活动地点：

活动类型：学术报告

主讲人：

活动时间：2021-11-16 09:00:00--2021-11-16 10:00:00

活动内容：

中心开展的网上讨论班信息（截至2022.12.20）

浙江大学数学中心：

第119周 复几何线上讨论班

时 间：北京时间12月20日上午9:00-10:00

地 点：腾讯会议（会议号: 905 081 790无密码)

报告人：张梦怡

题 目：Spin-Manifolds and Group Actions

摘 要：There are many old results that have played major roles in the development of the filed of elliptic genera. In this report, we will discuss Atiyah and Hirzebruch’s work about spin-manifolds and group actions, in which they showed that the existence of an S1-action implies that the \hat{A}-genus is zero. And we can apply their conclusions to the proof of the rigidty of elliptic genera.

第 118 周 复几何线上讨论班

时   间:  北京时间12月13日上午9:00-10:00

地   点:  腾讯会议（会议号: 473 068 458无密码)

报告人:  夏天澄

题   目：Two dimensional gauge theories revisited II

摘   要：In this talk, we will focus on the second part of Witten's paper (Two dimensional gauge theories revisited). We will first introduce the topological field theory and physical Yang-Mills theory, then make a detailed comparsion of  two dimensional Yang-Mills theory with the non-abelian localization theorem, which discussed in last talk.

第 117 周 复几何线上讨论班

时   间:  北京时间11月29日上午9:00-10:00

地   点:  腾讯会议（会议号: 671 073 817无密码）

报告人:  胡孟戈

题    目：Heat kernel and localization in symplectic geometry

摘    要：In this presentation, we mainly focus on the heat kernel proof of non-Abelian localization formula given by Liu. First, we review some developments of symplectic geometry in 1980s and 1990s. Then we talk about the Duistermaat-Heckman theory, which includes much materials needed in the heat kernel proof. Finally, we'll introduce group-valued moment map and quasi-Hamiltonian space for some possible further applications of this heat kernel method.

第 116 周 复几何线上讨论班

时   间:  北京时间11月22日上午9:00-10:00

地   点:  腾讯会议（会议号: 985 097 758无密码)

报告人:  周雷

题    目：The Novikov inequalities

摘    要：S.P. Novikov initiated a generalization of Morse theory in which instead of critical points of smooth functions one deals with closed 1-forms and their zeros. In his view of point, Morse function is related to exact 1-form, which could be generalized to closed 1-forms. He also invented inequalities which generalize the Morse inequalities. In this presentation, I will mainly introduce these inequalities by following a proof of Michael Farber.

第 115 周 复几何线上讨论班

时   间:  北京时间11月15日上午9:00-10:00

地   点:  腾讯会议（会议号: 257 045 116无密码)

报告人:  陈如通

题    目: Deformation of Atiyah class

摘    要: In this presentation, we first introduce the  Atiyah class of a holomorphic line bundle on a complex manifold X, then we talk about its relationship with Chern class and Bott-Chern class, last, we talk about the deformations of Atiyah class and the product of Atiyah class and Kodaira-Spencer class.

第 114 周 复几何线上讨论班

时   间:  北京时间11月10日上午9:00-10:00

地   点:  腾讯会议（会议号: 910 021 613无密码)

报告人:  乔琛凯（重庆理工大学）

题   目:  低维几何与广义相对论中若干问题

摘   要: 广义相对论的数学基础是3+1维Lorentz流形，但通过一些方式可以得到一些等效的低维流形。这类低维几何的数学性质和广义相对论的一些重要问题密切相关。例如，黑洞理论中需要研究Lorentz流形的零超曲面；对引力场能量的研究和数值广义相对论方法则需要研究Lorentz流形的类空超曲面。

在本报告中，我们介绍另一类低维等效流形---光学流形（Optical Geometry）。光学流形对求解引力场中光子轨迹很有帮助。本报告中主要介绍两方面工作。1.在光学流形中利用Gauss-Bonnet公式，求解光线在黑洞附近的引力偏转角；2.利用光学流形上的测地曲率和高斯曲率，确定黑洞附近光子圆轨道的位置和稳定性。

第 113 周 复几何线上讨论班

时   间:  北京时间11月8日上午9:00-10:00

地   点:  腾讯会议（会议号: 686 061 335无密码)

报告人:  刘   凯

题    目: Stability of Harmonic maps and Yang-Mills fields

摘    要: In this presentation, we first review the history of stability results about harmonic maps and Yang-Mills fields. Then we will talk about some recent works, which imply the earlier Morse index estimates in both cases. At last we will discuss some related problems.

