Title:
Sai-Kee Yeung (Purdue University), 7/, Room 412 Zhihua Building
Abstract:
Title: Initial degenerations of Grassmannians
Daniel Corey (University of Nevada), 6/10, 10:00AM-11:00AM, Room 412 Zhihua Building
Abstract: In this talk, I will describe a relationship between initial degenerations of the Grassmannian Gr(d,n) and inverse limits of matroid strata of the Grassmannian. For (d,n) = (2,n), (3,6), (3,7), and (3,8), this relationship, together with a large combinatorial analysis, is used to prove that Gr(d,n) is schön in the sense of Tevelev. The primary application of this result is a resolution of a conjecture of Keel and Tevelev, that the Chow quotient of Gr(d,n) by the diagonal torus of PGL(n) is the log canonical compactification of the moduli space of n lines in P^{d-1} (for the pairs (d,n) listed above). Recently, we showed that Gr(d,n) is not schön for an infinite number of pairs (d,n). Part of this work is joint with Dante Luber.
Title: Efficient quantum algorithms for state preparation: ground and thermal state
Zhiyan Ding (University of California, Berkeley), 6/3, 10:00AM-11:00AM, Room 412 Zhihua Building
Abstract: One of the most promising applications of quantum computers is to simulate the ground or thermal state properties of many-body quantum systems. We propose a framework for designing efficient ground/thermal state preparation quantum algorithms. As a Monte Carlo-style quantum algorithm, our method uses a Lindbladian where the target state is stationary, and its evolution can be efficiently implemented using just one ancilla qubit. We will mainly focus on presenting our algorithms for ground state preparation, with a brief mention of the extension to thermal state preparation. For the ground state preparation, our method stands out for its ability to prepare the ground state even when the initial state has zero overlap with the ground state, bypassing the most significant limitation of methods like quantum phase estimation. Some interesting open problems and future directions will also be mentioned at the end of the talk.
Title: Phase transitions with Allen-Cahn mean curvature bounded in Lp.
Shengwen Wang (Queen Mary University of London), 5/6, 10:00AM-11:00AM, Room 412 Zhihua Building
Abstract: We consider the varifolds associated to a phase transition problem whose first variation of Allen-Cahn energy is Lp integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded Lp mean curvature. This is joint work with Huy Nguyen.
Title: Geometry of Neighborhoods of Rational Curves (Distinguished Lecture)
Jun-Muk Hwang (IBS Center for Complex Geometry), 4/26, 14:00PM-15:00PM, Siyuan Room Zhihua Building
Abstract: The diffeomorphic equivalence type of the germ of neighborhoods of a compact complex submanifold in a complex manifold is determined by its normal bundle, but its biholomorphic equivalence type is much subtler and complicated. Even when the compact submanifold is just a smooth rational curve and its normal bundle is of the simplest type, the geometry of its neighborhoods can be very rich and puzzling. We discuss some recent results on this topic obtained by a combination of tools from algebraic geometry and differential geometry.
Title: The Kesten problem in higher dimensions
Matthew Welsh (University of Maryland), 4/3, 14:10PM-15:10PM, Room 412 Zhihua Building
Abstract: For an irrational $\alpha$ and an interval $ I = [x, x +u] \subset \RR/\ZZ$, consider the discrepancy $D_N(\alpha, x, u)$ between the number of $n \leq N$ with $n \alpha \pmod 1 $ in $I$ and the expected number, $uN$. In the 1960s, Kesten proved that if $\alpha$ and $x$ are taken uniformly at random, then $D_N$ normalized by $\rho(u) \log N$ converges to a Cauchy distribution. Similar results are known in higher dimensions due to work by Dolgopyat and Fayad; however with extra randomness akin to taking $u$ random as well. In work in progress with Dolgopyat and Fayad, we succeed in removing this extra randomness by proving extensions of equidistribution results on $\SL(d, \RR ) \ltimes \RR^d$ modulo $\SL(d, \ZZ) \ltimes \ZZ^d$ due to Strombergsson and Kim.
Title: Covariant derivative of eigenfunctions along parallel tensors
Fei Qi (University of Denver), 3/27, 14:10PM-15:10PM, Room 412 Zhihua Building
Abstract: The study is motivated by constructing the two-dimensional quantum nonlinear sigma model with the nonflat target manifold via vertex algebraic methods. When the target manifold is not flat, the zero modes of the vertex operators act via covariant derivatives. Thus, the vertex operators are expected to be noncommutative. To understand such vertex operators, the first step is to understand the covariant derivatives. In this talk, I will report my work on the covariant derivatives of eigenfunctions along the parallel tensors. Surprisingly, over a Riemannian manifold with constant nonzero sectional curvature, all the covariant derivatives are scalar multiples of the eigenfunction. We suspect that over Kahler and Calabi-Yau manifolds, the covariant derivatives of an eigenfunction might span a finite-dimensional space.
Title: Bounds on the Saturation Degree of a Smooth Projective Variety
Lawrence Ein (University of Illinois at Chicago), 3/21, 10:00AM-11:00AM, Room 412 Zhihua Building
Abstract: We discuss the results of Bertram-Ein-Lazarsfeld on effective Serre's vanishing theorem ant its applications. W'll also discuss recent joint work with Ha and Lazarsfeld on a question of Bombieri on a bound for the saturation degree. for a smooth projective variety in P^N.
Title: Asymptotic Syzygies of Algebraic Varieties
Lawrence Ein (University of Illinois at Chicago), 3/19, 10:00AM-11:00AM, Room 412 Zhihua Building
Abstract: We'll discuss how to study the geometry of algebraic varieties using the syzygies. We'll discuss the gonality conjecture of Green and Lazarsfeld on algebraic curves. We discuss the the various generalizations to higher dimensional varieties.
Title: Holomorphicity of totally geodesic isometries between bounded symmetric domains
Sung-Yeon Kim (IBS, Korea), 3/11, 13:00PM-14:00PM, Room 413 Zhihua Building
Abstract: The holomorphicity of isometries between complex manifolds is a classical topic in complex differential geometry going back to the work of Siu for strong rigidity of compact K¨ahler manifolds. In this talk, we investigate the properties of totally geodesic isometric embedding $f:\Omega\rightarrow\Omega'$ between two bounded symmetric domains $\Omega$ and $\Omega'$ with respect to their Bergman/Kobayashi metrics. In particular, we discuss the question of holomorphicity of totally geodesic isometric discs $f:\Delta\rightarrow\Omega$ that extend sufficiently smooth up to the boundary. As an application, we give a sufficient condition for a totally geodesic isometric embedding $f:\Omega\rightarrow\Omega'$ to be holomorphic or anti-holomorphic. This is a joint work with Aeryeong Seo.
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:
Course contents: I will be focusing on the following topics.
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
Title: Kudla Rapoport conjecture over the ramified primes
Yousheng Shi (University of Wisconsin-Madison), 5/27, 10:00AM-11:59AM