the study and application of algebraic and functional methods in the theory of smooth manifolds.

**Lecture 2 - Unipotent representations and quantization of classical nilpotent varieties**

**Speaker**: Prof. Daniel Wong (黄家裕）, Chinese University of Hongkong at Shenzhen.

**Time**: 2020-09-25 17:00-18:00 Beijing time (12:00-13:00 Moscow time)

**Bio**: Graduated at Cornell University in 2013. Research area is on Representation theory of real reductive Lie groups.

******

__Lecture Series V - July 10th, 2020__

**Video:**

https://disk.pku.edu.cn:443/link/55C58180540241608743998D28941762

Valid Until: 2025-08-31 23:59

**Lecture 1 - Limits of the Boltzmann equation.**

Speaker: **Feimin Huang**, Academy of Mathematics and Systems Science, CAS, China

Time: 2020-07-10 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: In this talk, I will present recent works on the hydrodynamic limits to the generic Riemann solutions to the compressible Euler system from the Boltzmann equation.

Bio: Prof. Huang, Feimin got Ph. D in Chinese Academy Sinica in 1997，and then did postdoc in ICTP, Italy and Osaka University. His research field is hyperbolic equations and conservative laws, including fluid dynamical systems, Navier-Stokes equations, and other various Partial Differential Equations. He was awarded the SIAM Outstanding Paper Prize by Society of American Industrial and Applied Mathematics in 2004. He won the Second Prize of National Natural Science Award in 2013.

**Lecture 2 - On the geometric solutions of the Riemann problem for one class of systems of conservation laws.**

Speaker: **Vladimir Palin**, Moscow State University

Time: 2020-07-10 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract: We consider the Riemann problem for a system of conservation laws. For non-strictly hyperbolic in the sense of Petrovskii step-like systems, a new method of constructing a solution is described. The proposed method allows us to construct a unique solution to the Riemann problem, which for each $t$ is a picewise smooth function of $x$ with discontinuities of the first kind. Moreover, for the scalar conservation law, the solution constructed by the proposed method coincides with the known admissible solution.

Bio: Vladimir Palin recieved higher education degree from Moscow State University in 2005, PhD degree from Moscow State University in 2009. He is now a senior lecturer in the Faculty of Mathematics and Mechanics, Moscow State University. His research interests include hyperbolic equations and systems, conservation laws and matrix equations.

__Lecture Series IV - June 26th, 2020__

Video： https://disk.pku.edu.cn:443/link/FCEAE0F38D5164A103721C6E93EB7A9F

Valid Until： 2025-01-01 00：00

**Lecture 1 - Tying Knots in Fluids**

Speaker: **Prof. Yue Yang**, Department of Mechanics and Engineering Science, College of Engineering, Peking University,Beijing

Time: 2020-06-26 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract:

We develop a general method for constructing knotted vortex/magnetic tubes with the finite thickness, arbitrary shape, and tunable twist. The central axis of the knotted tubes is determined by a given smooth and non-degenerated parametric equation. The helicity of the knotted tubes can be explicitly decomposed into the writhe, localized torsion, and intrinsic twist. We construct several knotted vortex/magnetic tubes with various geometry and topology, and investigate the effect of twist on their evolution in hydrodynamic or magnetohydrodynamic flows using direct numerical simulation. In addition, we illustrate a knot cascade of magnetic field lines through the stepwise reconnection of a pair of orthogonal helical flux tubes with opposite chirality.

Bio:

Yue Yang received BE degree from Zhejiang University in 2004, MS degree from the Institute of Mechanics, Chinese Academy of Sciences in 2007, and PhD degree from California Institute of Technology in 2011, then he was sponsored by the CEFRC Fellowship for postdoc research at Princeton University and Cornell University. Yang joined the Department of Mechanics and Engineering Science in College of Engineering, Peking University in 2013, and was promoted to full professor in 2020. He received the “National Distinguished Young Researcher” award and “Qiu Shi Outstanding Young Scholar Award”. His research interests include turbulence, transition, and combustion.

