# YMSC-2020暑期线上公开课-基础数学方向

## Venue: Online

Speaker: Dr. Fei Wei (Yau Mathematical Sciences Center, Tsinghua University)
Time: 13:30-15:30, every Mon. between Jul. 6 th and Jul. 27 th
Tencent Meeting ID: 869 422 2242
Title: An introduction to bounded gaps between primes
Abstract: In these lectures, we introduce the work of Yitang Zhang on bounded gaps between primes ([1]) and Maynard's improvement ([2]) on it.
[1] Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, 179(2014), 1121-1174.
[2] James Maynard, Small gaps between primes, Annals of Mathematics, 181(2015), 383-413.

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Speaker: Dr. Bin Zhao (Morningside Center of Mathematics, Chinese Academy of Sciences)
Time: 10:00-12:00, every Tue. & Thurs. between Jul. 7th and Aug. 27th
Tencent Meeting ID: 498 966 9233
Title: Abelian Varieties
Abstract: The course will give a general introduction to abelian varieties, including complex analytic theory, general algebraic theory (over an arbitrary field), duality, endomorphism algebras, and some special results over finite fields, global fields (and local fields if time permits). Emphasis will be put on the algebraic and arithmetic sides of the theory. Familiarity with the theory of schemes (e.g. Chapter 2 and 3 of Hartshorne’s Algebraic geometry) will be very helpful. We will use Mumford’s Abelian varieties as the textbook.

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Speaker: Prof. Vassily Manturov (Moscow State Technical University)
Time: 15:30-17:00, July 7th/10 th/ 14 th
Zoom Meeting ID: 831 5020 0580
Title: Groups G_{n}^{k}, \Gamma_{n}^{k} and Invariants
Abstract:
Lecture 1. The main principle. First examples. The braid groups and the G_{n}^{k} groups
Lecture 2. The Groups \Gamma, higher dimensional spaces, submanifolds, and higher-dimensional braids.

Lecture 3. Representations of \Gamma, cluster algebras, actions of braids on knots and groups of tilings.

Time: 15:30-17:00, July 28, Aug 1/ 4
Zoom Meeting ID: 831 5020 0580
Title: Framed 3-graphs and picture-valued invariants.
Abstract:
Lecture 1. Free knots and parity. Picture-valued invariants and the formula [K]=K.
Lecture 2. The group G_{n}^{3} and framed 3-graphs. Invariants of 3-free knots and maps from 3-free knots to free knots
Lecture 3. Link homotopy (after N.Habegger and X-S.Lin). Invariants of links up to link homotopy. Unsolved problems

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Speaker: Prof. Peng Shan (Yau Mathematical Sciences Center, Tsinghua University)
Time: 9:30-11:30, every Wed. between Jul. 8th and Jul. 29 th
Zoom Meeting ID: 184 3044 3981
Title: Quiver Hecke Algebras and applications
Abstract: Quiver Hecke algebras play a prominent role in representation theory - they provide algebraic categorification of quantum groups, and they control symmetries on representations of Lie algebras and algebraic groups. More and more applications of these algebras are being discovered. In particular, recent series of works by Kang-Kashiwara-Kim-Oh-Park reveal deep links between quiver Hecke algebras, quantum affine algebras and monoidal categorification of cluster algebras. In this course, we will give an introduction to these topics.

7.15 Tencent Meeting ID:  269 750 058，Password：100084

7.22  Tencent Meeting ID:  545 263 330，Password：100084

7.29  Tencent Meeting ID:  563 619 609，Password：100084

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Speaker: Dr. Qijun Yan (Yau Mathematical Sciences Center, Tsinghua University)
Time: 13:30-15:30, every Wed. & Fri. between Jul. 8th and Aug. 28 th
Zoom Meeting ID: 878 7444 1433
Title: Introduction to Shimura varieties
Abstract: The course will start with definitions and examples of Shimura varieties. We mainly focus on those types of Shimura varieties which admit moduli interpretations (e.g. moduli space of principally polarized abelian varieties with level structures).  We will also discuss canonical models and integral models for Shimura varieties.