第 112 周 复几何线上讨论班

时   间:  北京时间10月25日上午9:00-10:00

地   点:  腾讯会议（会议号: 584 058 718无密码)

报告人:  夏天澄

题    目：Two dimensional gauge theories revisited I

摘    要：In this talk, we will focus on the first part of Witten's paper (Two dimensional gauge theories revisited), that is the Non-abelian localization. First, we will review the partition function and theDuistermaat-Heckman integration formula, then we explain the localization principle and consider the contribution of μ^{-1}(0) in the end.

第 111 周 复几何线上讨论班

时   间:  北京时间10月11日上午9:00-10:00

地   点:  腾讯会议（会议号: 214 066 523无密码）

报告人:  胡孟戈

题    目:  Adiabatic limits, leafwise positive scalar curvature and Seiberg-Witten invariants

摘    要: In this presentation, we'll talk about a proof of the vanishing theorem of Seiberg-Witten invariants with leafwise positive scalar curvature, which partialy solved a conjecture of Zhang. First, we'll introduce the backgrounds, basic concepts, and the main tools, the Adiabatic limits. Then we would review the Seiberg-Witten theory and Seiberg-Witten invariants. Finally, we give a sketch of proof of the vanishing theorem for rank 2. If time permits, we'll also talk about some attemps for the codim-1 case.

第 110 周 复几何线上讨论班

时   间:  北京时间10月4日上午9:00-10:00

地   点:  腾讯会议（会议号: 348 065 768无密码)

报告人:  周雷

题    目：Index theorem, Loop space and Witten genus

摘    要：In this presentation, I want to briefly introduce the proof of the Index theorem from the perspective of K-theory, which involves a  relatively profound analysis towards the topology of a vector bundle. Then I also want to talk about the Witten genus from the point of view of the loop space, which is an utilizing of equivariant Index theorem.

第 109 周 复几何线上讨论班

时   间:  北京时间9月27日上午9:00-10:00

地   点:  腾讯会议（会议号: 869 014 770无密码)

报告人:  陈如通

题    目: Lefschetz theorem on (1,1) classes for ddbar-manifolds

摘    要: In this presentation, we will define a map from the Picard group to integral Bott-Chern (1,1) cohomology group for a compact complex manifold X, then we will prove that this map is surjective if and only if X satisfies certain type of ddbar-lemma, which can be treated as a generalization of classical Lefschetz theorem on (1,1) classes for  compact Kähler manifolds. Last, we will talk about some related topics to it.

第 108 周 复几何线上讨论班

时   间:  北京时间9月20日上午9:00-10:00

地   点:  腾讯会议（会议号: 338 034 881无密码)

报告人:  刘   凯

题    目: On the curvature operator of the second kind II

摘    要: In this presentation, we will discuss some recent progress on the study of curvature operator of the second kind on Kahler manifolds. Especially, we will talk about some recent rigidity theorems relating to this operator.

第 107 周 复几何线上讨论班

时   间:  北京时间9月13日上午9:00-10:00

地   点:  腾讯会议（会议号: 689 040 403无密码)

报告人:  夏天澄

题    目：On Bismut's proof of the Verlinde formula

摘    要：In this talk, at first, we briefly review the definition of the Verlinde formula and the main idea of Bismut's proof of it. Inspired by Liu's proof of Witten's symplectic volume formula, we try to prove an important step in Bismut's proof in a different way.

第 106 周 复几何线上讨论班

时   间:  北京时间9月6日上午9:00-10:00

地   点:  腾讯会议（会议号: 243 078 074无密码)

报告人:  沈 洋

题    目：Compact quotients of non-classical period domains are not Kahler II

摘    要：In this talk, we introduce the work of Sampson and Siu on harmonic maps, and the work of Carlson and Toledo on the applications of harmonic maps to Hodge theory. From the work of Sampson and Siu, we have that a harmonic map of sufficiently high maximum rank of a compact Kahler manifold to a quotient of an irreducible bounded symmetric domain must be holomorphic or anti-holomorphic, which is known as Siu’s rigidity theorem. Then Carlson and Toledo used their trick in Hodge theory to study the geometry of the period domains. Recently, they proved that there are no Kahler structures, and hence no algebraic structures, on the compact quotients of non-classical period domains.

