##### Description

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

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

(2) Liang Xiao (BICMR, number theory and arithmetic geometry)

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

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

(5) Baohua Fu (AMSS, algebraic geometry: singularity 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 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.

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**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.

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**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.

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**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.

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**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.

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**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).

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**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.

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__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: **

**https://disk.pku.edu.cn:443/link/0B14E32445D33A9394769538904C939A**

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

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**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