Lecture Series IV - July 14th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）
Lecture 1 - WORD MAPS ON SIMPLE ALGEBRAIC GROUPS AND RELATED TOPICS
Speaker: Prof. Nikolai Gordeev, Saint Petersburg State Pedagogical University
Time：2020-07-14 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)
Lecture 2 - Structure of Commutator Subgroups
Speaker: Prof. Zuhong Zhang， Beijing Institute of Technology
Time：2020-07-14 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)
This talk is based on the joint works with R. Hazrat, A. Stepanov and N. Vavilov in the past decade. In his seminal paper, more than half a century ago, Hyman Bass initialed the study of commutator subgroups and commutator formulas over rings. Since then, it attracted great attend of many leading experts including A. Bak, A.A. Suslin, L.N. Vaserstein, etc.. Various commutator formulas have been obtained in stable and non-stable settings and for a range of classical and algebraic like-groups.
In this talk, we will describe some recent results on the study (higher/birelative) commutators in general linear groups $GL(n,A)$ as well as their elementary generators. we will also discuss some further related research and applications.
Lecture Series III - June 30th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）
Valid Until: 2025-08-31 23:59
Lecture 1 - Dynamics in the space of metrics and new invariants in ergodic theory
Speaker: Prof. Anatoly Vershik, St. Petersburg Department of Steklov Institute of Mathematics, Russian Academy of Sciences
Time：2020-06-30 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)
1. Admissible metrics on the measure space: new trend in the theory of mm spaces.
2. Classification of mm-spaces. Matrix distributions.
3.Thе actions of measure preserving groups in the space of admissible metrics in measure space. ($\ca M \subset L^1(X\otimes X, \mu \otimes \mu).
4.Average metrics, ergodic limit and asymptotic invariants,
5.Sclaing entropy. Scaling entropy function. Examples.
6.Theorem. Bounded scaling entropy and discrete spectra. Sequential entropy by Kushnirenko. All possible scalings.
7.New geometrical problems.
Valid Until: 2025-07-31 23:59
Prof. Vershik is the Head of the Laboratory of Representation Theory and Dynamical Systems at Saint Petersburg Department of Steklov Mathematical Institute and professor at Saint Petersburg State University. He was President of the Saint Petersburg Society from 1998 to 2008. He is a member of European Academy of Sciences (since 2015), he was a member of Executive Committee of European Mathematical Society (1996-2000), Laureate of Humboldt Research Award - 2007 and an ICM invited speaker (1974 and 1994), Miller-professor (Berkeley, 1995) and Simons-professor (MSRI, 2008). His research interests include but are not limited to
- infinite-dimensional groups and their Asymptotic Representation Theory;
- Lie groups;
- New methods of Representation Theory for finite symmetric groups;
- Combinatorial Probability Theory and limit forms for configurations (the Vershik-Kerov theorem on limit forms of Young diagrams, Bratelli-Vershik diagrams, etc) ;
- Universal objects in Combinatorics Geometry and Dynamical Systems;
- Dynamical Systems and Ergodic Theory;
- Non-holonomic Geometry and Mechanics;
- Random processes, random walks and random matrices;
Lecture 2 - The characteristic factors in dynamical systems
Speaker: Prof. Ye Xiangdong, University of Science and Technology of China
Time：2020-06-30 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)
It is an ideal situation if we can reduce a problem P stated for general ergodic or minimal systems to the same problem in their "simple factors". We will explain how one did so for the problem on the convergence of multiple ergodic averages in ergodic theory. Moreover, we will present a recent work by Glasner-Huang -Shao-Weiss-Ye on the similar problem in the topological setup.
On the way to do so, we will address the parallels between topological dynamics and ergodic theory, and their applications to combinatoric number theory.
Prof. Ye got his Ph. D. in Mechanics and Mathematics Department of Moscow State University in 1991, and then did postdoc work in ICTP during 1991-1993. He was a faculty of Mathematics School in USTC since 1993. He was selected as a member of the Chinese Academy Sinica in 2019. His research interests include topological dynamics, ergodic theory and combinatoric number theory.
Lecture Series II - June 16th, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）
Lecture 1 - Multiple structures for quasilinear equations by the variational method
Speaker: Alexander Nazarov, PDMI RAS and Math&Mech Faculty, St. Petersburg State University
Time：2020-06-16 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)
We study entire bounded solutions to the equations of variational nature. The model example here is $\Delta u - u + u^3 = 0$ in $\mathbb R^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in an unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.),both positive and sign-changing. It is also applicable for more general equations in any dimension.
The talk is based on the joint paper Lerman L.M., Naryshkin P.E., Nazarov A.I., Abundance of entire solutions to nonlinear elliptic equations by the variational method, Nonlinear Analysis -- TMA. 190 (2020), DOI 10.1016/j.na.2019.111590, 1-21.
Lecture 2 - Periodic and quasi-periodic solutions of 1-d Q-curvature equation.
Speaker: Jiang Meiyue, School of Mathematical Sciences, Peking University
Time：2020-06-16 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)
Expiration Time：2025-07-31 23:59
Lecture Series I - June 2nd, 2020 （20:00-22:00 Beijing time or 15:00-17:00 St Petersburg time）
Lecture 1 - Spectral synthesis for systems of exponentials and reproducing kernels
Speaker：Anton Baranov (Saint Petersburg State University)
Time：2020-06-02 20:00-21:00 Beijing time (15:00-16:00 St Petersburg time)
Let $x_n$ be a complete and minimal system of vectors in a Hilbert space $H$. We say that this system is hereditarily complete or admits spectral synthesis if any vector in $H$ can be approximated in the norm by linear combinations of partial sums of the Fourier series with respect to $x_n$. It was a long-standing problem whether any complete and minimal system of exponentials in $L^2(-a,a)$ admits spectral synthesis. Several years ago Yu. Belov, A. Borichev and myself gave a negative answer to this question which implies, in particular, that there exist non-harmonic Fourier series which do not admit a linear summation method. At the same time we showed that any exponential system admits the synthesis up to a one-dimensional defect. In the talk we will also discuss related problems for systems of reproducing kernels in Hilbert spaces of entire functions (such as Paley-Wiener or Fock).
Expiration Time：2025-07-31 23:59
Lecture 2 - On gauged linear sigma model and related problems
Speaker: Prof. Huijun Fan (Peking University)
Time：2020-06-02 21:00-22:00 Beijing time (16:00-17:00 St Petersburg time)
Gauged linear sigma model was proposed by Witten in the early of 90's to explain the mirror symmetry phenomenon and the CY/LG correspondence conjecture. In this lecture, I will firstly formulate the mathematical framework of the GLSM, and then describe an algebraic way to construct the quantum invariants of GLSM in narrow case (for general gauge group) via quasimaps.
This was a joint work with Jarvis and Ruan. Finally I will report the recent progress in this field and related problems.
Expiration Time：2025-07-31 23:59