• Title: Hilbert quotients and Hilbert families of Grassmannians
    Yi Hu (University of Arizona), 1/16, 14:00PM-16:00PM, Room 412 Zhihua Building

    Abstract: By Mnev’s universality theorem, every singularity type naturally occurs in matroid strata of Grassmannians. This motivates to study the Hilbert quotients and Hilbert families of Grassmannians.  In this talk, we begin with a quick review of Mnev’s universality, a gentle introduction of Hilbert quotient, then introduce the structures of the Hilbert quotients and Hilbert families of Grassmannians and explain their implications. The talk is made accessible for the general audience with a background in algebraic geometry.

  • Title: Intersection theoretic inequalities via Lorentzian polynomials
    Jian Xiao (Tsinghua University), 12/27, 15:30PM-16:30PM, Room 312 Zhihua Building

    Abstract: The theory of Lorentzian polynomials was recently introduced and systematically developed by Braden-Huh and independently (with part overlap) by Anari-Liu-Gharan-Vinzant. It has many important applications in combinatorics, including a resolution of the strongest version of Mason conjecture and new proofs of the Heron-Rota-Welsh conjecture. In this talk, we explore its applications to geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property. We will discuss the origin of the rKT property in analytic geometry, and its connections with the submodularity for numerical dimension type functions and the sumset estimates for volume type functions. Joint work with J. Hu.

  • Title: Plateau问题,极小集,以及奇点分类
    Xiangyu Liang (Beihang University), 12/20, 13:30PM-14:30PM, Room 412 Zhihua Building

    Abstract: Plateau问题是19世纪同名物理学家提出的给定边界能量极小物理对象存在性及局部结构 刻划的著名古典问题。至今一直备受关注,其在三维空间最特殊情况下的解使Douglas获首届菲尔兹奖; 此后为应对一般情况下广泛存在的奇点现象,许多顶尖数学家针对该问题持续研究,发展出几何测度论。因其复杂性,人们对Plateau问题解局部结构的认识仍非常有限,很多基本问题未解决,包括正则性,奇点分类,切结构唯一性。 在这次报告中,我们将首先介绍Plateau问题,并讨论一些经典的数学模型。 这些模型中,Almgren极小集模型所得到的奇点类型恰与物理中所能观测到的肥皂膜奇点类型相吻合。 因此随后我们将以Almgren极小集模型为例,进一步介绍局部结构的研究思路。如果时间允许,我们将进一步讨论奇点分类以及相关的最新进展。

  • Title: Hodge-Riemann property of Griffiths positive matrices with (1,1)-form entries
    Zhangchi Chen (CAS), 12/13, 15:30PM-16:30PM, Room 412 Zhihua Building

    Abstract: The classical Hard Lefschetz theorem (HLT), Hodge-Riemann bilinear relation theorem (HRR) and Lefschetz decomposition theorem (LD) are stated for a power of a Kähler class on a compact Kähler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative. Explicit counterexamples of bidegree (2,2) classes in dimension 4 can be found in Timorin (1998) and Berndtsson-Sibony (2002). Dinh-Nguyên (2006, 2013) proved the mixed HLT, HRR, LD for a product of arbitrary Kähler classes. Instead of products, they asked whether determinants of Griffiths positive $k\times k$ matrices with (1,1) form entries in C^n satisfies these theorems in the linear case. In a recent work I gave positive answer when k=2 and n=2,3. Moreover, assume that the matrix only has diagonalized entries, for k=2 and $n\geq 4$, the determinant satisfies HLT for bidegrees (n−2,0), (n−3,1), (1,n−3) and (0,n−2). In particular, Dinh-Nguyên's question has positive answer when k=2 and n=4,5 with this extra assumption. The proof uses a Heron's formula type factorization, observed by computer (Mathematica).

  • Title: The regularity problem for axially symmetric Navier-Stokes equations on some bounded regions
    Xin Yang (Southeast University), 11/29, 15:30PM-16:30PM, Room 412 Zhihua Building

    Abstract: The regularity problem of the Navier-Stokes equations (NS) in $\mathbb R^3$ asks whether a global smooth solution exists for any initial velocity $v_0$ that is divergence free and lies in the Schwartz class $\mathcal S(\mathbb R^3)$. This problem is still wide open in general for large initial values, and one of the essential barriers is the supercriticality of the (NS). If confining attention to axially symmetric vector fields, it was observed that the axially symmetric Navier-Stokes equations (ASNS) are critical after some proper transformations, which raises some hope to settle the regularity problem for (ASNS). Despite this problem on $\mathbb R^3$ remains open as well, it was solved recently on some particular bounded cusp domains with a Navier-slip boundary condition. Motivated by this work, we continue to study the regularity problem for (ASNS) on more regular and more realistic bounded regions than those cusp domains under the Navier-Hodge-Lions (NHL) boundary condition.

