About GAGP Seminar
The main objective of the GAGP seminar is to introduce advanced topics mainly on algebra (A), algebraic and symplectic geometry (G), and mathematical physics(P), to jonior researchers and also to enhance interactions among scholars working on related fields.
Each GAGP seminar talk consists of two parts. In the first 30-minute talk, the speaker is encouraged to deliver an elementary introduction on the topics accessible to broad math graduate students or even senior undergraduate students. Then, a 10-minute break will be followed. After that, the speaker will spend one hour or so on more technical details.
TALKS IN 2017
Venue: Room 403, New Math. Building. Time: 16:00-17:30 Date: 09/11 (Room 416 )
Speaker: Dafeng Zuo (University of Science and Technology of China)
Title: Extended affine Weyl groups of BCD-type: their Frobenius manifolds and Landau-Ginzburg superpotentials
Abstract: For the root systems of type B_l, C_l and D_l, we generalize the result of B.Dubrovin and Youjin Zhang by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made by Dubrovin and Zhang. It also depends on certain additional data. We also construct Landau-Ginzburg superpotentials for these Frobenius manifold structures. This is a joint work with B.Dubrovin, Ian.Strachan and Youjin Zhang
Date: 8/04 (Room 416 Time: 14:30-17:30)
Speaker: Yongqiang Liu (KU Leuven) Title: The monodromy theorem for compact Kähler manifolds and smooth quasi-projective varieties Abstract: Given any connected topological space X, assume that there exists an epimorphism from the fundamental group of X to the free ableian group Z. The deck transformation group Z acts on the associated infinite cyclic cover of X, hence on the homology group of the covering space with complex coefficients. This action induces a linear automorphism on the torsion part of the homology group as a module over the complex Laurent polynomial ring, which is a finite dimensional complex vector space. We study the sizes of the Jordan blocks of this linear automorphism. When X is a compact K\"ahler manifold, we show that all the Jordan blocks are of size one. When X is a smooth complex quasi-projective variety, we give an upper bound on the sizes of the Jordan blocks, which is an analogue of the Monodromy Theorem for the local Milnor fibration. This is a joint work with Nero Budur and Botong Wang.
Speaker: Wenhao Ou (UCLA) Title: Positivity of Tangent bundles Abstract: During the last few decades, much progress has been made in classification of complex algebraic varieties. From the view point of Minimal Model Program, projective manifolds should be classified according to `sign' of their canonical class K_X. It is then natural to ask how far we can lift the positivity or the negativity of K_X to the cotangent bundle. In the positive case, Miyaoka showed that if K_X is pseudo effective, then the cotangent bundle is generically nef, that is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. As a consequence, she also showed that the second Chern class of X has non-negative intersection numbers with ample divisors in this case. We are interested in the negative case. We show that if K_X is nef, then the tangent bundle is generically nef, and the second Chern class of X has the same positivity as before. We also investigate under which conditions the postivities would be strict.
Speaker: Chengjian Yao (Universite liber de Bruxelle) Title: The extension of hypersymplectic flow under bounded torsion and convergence of global solution Abstract: A triple of symplectic forms on a differentiable 4-manifold, spanning a maximal positive subspace of \Lambda^2 at each point is called a hypersymplectic structure. This notion was introduced by Donaldson in his programme of studying the adiabatic limits of G_2 manifolds. HyperKaehler manifolds of dimension 4 give rich sources of hypersymplectic structures, and were conjectured by Donaldson to be the "only sources" in the compact case. We study a geometric flow---hypersymplectic flow-- of such structures, designed to deform a given hypersymplectic structure to a hyperKaehler one in the same cohomology class. This flow is a dimensional reduction of the more well-known G_2 Laplacian flow introduced by Hitchin and Bryant to study the existence of metrics with G_2 holonomy. We show that the hypersymplectic flow does not develop finite time singularity with uniformly bounded torsion. We also study the convergence of the global solution of the flow under various geometric conditions. This is joint work with Joel Fine.