第 105 周 复几何线上讨论班

时   间:  北京时间8月30日上午9:00-10:00

地   点:  腾讯会议（会议号: 794 059 267无密码)

报告人:  沈 洋

题    目：Compact quotients of non-classical period domains are not Kahler

摘    要：In this talk, we introduce the work of Sampson and Siu on harmonic maps, and the work of Carlson and Toledo on the applications of harmonic maps to Hodge theory. From the work of Sampson and Siu, we have that a harmonic map of sufficiently high maximum rank of a compact Kahler manifold to a quotient of an irreducible bounded symmetric domain must be holomorphic or anti-holomorphic, which is known as Siu’s rigidity theorem. Then Carlson and Toledo used their trick in Hodge theory to study the geometry of the period domains. Recently, they proved that there are no Kahler structures, and hence no algebraic structures, on the compact quotients of non-classical period domains.

第 104 周 复几何线上讨论班

时   间:  北京时间8月23日上午9:00-10:00

地   点:  腾讯会议（会议号: 546 093 107无密码)

报告人:  胡孟戈

题    目：Seiberg-Witten theory and Donaldson's diagonalisable theorem

摘    要：In this talk, we first modeling Spin^c-structure by using quaternions, which is necessary for defining Seiberg-Witten equations. Then we give some important properties of the moduli space of solutions of them. At last, we apply it to prove Donaldson's celebrated theorem in topology of smooth 4-manifold.

第 103 周 复几何线上讨论班

时   间:  北京时间8月16日上午9:00-10:00

地   点:  腾讯会议（会议号: 648 053 029无密码)

报告人:  张良迪

题    目：On the parabolic Donaldson's equation over a compact complex manifold

摘    要：We prove the uniqueness and long time existence of the smooth solution to a parabolic Donaldson's equation on a compact complex manifold.Then we show smooth convergence of a suitably normalized solution. 

第 102 周 复几何线上讨论班

时   间:  北京时间8月9日上午9:00-10:00

地   点:  腾讯会议（会议号: 471 013 452无密码)

报告人:  周 雷

题    目：The Heat kernel proof of the Local Index theory

摘    要：When M is of even-dimension, the index of the Dirac operator associated with a Clifford module on M is revealed by the local index theory. In this talk, I will introduce the heat kernel proof,  which shows how to relate the long-time and short-time behaviors of the heat kernel and use it to get the relation between the topology and the geometry of a manifold.

第 101 周 复几何线上讨论班

时   间:  北京时间7月26日上午9:00-10:00

地   点:  腾讯会议（会议号: 212-004-523无密码)

报告人:  陈如通

题    目: Stability theorems in deformation theory

摘    要: In this presentation, we first review Kodaira-Spencer's original proof of stability theorem of Kähler structures, then we talk about C.-C. Wu' proof of stability of balanced ddbar structures. Last, we introduce Rao-Wan-Zhao's work on stabilities of p-Kähler structures.

第 100 周 复几何线上讨论班

时   间:  北京时间7月20日上午9:00-10:00

地   点:  腾讯会议（会议号: 230 079 561无密码)

报告人:  刘   凯

题    目: On the curvature operator of the second kind

摘    要: In this presentation, we first review the history of curvature operator of the second kind. Then we will talk about some recent vanishing and rigidity theorems relating to this operator.

第 99 周 复几何线上讨论班

时   间:  北京时间7月12日上午9:00-10:00

地   点:  腾讯会议（会议号: 430 016 188无密码)

报告人:  夏天澄

题   目：Witten's symplectic volume formula and heat kernel

摘   要：In this talk, firstly, we will briefly review Bismut's proof of the Verlinde formula .Then the definition and basic properties of the heat equation and heat kernel will be listed. Finally, we will give Liu's proof of Witten's formula.