**Lecture 2 - Supercomputer simulations of aerodynamics and aeroacoustics problems using high-accuracy schemes on unstructured meshes.**

Speaker: **Prof. Andrey Gorobets**, Keldysh Institute of Applied Mathematics of RAS, Moscow

Time: 2020-06-26 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract:

This talk is devoted to scale-resolving simulations of compressible turbulent flows using edge-based high-accuracy methods on unstructured mixed-element meshes. The focus is on parallel computing. Firstly, the family of edge-based schemes that we are developing will be outlined. Then our simulation code NOISEtte will be presented. It has multilevel MPI+OpenMP+OpenCL parallelization for a wide range of hybrid supercomputer architectures. A description of the parallel algorithm will be provided. Finally, our supercomputer simulations of aerodynamics and aeroacoustics problems will be demonstrated.

Bio:

Andrey Gorobets graduated from Moscow State University in 2003. He then outlived three thesis defenses: 2007, Candidate of Sciences (к. ф.-м. н., equivalent to Ph.D.) at IMM RAS; 2008, European Ph.D. degree at UPC, Barcelona, Spain; 2015, Doktor nauk (д. ф.-м. н., higher doctoral degree) at the Keldysh Institute of Applied Mathematics of RAS (KIAM), Moscow, Russia. He is now a leading researcher at KIAM. His work is focused on algorithms and software for large-scale supercomputer simulations of turbulent flows.

__Lecture Series III - June 12th, 2020__

Video： Part I**： https://disk.pku.edu.cn:443/link/8C9A18E37EAFA09F91A1103558A646F7**

** **Part II**： https://disk.pku.edu.cn:443/link/A52FCD8F4ACF2B51F7095A4D2BDB3BC8**

Valid Until：2025-01-01 00：00

**Lecture 1 - Slopes of modular forms and ghost conjecture of Bergdall and Pollack**

Speaker：**Prof. Xiao Liang **（Beijing International Center for Math. Research ）

Time: 2020-06-12 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: In classical theory, slopes of modular forms are p-adic valuations of the eigenvalues of the Up-operator. On the Galois side, they correspond to the p-adic valuations of eigenvalues of the crystalline Frobenius on the Kisin's crystabelian deformations space. I will report on a joint work in progress in which we seems to have proved a version of the ghost conjecture of Bergdall and Pollack. This has many consequences in the classical theory, such as some cases of Gouvea-Mazur conjecture, and some hope towards understanding irreducible components of eigencurves. On the Galois side, our theorem can be used to prove certain integrality statement on slopes of crystalline Frobenius on Kisin's deformation space, as conjectured by Breuil-Buzzard-Emerton. This is a joint work with Ruochuan Liu, Nha Truong, and Bin Zhao.

**Lecture 2 -**** ****Higher-dimensional Contou-Carrere symbols**

Speaker：**Prof. RAS Denis V. Osipov** (Steklov Mathematical Institute of Russian Academy of Sciences)

Time: 2020-06-12 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract: The classical Contou-Carrere symbol is the deformation of the tame symbol, so that residues and higher Witt symbols naturally appear from the Contou-Carrere symbol. This symbol was introduced by C. Contou-Carrere itself and by P. Deligne. It satisfies the reciprocity laws. In my talk I will survey on the higher-dimensional generalization of the Contou-Carrere symbol. The n-dimensional Contou-Carrere symbol naturally appears from the deformation of a full flag of subvarieties on an n-dimensional algebraic variety and it is also related with the Milnor K-theory of iterated Laurent series over a ring. The talk is based on joint papers with Xinwen Zhu (when n=2) and with Sergey Gorchinskiy (when n>2).