Date: 7/03, Speaker: Siu-Hung Ng (Louisiana State University) Title: Frobenius-Schur indicators and exponents for finite groups, Hopf algebras and fusion categories Abstract: The exponent of a finite group is an invariant of the tensor category of its complex representations. Using Frobenius-Schur indicators, one can rediscover the Cauchy theorem for finite groups in terms of their exponents and orders. In this talk, we will discuss the Frobenius-Schur indicators for Hopf algebras as well as fusion categories in terms of categorical invariants. Moreover, we discuss some of their arithmetic properties and applications to the congruence subgroup theorem and the Cauchy theorem for fusion categories.
Date: 6/19, Speaker: Xiaokui Yang (Chinese Academy of Sciences) Title: Characterizations of projective spaces and quadrics by strictly nef bundles Abstract: In this talk, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety X^n (n\geq 3) has strictly nef \Lambda^2 TX, then it is isomorphic to the projective space P^n or the quadric Q^n. We also prove that on elliptic curves, strictly nef vector bundles are ample, whereas there exist Hermitian flat and strictly nef vector bundles on any smooth curve with genus g\geq 2. This is a joint work with Duo Li.
Date: 6/15 (Special Seminar. Venue Room 416. Time: 10:30-11:30), Speaker: Changlong Zhong (State University of New York at Albany) Title: Hecke-type algebras and equivariant cohomology of flag varieties Abstract: The so-called Schubert calculus deals with (equivariant) cohomology of flag varieties. A systematic way of dealing with Schubert calculus by using Hecke-type algebras was started by Demazure and Bernstein-Gelfand-Gelfand, and later continued and generalized by Arabia, Kostant-Kumar, Lusztig, Bressler-Evens. In the last 10 years it was then generalized by myself with Calmes, Savage and Zainoulline. Our work studies general oriented cohomology theory of flag varieties, and can be applied to simplify and generalize classical results on singular cohomology and K-theory of flag varieties. In this talk I will explain the main idea in this series of work.
Date: 6/12, Speaker: Ling Long (Louisiana State University) Title: Hypergeometric functions over finite fields Abstract: Hypergeometric functions are special functions with lot of symmetries. In this talk, we will introduce hypergeometric functions over finite fields, originally due to Greene, Katz and McCarthy, in a way that is parallel to the classical hypergeometric functions, and discuss their properties and applications to character sums and the arithmetic of hypergeometric abelian varieties. Most of the talk will be accessible to senior undergraduate students. This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu.
Date: 6/05, Speaker: 郭旗 (华南师范大学) Title: 光孤子的传输特性 Abstract: 光孤子是光包络（时间光脉冲或者空间光束）在非线性介质中传输时由于线性光学效应 和非线性光学效应作用达到平衡时的稳定传输状态。光孤子的研究不仅可以使我们扩展 对基本物理现象的理解，而且更重要的是光孤子本身在光子（全光）信息处理方面具有 广泛的潜在应用。 本报告的内容包括： 1. 如何从Maxwell方程（描述电磁场运动规律的基本方程）得出光包络函数满足 的慢变包络方程――（非局域）非线性Schrodinger方程； 2. 非局域非线性Schrodinger方程的孤子解及其特性； 3. 关于非局域非线性Schrodinger方程求解的一个猜想。
Date: 5/22, Speaker: Zhan Li (Beijing International Center for Mathematical Research, Peking Unviersity) Title: A construction of non-commutative mirror symmetry and derived equivalences Abstract: I will explain a construction of non-commutative Calabi-Yau varieties. The motivations for such construction are twofold: first, it generalizes the classical Batyrev-Borisov mirror symmetry construction to non-commutative setting; second, it unifies many sporadic examples of derived equivalent non-commutative varieties. This construction is conjectured to give mirror pairs. As a first test, we show their derived equivalences. This work is partially joint with Lev Borisov.
Date: 5/15, Speaker: Duo Li (Yau Mathematical Sciences Center, Tsinghua Unviersity) Title: On certain K-equivalent birational maps Abstract: We study K-equivalent birational maps which are resolved by a single blowup. Examples of such maps include standard flops and twisted Mukai flops. We give a criterion for such maps to be a standard flop or a twisted Mukai flop. As an application, we classify all such birational maps up to dimension 5.