第 98 周 复几何线上讨论班

时   间:  北京时间7月5日上午9:00-10:00

地   点:  腾讯会议（会议号: 268 021 696无密码)

报告人:  胡孟戈

题    目：Topology of moduli space of stable bundles over Riemann surfaces.

摘    要：In this talk, we mainly focus on the computation of the Poincare series of the moduli space of stable bundles. It has been calculated by using tools in algebraic geometry and number theory. However, Atiyah-Bott found a Morse stratification of the moduli space and gave a inductive formula to calculate the Poincare series of moduli space.

第 97 周 复几何线上讨论班

时   间:  北京时间6月28日上午9:00-10:00

地   点:  腾讯会议（会议号: 571 041 526无密码)

报告人:  张梦怡

题   目：On framings of 3-manifolds—Michael Atiyah, 1989

摘   要：Define the notion of a 2-framing and introduce some of its properties. 

第 96 周 复几何线上讨论班

时   间:  北京时间6月21日上午9:00-10:00

地   点:  腾讯会议（会议号: 631 076 031无密码)

报告人:  李 逸

题   目：Donaldson's proof of the Narasimhan - Seshadri theorem III

摘   要：Along the outline of Donaldson's 1983 paper, I try to supply more detail.

第 95 周 复几何线上讨论班

时   间:  北京时间6月14日上午9:00-10:00

地   点:  腾讯会议（会议号: 940 050 898无密码)

报告人:  李 逸

题   目：Donaldson's proof of the Narasimhan - Seshadri theorem II

摘   要：Along the outline of Donaldson's 1983 paper, I try to supply more detail.

第 94 周 复几何线上讨论班

时   间:  北京时间6月7日上午9:00-10:00

地   点:  腾讯会议（会议号: 482 017 533无密码)

报告人:  李 逸

题   目：Donaldson's proof of the Narasimhan - Seshadri theorem

摘   要：Along the outline of Donaldson's 1983 paper, I try to supply more detail.

第 93 周 复几何线上讨论班

时   间:  北京时间5月31日上午9:00-10:00

地   点:  腾讯会议（会议号: 215 072 492无密码)

报告人:  胡孟戈

题   目：Narasimhan-Seshadri theorem and Yang-Mills equations over Riemann surfaces.

摘   要：In this presentation, we will introduce the celebrated Narasimhan-Seshadri theorem which states the relationship between stable bundles on Riemann surface and the unitary representation of fundamental groups of Riemann surface. Then we talk about Atiyah-Bott's reinterpretation of this theorem by applying Yang-Mills equations over Riemann surfaces. More precisely, they constructed a correspondence between the solutions of Yang-Mills equations and the above two objects, which demonstrated an analytic point of view. Following Atiyah-Bott's idea, Donaldson gave an analytic proof of this new formulation of Narasimhan-Seshadri theorem.

第 92 周 复几何线上讨论班

时   间:  北京时间5月24日上午9:00-10:00

地   点:  腾讯会议（会议号: 325 077 423无密码)

报告人:  周 雷

题    目：Thom-Smale and Witten Complexes

摘    要：In this presentation, I want to talk about the construction of Thom-Smale complex appears in Witten's paper 《Supersymmetry and Morse theory》. In this paper, he used an analytic method to figure out the topological information of a manifold. I will introduce the construction and mainly refer to the book 《Lectures on Chern-Weil Theory and Witten Deformations》. Firstly, I want to describe the Thom-Smale complex by a topological method, and then talk about the connection between the two complexes.

第 91 周 复几何线上讨论班

时   间:  北京时间5月17日上午9:00-10:00

地   点:  腾讯会议（会议号: 488 088 445无密码)

报告人:  毕翔程

题    目: Geometry of data 

摘    要: Topological data analysis asks when balls in  a metric space (X,d) intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to traditional core geometric concept of curvature to reconceptualize curvature and link it to the geometric notion of hyperconvexity.

第90周 复几何线上讨论班

时   间:  北京时间5月10日上午9:00-10:00

地   点:  腾讯会议（会议号:777 042 774无密码）

报告人:  徐  飞

题   目：supersymmetry and morse theory II

摘   要：In this talk we will continue to discuss how witten proved Conner's theorem and how to use witten deformation to give an analytic proof of  Poincare-Hopf index theorem.

