Wenhan Dai

PKU Mathematics Forum on Pure Mathematics

• Time: August 2 to 5, 2023.
• Venue: Big Hall, Jia-yi-bing Building, Jingchunyuan 82, BICMR, Peking University.

Lecture Archives (in chronological order)

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August 1

• Yi Ni 倪忆 (CalTech) – The next-to-top term in knot Floer homology.
  Knot Floer homology is a knot invariant which categorifies the Alexander polynomial. This invariant contains a lot of information about the topology of the knot. For example, the topmost Alexander grading of the knot Floer homology of a knot is the Seifert genus of this knot, and the summand of the knot Floer homology at this grading has rank one if and only if the knot is fibered. In recent years, it became clear that the summand of the knot Floer homology at the next-to-top Alexander grading also contains information about the topology of the knot. In this talk, we will address the question whether this summand is nontrivial, and discuss the information contained in this summand about the monodromy when the knot is fibered.

• Xinwen Zhu 朱歆文 (Stanford) – The p-adic Borel Hyperbolicity of Ag.
  A theorem of Borel says that any holomorphic map from a smooth complex algebraic variety to a smooth arithmetic variety is automatically an algebraic map. The key ingredient is to show that any holomorphic map from the punctured disc to the arithmetic variety has no essential singularity. I will discuss some work towards a p-adic analogue of this theorem for moduli of abelian varieties. Joint with Abhishek Oswal and Ananth Shankar.

• Zhipeng Liu 刘志鹏 (Kansas) – Some exact formulas of the KPZ fixed point and directed landscape.
  In the past twenty years, there have been huge developments in the study of the Kardar-ParisiZhang (KPZ) universality class, which is a broad class of phy ical and probabilistic models including one-dimensional interface growth processes, interacting particle systems and polymers in random environments, etc. It is broadly believed and partially proved, that all the models share the universal scaling exponents and have the same asymptotic behaviors. The height functions of models in the KPZ universality class are expected to converge to a limiting space-time fluctuation field, the KPZ fixed point. Moreover, there is a random “directed metric” on the space-time plane that is expected to govern all the models in the KPZ universality class. This “directed metric” is called the directed landscape. Both the KPZ fixed point and the directed landscape are central objects in the study of the KPZ universality class, while they were only characterized/constructed very recently [MQR21, DOV18]. In this talk, we will discuss some exact formulas of distributions in these two random fields and their analogs in the periodic domain. These exact formulas are given by Fredholm determinant or their analogs. We will show some surprising probabilistic properties of the KPZ fixed point and the directed landscape using the exact formulas. Some of the results are based on joint work with Jinho Baik, Yizao Wang, and Ray Zhang.

• Yao Yao 姚珧 (NUS) – Small scale formations in fluid equations with gravity.
  In this talk, we discuss some PDEs that describe fluid motion under the influence of gravity, including the incompressible porous media equation and incompressible Boussinesq equation in two dimensions. Using an interplay between various monotone and conserved quantities, we construct rigorous examples of small scale formations as time goes to infinity. These growth results work for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth. (Based on joint works with Alexander Kiselev and Jaemin Park).

August 3

• Chenyang Xu 许晨阳 (Princeton) – Kähler-Einstein metric, K-stability and moduli spaces.
  The question of whether a smooth complex variety with a positive first Chern class, called a Fano variety, has a Kähler-Einstein metric has been a major topic in complex geometry since the 1980s. In the last decade, algebraic geometry, or more specifically higher dimensional geometry has played a surprising role in advancing our understanding of this problem. The interplay between complex geometry and algebraic geometry has also provided deep insights into higher dimensional algebraic geometry itself, peaked by the construction of a projective moduli space that parametrizes Fano varieties. More precisely, the moduli space parametrizes Fano variety satisfying the stability condition which is used to characterize the existence of a Kähler-Einstein metric - known as K-stability. In the lecture, I will explain the main ideas behind the recent progress of the field.

• Yue Yu 余越 (CalTech) – Moduli space of non-archimedean holomorphic disks.
  I will describe the moduli space of non-archimedean holomorphic disks in affine log Calabi-Yau varieties, which is foundational to the non-archimedean mirror symmetry program. I will discuss boundary conditions, smoothness, dimension and properness. Smoothness relies on the nonarchimedean deformation theory joint with M. Porta. Properness relies on formal models and Temkin’s s theory of reduction of germs. Work in progress with S. Keel.

• Gang Liu 刘钢 (East China Normal Univ.) – Gromov-Hausdorff convergence of Kähler manifolds.
  We survey recent progress on Gromov-Hausdorff convergence of Kähler manifolds with geometric applications.

• Zhouli Xu 徐宙利 (UCSD) – The Adams differentials on the classes $h(j,3)$. (article)
  In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes $h(j)$, resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill-Hopkins-Ravenel proved that the classes $h(j,2)$ support non-trivial differentials for $j \geq 7$, resolving the celebrated Kervaire invariant one problem.