  • Title: Entire curves generating all shapes of Nevanlinna currents
    Song-Yan Xie (CAS), 11/22, 15:30PM-16:30PM, Room 208 Zhihua Building

    Abstract: First, we show that every complex torus $\mathbb{T}$ contains some entire curve $g: \mathbb{C}\rightarrow \mathbb{T}$ such that the concentric holomorphic discs $\{g\restriction_{\overline{\mathbb D}_{r}}\}_{r>0}$ can generate all the Nevanlinna/Ahlfors currents on $\mathbb T$ at cohomological level. This confirms an anticipation of Sibony. Developing further our new method, we can construct some twisted entire curve $f: \mathbb{C}\rightarrow \mathbb{CP}^1\times E$ in the product of the rational curve $\mathbb{CP}^1$ and an elliptic curve $E$, such that, concerning Siu's decomposition, demanding any cardinality $|J|\in \mathbb{Z}_{\geqslant 0}\cup \{\infty\}$ and that $\mathcal{T}_{\mathrm{diff}}$ is trivial ($|J|\geqslant 1$) or not ($|J|\geqslant 0$), we can always find a sequence of concentric holomorphic discs $\{f\restriction_{\overline{\mathbb D}_{r_j}}\}_{j\geqslant 1}$ to generate a Nevanlinna/Ahlfors current $\mathcal{T}=\mathcal{T}_{\mathrm{alg}}+\mathcal{T}_{\mathrm{diff}}$ with the singular part $\mathcal{T}_{\mathrm{alg}}=\sum_{j\in J} \,\lambda_j\cdot[\mathsf C_j]$ in the desired shape. This fulfills the missing case where $|J|=0$ in the previous work of Huynh-Xie. By a result of Duval, each $\mathsf C_j$ must be rational or elliptic. We will show that there is no a priori restriction on the numbers of rational and elliptic components in the support of $\mathcal{T}_{\mathrm{alg}}$, thus answering a question of Yau and Zhou. Moreover, we will show that the positive coefficients $\{\lambda_j\}_{j\in J}$ can be arbitrary as long as the total mass of $\mathcal{T}_{\mathrm{alg}}$ is less than or equal to $1$. Our results foreshadow striking holomorphic flexibility of entire curves in Oka geometry, which deserves further exploration. This is joint work with Hao Wu (NUS).

  • Title: Concepts of Geometry in Condensed Matter Physics
    Xu Yang (The Ohio State University), 7/27, 7/29, 7/31, 8/2, 8/4, 8/6, 8/8, 20:00-22:00

    Course Overview: I will focus on the role of geometry in condensed matter physics. I will develop necessary mathematical techniques in solving physics-motivated problems along the way. Topics including the Fermi liquid theory, quantum Hall effects, response theories and quasi-crystals will be discussed. Mathematical tools of Morse theory and algebraic topology will be introduced. This short course will be lectures aiming at providing students with a perspective of the role of geometry in condensed matter physics. Students with a general physics and math background (~level of sophomore) and a certain degree of mathematical maturity are encouraged to enroll.

    Suggested Reading:

    1. Morse Theory, J. Milnor
    2. Solid State Physics, N. Ashcroft and D. Mermin
    3. Many-Body Physics, Topology and Geometry, S. Sen and K. Gupta
    4. Geometric Phases in Physics, A. Shapere and F. Wilczek (ed.)
    5. Topology in Condensed Matter, M. I. Monastyrsky

    Course contents: I will be focusing on the following topics.

    1. Introduction to Morse theory: Morse inequality
    2. Physics of van Hove singularities: quantum oscillations, Lifshitz transitions, shape of Fermi surfaces
    3. Weyl semimetals and the idea of Berry phases
    4. Insulators: crystals, quasi-crystals and topological defects
    5. Insulators: band topologies

    Homework Policy: There will be homework problems to help students get familiar with ideas and calculations. In addition, I would recommend students write a term paper about interesting topics related to geometry in condensed matter physics. I will provide references if students need guidance/advice on the potential topics of their interest.

  • Title: Fine-scale distribution of roots of quadratic congruences
    Matthew Welsh (University of Bristol), 11/5, 10:00AM-11:59AM

  • Title: Invariance of plurigenera for generalized pairs with abundant nef parts (I)&(II)
    Zhan Li (SUSTech), 10/22, 10/24, 10:00AM-11:59AM

  • Title: Big quantum cohomology of Fano complete intersections
    Xiaowen Hu (Sun Yat-sen University), 10/15, 10:00AM-11:59AM

2021 Spring

  • Title: Kudla Rapoport conjecture over the ramified primes
    Yousheng Shi (University of Wisconsin-Madison), 5/27, 10:00AM-11:59AM