第89周 复几何线上讨论班

时   间:  北京时间5月3日上午9:00-10:00

地   点:  腾讯会议（会议号:391 055 984无密码）

报告人:  徐  飞

题   目：supersymmetry and morse theory

摘   要：In this talk we will discuss the Witten‘s analytic proof of morse inequality and how he proved Conner’s theorem，Hopf index theorem，and Hirzebruch signature formula using analogue approach.

第 88 周 复几何线上讨论班

时   间:  北京时间4月26日上午9:00-10:00

地   点:  腾讯会议（会议号: 866 094 567无密码)

报告人:  曹诗禹

题   目：On the log d-bar equations and the logarithmic deformation of compact Kahler manifolds

摘   要：In this talk we will discuss the new method developed by Liu-Rao-Wan to solve certain d- bar equations for logarithmic differential

forms on compact Kahler manifolds by using harmonic integral theory for currents and another result ontained by X. Wan on solving log d-bar equation associated to a smooth divisor by using the cyclic covering trick. Then we will talk about some results in their papers on the log deformation theory. If time permits, we will also talk about some other apllications in algebraic geometry such as the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at E1, etc.

第 87 周 复几何线上讨论班

时   间:  北京时间4月19日上午9:00-10:00

地   点:  腾讯会议（会议号:483 017 459无密码)

报告人:  夏天澄

题   目：Symplectic Geometry and the Verlinde formulas

摘   要：In this talk, we firstly introduce the background of the Verlinde formulas and the main ideas in the methods of symplectic geometry, then the main steps of Bismut's proof are given. Finally, we will focus on a volume theorem which can deduce the Witten's formula for the symplectic volume of the reductions of M/G.

第 86 周 复几何线上讨论班

时   间:  北京时间4月12日上午9:00-10:00

地   点:  腾讯会议（会议号: 238 056 699无密码)

报告人:  毕翔程

题目: Mean curvature flow of pinched submanifolds to spheres

摘要: In this presentation, I will introduce the mean curvature flow of pinched submanifolds. Considering compact submanofolds of dimension n≥2 in R^n+k with nonzero mean curvature vector everywhere and some pinching condition satisfied, we prove that the mean curvature flow deforms such a submanifold to a point in finite time.

第 85 周 复几何线上讨论班

时   间:  北京时间4月5日上午9:00-10:00

地   点:  腾讯会议（会议号: 915 018 958无密码)

报告人:  陈如通

题    目：Variations of Hodge structure for ddbar-manifolds

摘    要：In this report, we'll introduce Xia Wei's work on Variations of Hodge structure for ddbar-manifolds. First, we will give a brief introduction on the theory of deformed Bott-Chern cohomology, then we use it to establish the theory of VHS for ddbar manifolds. Last, we talk about some applications of it.

第 84 周 复几何线上讨论班

时   间:  北京时间3月29日上午9:00-10:00

地   点:  腾讯会议（会议号: 146 006 741无密码)

报告人:  胡孟戈

题    目：Localization formula of Lie algebroid and related residue formulas

摘    要：In this presentation, we'll introduce the basic concept and representations of Lie algebroid, which includes many classical algebraic and geometric objects. Then we would talk about the equivariant cohomology and localization formula for Lie algebroid. As an application, we may focus on some related residue formulas in the context of equivariant cohomology of Lie algebroid.

第 83 周 复几何线上讨论班

时   间:  北京时间3月22日上午9:00-10:00

地   点:  腾讯会议（会议号: 877 035 666无密码)

报告人:  刘 荣

题    目：Some classical results of Chern's conjecture

摘    要：Affine manifold is a special differential manifold, and its specialized function is affine transformation. Around 1955 Chern conjectured that the Euler characteristic of any compact affine manifold has to vanish. Klingler proved the conjecture for special affine manifolds.

第 82 周 复几何线上讨论班

时   间:  北京时间3月15日上午9:00-10:00

地   点:  腾讯会议（会议号: 370 049 141无密码)

报告人:  周 雷

题    目:  Equivariant Dolbeault cohomology and localization formula

摘    要:  In this presentation I want to talk about the equivariant formality of the equivariant Dolbeault cohomology, which is a major problem I met in my dissertation. Next, I want to introduce the localization theorem of the holomorphic equivariant cohomology. Finally, if the time permits, we can use a classic method to prove a localization formula for equivariant Dolbeault cohomology. 