• Xin Zhou 周鑫 (Cornell) – Some recent development in minimal surface theory.
  We will present some recent progress on two problems in minimal surface theory posed by S. T. Yau in 1982. In particular, we will discuss the existence of infinitely many closed minimal hypersurfaces in a closed Riemannian manifold and the existence of four closed minimal twospheres in a Riemannian three-sphere.

August 4

• Wei Zhang 张伟 (MIT) – Algebraic cycles over number fields and L-functions: conjectures and results..
  Algebraic cycles are among the most fundamental mathematical objects. Those defined over number fields are of particular interest. There are also analytic invariants, the Hasse-Weil L-functions, attached to algebraic varieties over number fields. We will review the history of a few conjectures on the connection between them, focusing on that of Tate and Birch-SwinnertonDyer. We will then state a few recent results towards these conjectures, proved by studying automorphic period integral and arithmetic intersection theory on Shimura varieties.

• Ziyang Gao 高紫阳 (Hannover) – Sparsity of Rational and Algebraic Points.
  It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points. In my talk, I will explain the historical and recent developments of this problem according to the different grades. Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.

• Ruixiang Zhang 张瑞祥 (UCB) – Fourier restriction type problems: New developments in the last 15 years.
  Fourier restriction problems are a significant class of problems in harmonic analysis. Given a function in Euclidean space, whose Fourier transform is supported on a curved submanifold, the Fourier restriction problem is concerned with obtaining optimal estimates for certain L^p norms of such functions. In addition to harmonic analysis, restriction problems have close connections to analytic number theory, partial differential equations, geometric measure theory, and mathematical physics. In the past fifteen years, mathematicians have made significant progress in the field of restriction problems. In this report, I will introduce this type of problem, discuss some recent breakthroughs from the past fifteen years, and finally explore some future directions worth considering in this field.

• Hong Wang 王虹 (UCLA) – Incidence estimates and applications.
  Let $P$ be a set of points and $L$ be a set of lines in the plane, what can we say about the number of incidences between $P$ and $L$, $I(P, L):=|{(p, l) \in P \times L, p \in L}|$? The problem changes drastically when we consider a thickening version, i.e. when $\mathrm{P}$ is a set of unit balls and $L$ be a set of tubes of radius 1. We will survey some classical and modern incidence theorems and discuss their connection to combinatorics, geometric measure theory, harmonic analysis, and dynamics.

August 5

• Xuhua He 何旭华 (CUHK) – Towards a Geometric Theory of Characters.
  In the field of representation theory, understanding the behavior of characters has long been a central pursuit. Characters are fundamental objects that encode crucial information about the symmetries inherent in mathematical structures. In this talk, we embark on an exciting journey towards a geometric theory of characters, a captivating framework that reveals hidden connections between algebraic geometry, combinatorics, and the vast landscape of representation theory. We will begin by exploring the original algebraic definition of characters, focusing on their significance in groups and group algebras. From there, we will delve into Lusztig’s theory of character sheaves on $GL(n,\bar{Fq})$, which serves as a geometric counterpart to characters of the finite group $GL(n,Fq)$. This geometric perspective unveils remarkable connections between algebraic geometry of algebraic groups and their flag variety and the study of characters of finite groups of Lie type. In the end, we will embark on an ongoing project aimed at extending the theory of character sheaves to loop groups.

• Zhiwei Yun 恽之玮 (MIT) – Functions on the commuting scheme via Langlands duality.
  I will explain how ideas from the (geometric) Langlands program help solve the following purely algebraic problem: describe the ring of conjugation-invariant functions on the scheme of commuting pairs in a complex reductive group. The answer was known up to nilpotents, and we show that this ring is indeed reduced. We also describe the ring of invariant functions on the derived version of the commuting scheme. The proof brings in seemingly unrelated objects such as the affine Hecke category and character sheaves (of the Langlands dual group). This is joint work with Penghui Li and David Nadler.

• Botong Wang 王博潼 (Wisconsin-Madison) – Perverse sheaves and positivity of Euler characteristics.
  Some geometric or topological properties of a space have or conjecturally have implications on the sign of its Euler characteristics. For example, the Singer-Hopf conjecture predicts that if a closed 2d-dimensional manifold M has non-positive sectional curvature, or more generally has contractible universal cover, then $(-1)^d\chi(M)\geq 0$. We will discuss a few such examples and their generalizations in terms of perverse sheaves.

• Hao Shen 申皓 (Wisconsin-Madison) – Stochastic quantization of Yang-Mills.
  Stochastic quantization is a bridge between quantum field theory on one hand and stochastic analysis on the other hand Through this connection, it is possible to put certain functional integrals in quantum field theory on rigorous footing, and prove various properties of them; I will discuss some recent progress and ongoing efforts. These studies also foster the developments of stochastic analysis so the two fields cross-fertilize. I will focus on our work with M. Hairer et al on the Yang-Mills model in 2 and 3 dimensions, discuss the construction of singular orbit space, local stochastic dynamics, singular holonomies etc. On lattice, stochastic quantization can be also used to prove interesting properties such as mass gap at strong coupling.