GAP Seminar @ USTC (2014 Spring)

Date: May 26, 2014
Speaker: Weiyi ZHANG （University of Warwick, UK）
Title: Geometric structures, Gromov norm and Kodaira dimensions
Abstract: Kodaira dimension provides a very successful classification scheme for complex manifolds. The notion was extended to symplectic 4-manifolds. In this talk, we will define the Kodaira dimension for 3-manifolds through Thurston's eight geometries. This is compatible with other Kodaira dimensions in the sense of "additivity". We will then explore the relations of geometric structures and mapping orders with various Kodaira dimensions and other invariants like Gromov norm.

Date: May 19, 2014
Speaker: Daniel BURNS (University of Michigan)
Title: Some applications of symplectic reduction to complex analysis and geometry
Abstract: We review the problem of algebraicizing a closed analytic submanifold $X \subset C^N$. One formulation of this is to find very slowly growing entire functions on $X$ which serve to give a polynomial embedding of $X$ into another $C^{N_1}$. To measure slowness, we use plurisubharmonic solutions $u$ of the homogeneous complex Monge-Ampere equation (HCMA) $(\partial\bar\partial u)^n = 0$. The polynomial growth condition on an entire holomorphic function $f$ will be $|f| \le C(1 + \tau)^{N'}$, where $\tau:= e^u$, and $N' \in N^+$, for some positive integer $N'$ which depends on $f$. A class of complex manifolds with such HCMA exhaustions is given by entire Grauert tubes, manifolds which are complex structures placed on the tangent bundle of a Riemannian manifold $M$. Aguilar has shown a construction via symplectic reduction of all known examples of such entire tubes. His work shows they come in continuous biholomorphic families. In joint work we show that these entire tubes are affine algebraic varieties, and if they are biholomorphic, they are isomorphic as algebraic varieties. This is joint work with Zhou Zhang (University of Sydney), and relies on earlier work of the speaker with Victor Guillemin and Zuoqin Wang on stability functions in symplectic reduction.

Date: May 12, 2014
Speaker: 向昭银（电子科技大学）
Title: On the Cauchy problem for the (modified) two-component Euler-Poincare equations
Abstract: In this talk, we first use the classical energy methods to establish the local existence of unique classical solutions as well as two blow-up criteria for the Cauchy problem of the two-component Euler-Poincare equations. Then by using the particle trajectory and the Littlewood-Paley decomposition theory, we show that for a large of smooth initial data with some concentration property, the corresponding solutions will blow up in finite time. In the case of one component, we also obtain the precise blow-up rate estimates and global existence for the initial data with some non-positive property at the original. Next, we investigate the zero density limit and the zero dispersion limit. At the end of the talk, we also briefly demonstrate a Liouville type theorem for the stationary weak solutions. This is a joint work with Dr. Renjun Duan.

Date: May 05, 2014
Speaker: 胡红钢（中国科学技术大学信息学院）
Title: 密码学中的几个核心数学问题
Abstract: 密码学是信息安全的核心。在本报告中，我们将介绍现代密码学的诞生和发展过程，以及取得的主要成就，其中着重介绍几个与密码学有关的核心数学问题，以及人们对这些问题的研究进展，比如：大整数分解问题。最后介绍我们最近一个运用数学工具较多的工作。

Date: April 28, 2014
Speaker: 安金鹏（北京大学）
Title: Dimension data of subgroups and isospectral manifolds
Abstract: Given a compact Lie group, the dimension datum of its subgroup is an invariant of spectral nature and is related to geometry and number theory. We will discuss main properties of dimension data and applications to spectral geometry. If time permits, we will also talk about ideas of some proofs. This is a joint work with Jiu-Kang Yu and Jun Yu.

Date: April 21, 2014
Speaker: 王作勤（中国科学技术大学）
Title: An Invitation to Spectral Geometry
Abstract: Spectral geometry is the branch of mathematics that studies the eigenvalues and eigenfunctions of Laplace (type) operator on Riemannian manifolds. In this talk I will start with the background of this subject. Then I will summarize many beautiful known results, and will also introduce many open problems in this area.

Date: April 14, 2014 @ 4:00-5:30 pm.
Speaker: 洪伟（武汉大学）
Title: Unifying hypercomplex and holomorphic symplectic structures
Abstract: In this talk, we unify hypercomplex and holomorphic symplectic structures in the framework of Courant algebroids. Basic properties of such structures are established.

Date: April 14, 2014 @ 2:00-3:30 pm.
Speaker: Jiangwei XUE （台湾中央研究院数学研究所）
Title: Endomorphism algebras of factors of certain hypergeometric Jacobians
Abstract: We classify the endomorphism algebras of factors of the Jacobian of certain hypergeometric curves over a field of characteristic zero. Other than a few exceptional cases, the endomorphism algebras turn out to be either a cyclotomic field $E=\mathbb{Q}(\zeta_q)$, or a quadratic extension of $E$, or $E\oplus E$. This result may be viewed as a generalization of the well known results of the classification of endomorphism algebras of elliptic curves over $\mathbb{C}$.

Date: March 31, 2014
Speaker: Xiuxiong CHEN （中国科学技术大学）
Title: Liouville theorem on C^n
Abstract: In 1956, E.Calabi proved a striking theory that any convex function which solves $\det (u_{ij}) =1$ globally in $R^n$ must be a quadratic polynominal. In this paper, Calabi assumed the potential function $u \in C^5(R^n)$ and proved an ingenious $C^3$ estimates which influence geometric analysts from generations to come. Recently, with my student Yuanqi Wang, we prove a similar Liouville theorem in $C \times C^{n-1}$ for Kaehler metrics with conical singularities along a trivial divisor $\{0\} \times C^{n-1}$ with complete different yet elementary method. The talk shoule be accessible to graduate students and young faculties.

Date: March 17, 2014
Speaker: Yi LI (上海交通大学)
Title: Complex Monge-Ampere type equations over compact Hermitian manifolds
Abstract: In this talk, I present some recent progress on the complex Monge-Ampere type equations over compact Hermitian manifolds, including Calabi-Yau equation, Donaldson equation and J-flow.