Date: 5/08, Speaker: Shengda Hu (Wilfrid Laurier University-Waterloo) Title: Generalized Kaehler geometry on Lie groups Abstract: We will give introductions to Lie groups and generalized Kaehler geometry. We then describe a particular class of generalized Kaehler structures on compact Lie groups defined by invariant complex structures. We then describe some properties of these structures which have analogues in both Kaehler geometry and Poisson geometry.
Date: 4/24, Speaker: Xiaowen Hu (Sun Yat-sen University) Title: An introduction to algebraic cobordism Abstract: I will make an introduction to the theory of algebraic cobordism developped by Morel, Levine and also Pandharipande, and its applications in enumerative geometry in the work of Tzeng and Shen. Some problems and difficulties will be mentioned. will give introductions to Lie groups and generalized Kaehler geometry. We then describe a particular class of generalized Kaehler structures on compact Lie groups defined by invariant complex structures. We then describe some properties of these structures which have analogues in both Kaehler geometry and Poisson geometry.
Date: 4/10, Speaker: 林牛(Ngau Lam) 教授 台湾成功大学数学系 Title: 李代數、李超代數及超對偶 Abstract: 首先我們會介紹李代數及李超代數。在這演講中我們主要討論對象為有限維及無限維一般線性李代數(general linear Lie algebra)及一般線性李超代數(general linear Lie superalgebra) 的表現。我們會談談玻色子場(boson field)及費米子場 (fermion field)所產生的Fock空間,討論一般線性李代數及一般線性李超代數在Fock空間的表現為何。我們談談對稱函數和李(超)代數的特徵標, 最後我們描述無限維一般線性李代數的表現及無限維一般線性李超代數的表現之超對偶性。
Date: 3/27, Speaker: Zhiwei Wu (Ningbo University) Title: Geometric curve flows and integrable systems Abstract: The theory of Integrable system is involved in many fields in mathematics, such as PDEs, group theory, differential geometry and algebraic geometry. In this talk, I will start with a brief review of the research of the connection between differential geometry and integrable system. Then we will talk about hierarchies of geometric curve flows, whose invariants are solutions of soliton equations. The Cauchy problem can be solved for both rapidly decaying and periodic initial data. Such curve flows are Hamiltonian, with a Bi-Hamiltonian structure and a family of infinitely many conservation laws. Explicit soliton and rational solutions are constructed by Backlund transformation and a permutability formula.
Date: 3/20, Speaker: Yalong Cao (Kavli IPMU, University of Tokyo) Title: Gopakumar-Vafa type invariants for Calabi-Yau 4-folds Abstract: As an analogy of Gopakumar-Vafa conjecture for CY 3-folds, Klemm-Pandharipande proposed GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this talk, we propose a sheaf theoretical interpretation to these invariants using Donaldson-Thomas theory on CY 4-folds. This is a joint work with Davesh Maulik and Yukinobu Toda.
Date: 3/13, Speaker: Guowu Meng (Hong Kong University of Science and Technology) Title: 开普勒问题和罗伦兹变换 Abstract: 牛顿在17世纪引入的开普勒问题是个非相对论性的经典力学模型，其解析解给出了行星运行的开普勒三大定律的完美解释。罗伦兹变换是让麦克斯韦电磁学方程组保持形式不变的时空坐标变换，对其几何与物理意义的理解引发了20世纪初的物理学上的相对论革命。近年来的研究发现二者有紧密的数学联系。本讲座的目的是给这一数学联系的背景资料做个轻松而详细的介绍，力图让只懂多元微积分的学生也能听懂报告的绝大部分内容。如果时间容许，我会说明为什么开普勒问题以及谐振子的数学秘密都隐藏在来自于量子力学的欧式约当代数中。
Date: 2/27, Speaker: Siye Wu (National Tsing Hua University) Title: Global aspects of quantum gauge theories Abstract: We revisit a few aspects of gauge theories in four dimensions related to the topology of principal gauge bundles. We found that the usual concept of discrete electric and magnetic fluxes of 't Hooft requires a modification when the gauge group is an arbitrary compact semisimple Lie group and when the spatial slice is an arbitrary compact 3-manifold. We investigate quantum gauge theory, S-duality, and dimensional reduction in light of this adjustment. |