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Speaker: Leonardo Trivellato Rolla (University of Warwick)

Time: 15:00-17:00 (Beijing time) July 20th, 22nd, 24th, 27th, 29th.

Title: Activated random walks on Z^d

Abstract: Some stochastic systems are particularly interesting as they exhibit critical behavior without fine- tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with of a framework of tools and methods which is finally becoming more structured. In this mini-course we present the existing results and explain the techniques developed so far.

July 20th

Tencent: 808 254 867

Zoom: 646 447 42718 (密码: 350370)

July 22nd.

Tencent: 629 962 219

Zoom: 681 483 43498 (密码: 333331)

July 24th.

Tencent: 839 854 739

Zoom: 623 283 99967 (密码: 582848)

July 27th.

Tencent: 628 429 272

Zoom: 679 795 91392 (密码: 788589)

July 29th.

Tencent: 179 352 499

Zoom: 653 957 61981 (密码: 649848)

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Speaker: Yilin Wang (Massachusetts Institute of Technology)
Time: 9:00-10:30 am. Aug. 3rd / 5th/ 10th/ 12th
Title: Large deviation principles of Schramm-Loewner evolutions
Abstract: The theory of large deviations (LDP) concerns the asymptotic behavior of the probability of rare events under a sequence of probability measures. Schramm-Loewner evolution (SLE) is a one-parameter family of random fractal non-self-crossing curves that arise naturally as interfaces in two-dimensional conformally invariant systems. After explaining the basic ideas and results on LDP and SLE, we describe the large deviations of SLE in two regimes: when the parameter goes to 0 and to infinity. The rate functions of these LDPs are interesting in their own, especially from the point of view of geometric function theory. The LDPs of SLEs then allow us to derive several new results in geometric function theory, by giving probabilistic interpretation to deterministic objects.
Aug. 3rd.
Tencent meeting ID: 382 712 091
Zoom ID: 683 926 47011 (Password: 583699)
Aug. 5th.
Tencent meeting ID: 140 192 150
Zoom ID: 694 678 95296 (Password: 758495)
Aug. 10th.
Tencent meeting ID: 175 211 271
Zoom ID: 613 257 16599 (Password: 352330)
Aug. 12th.
Tencent meeting: 196 507 842
Zoom ID: 648 833 71630 (Password: 149231)

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Speaker: Dr. Zhenfu Wang (University of Pennsylvania)
Time: 9:00-10:30, every Mon.& Wed. between Aug. 17 th and Sep. 2nd
Meeting ID: TBA
Title: Topics in Analysis of Many Particle Systems

Abstract: This summer course will present some mathematical tools and concepts for the rigorous derivation and study of nonlinear partial differential equations (PDE's) arising from many-particle limits: (McKean-)Vlasov type equations, the vorticity formulation of the 2D incompressible Euler/Navier-Stokes equations, Boltzmann collision equations, nonlinear diffusion equations, quantum Hartree equations... Depending on time and interest it will include part or all of the following topics: the Liouville/Master equations of N-particle systems, the notion of empirical measures, the BBGKY hierarchy,the Hewitt-Savage theorem, the Dobrushin's stability estimate, the coupling method, the concepts of chaos and entropic chaos, the recent progresses on the mean-_eld limit, in particular, the relative entropy/modulated potential energy/modulated free energy methods as introduced in：

——P.-E. Jabin and Z. Wang, Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces. J. Funct. Anal. 271 (2016) 3588-3627.

——P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with W-1，∞ kernels. Inventiones mathematicae 214(1) (2018) 523-591.

——S. Serfaty (appendix with M. Duerinckx), Mean Field Limit for Coulomb-Type Flows. To appear in Duke Math. J.

——D. Bresch, P.-E. Jabin and Z. Wang. On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak-Keller-Segel model. Comptes Rendus Mathematique 357(9) (2019) 708-720.