__Lecture Series II - May 29th 2020__

**Recording: **

https://disk.pku.edu.cn:443/link/36F474E90952433E4B75BED76283A758

ExpirationTime：2021-07-31 23:59

**Lecture 1 - Geometry of Landau--Ginzburg models.**

Speaker：Prof. **Victor V. Przyjalkowski** (Steklov Mathematical Institute of Russian Academy of Sciences)

Time：2020-05-29 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: We discuss geometric and numerical properties of Landau--Ginzburg models of Fano varieties that reflect geometric and numerical properties of the initial Fano varieties. The main example is the threefold case.

**Lecture 2 -**** ****Deformation theory of Schroedinger equation arising from singularity theory**

**Recording: **

https://disk.pku.edu.cn:443/link/17D6C8D96B3F620DFAC11DA76F40288D

ExpirationTime：2021-07-31 23:59

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

Time：2020-05-29 21:00-22:00 Beijing time (16:00-17:00 Moscow time)
Abstract: Mirror symmetry phenomenon relates many mathematical branches in a mysterious way. For example, it is conjectured that the quantum geometry of a Calabi-Yau hypersurface is equivalent to the quantum singularity theory of the corresponding defining function. When we consider the complex structure deformation of the two sides, we get the B model mirror conjecture, where the exciting structures of deformation moduli space, Gauss Manin connection, period mapping and etc.will appear. In this lecture, I will report another way to study the deformation theory of singularity via Schroedinger equation. By study the spectral theory of Schroedinger equation, we can build the variation of Hodge theory, Gauss-Manin connection by wave function, Frobenius manifold for some cases and even BCOV type torsion invariants for singularity.

__Lecture Series I - May 15th 2020__

**Recording: **

ExpirationTime：2021-08-01 23:59

**Lecture 1 - ****Representation volumes and dominations of 3-manifolds**

Speaker：**Shicheng Wang** (Peking University)

Time：2020-05-15 20:00-21:00 Beijing time (15:00-16:00 Moscow time)

Abstract: We will discuss recent results on virtual representation volumes and finiteness of the mapping degree set on 3-manifolds.

**Lecture 2 - ****Topology of integrable systems on 4-manifolds**

Speaker：**Elena Kudryavtseva** (Moscow State University)

Time：2020-05-15 21:00-22:00 Beijing time (16:00-17:00 Moscow time)

Abstract: We will give a survey on the topology of integrable Hamiltonian systems on 4-manifolds. Open questions and problems will be also discussed. Recall that, from a topological point of view, an integrable Hamiltonian system can be treated as a singular Lagrangian fibration on a smooth symplectic 2n-manifold whose generic fibres are n-dimensional tori. By a singularity, we mean either a singular point or a singular fibre of the fibration. The topological structure of such singularities is very important for understanding the dynamics of integrable systems both globally and locally. Our goal is to describe topological invariants of such singularities and obtain their classification up to fibrewise homeomorphism (for time being we forget about symplectic structure). The next step is to combine these singularities together to study the global structure of the fibration. For many integrable systems, this structure is completely determined by topological properties of singularities.

**For more information, please click on the following link：**

https://disk.pku.edu.cn:443/link/3FFA6A7EF0B27A568A9C20B99FA076F6

**************

**Organized by:**

◆ Sino-Russia Mathematical Center:

◆ Mathematics Department, School of Mathematical Sciences (SMS), Peking University

◆ Beijing International Center for Mathematical Research (BICMR), Peking University

◆ Department of Mechanics and Engineering Science, College of Engineering (EC), Peking University

◆ Mathematics Institute of Academy of Mathematics and Systems Science of Chinese Academy of Sciences (AMSS)

◆ Moscow State University (MSU)

◆ Steklov Mathematical Institute （MIAN)

◆ Steklov International Mathematical Center

◆ Moscow Center of Fundamental and applied Mathematic

Logo and website of Moscow Center of Fundamental and applied Mathematic

https://mathcenter.ru/en