第 81 周 复几何线上讨论班

时   间:  北京时间 3 月 8 日上午9:00-10:00

地   点:  腾讯会议（会议号: 230 030 129无密码)

报告人:  刘   凯

题    目:  Tachibana-type theorems

摘    要: In this presentation, we first review the history of the classical Tachibana theorem, which is a rigidity result for compact manifolds. Then we will talk about some generalizations due to Petersen and Wink. After that we talk about the work of Colombo, Mariani and Rigoli which is a refinement of Petersen and Wink’s estimate. These works are all based on the famous Bochner technique.

第 80 周 复几何线上讨论班

时   间:  北京时间3月1日上午9:00-10:00

地   点:  腾讯会议（会议号: 487 099 908无密码)

报告人:  曹诗禹

题    目：Global methods on variations of complex structures

摘    要：In this talk we will talk about the paper written by K. Liu and S, Zhu with the same title, in which they gave a much simpler treatment to the deformation of complex structures by using Banach fixed point theorem and Hodge theorems. We will also talk about some applications in deformation theory and extensions of holomorphic forms.

第 79 周 复几何线上讨论班

时   间:  北京时间2月22日上午9:00-10:00

地   点:  腾讯会议（会议号: 256 052 240无密码)

报告人:  闫嘉琦

题    目：Structure of (P_2(M), W_2) 

摘    要：In this report, we wish to give a comprehensive description of the structure of the Riemannian manifold (P_2(R^d), W_2) .

第 78 周 复几何线上讨论班

时   间:  北京时间2月15日上午9:00-10:00

地   点:  腾讯会议（会议号: 350 016 292无密码)

报告人:  刘 荣

题    目：Some classical results of Chern's conjecture

摘    要: Affine manifold is a special differential manifold, and its specialized function is affine transformation. Around 1955 Chern conjectured that the Euler characteristic of any compact affine manifold has to vanish. Kostant and Sullivan  proved Chern’s conjecture in the case where the manifold is complete；Smillie proved that the condition that the connection is torsion free matters. For each even dimension greater than 2, Smillie constructed closed manifolds with non-zero Euler characteristic that admit a flat connection on their tangent bundle.

第 77 周 复几何线上讨论班

时   间:  北京时间2月8日上午9:00-10:00

地   点:  腾讯会议（会议号: 193 001 882无密码)

报告人:  闫嘉琦

题    目：Optimal transport and Gradient Flow on Wasserstein space 

摘    要：In this report, we review some basic knowledge about gradient flow, then give a framework of the gradient flow in Wasserstein space using optimal transport. 

第 76 周 复几何线上讨论班

时   间:  北京时间1月25日上午9:00-10:00

地   点:  腾讯会议（会议号: 176 087 356无密码)

报告人:  朱家林

题    目：Quantization of symplectic fibration and canonical metrics

摘    要: This talk is on the recent work of Louis Ioos and Leonid Polterovich (arXiv:2112.00419). We will talk about the relation between Berezin-Toeplitz quantization of higher rank vector bundles

and quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We will show its application in the spectral gap of the Berezin transform and the convergence rate of Donaldson's 

iterations towards balanced metrics on stable vector bundles.

第 75 周 复几何线上讨论班

时   间:  北京时间1月18日上午9:00-10:00

地   点:  腾讯会议（会议号: 815 057 552无密码)

报告人:  刘  荣

题    目：Some classical results of Chern's conjecture

摘    要：Affine manifold is a special differential manifold, and its specialized function is affine transformation. Around 1955 Chern conjectured that the Euler characteristic of any compact affine manifold has to vanish. In this report, we will introduce inequalities Milnor proved which completely characterise those oriented rank two bundles over a surface that admit a flat connection.

第 74 周 复几何线上讨论班

时   间:  北京时间1月11日上午9:00-10:00

地   点:  腾讯会议（会议号: 791 095 102无密码)

报告人:  闫嘉琦

题    目：Review of optimal transport theory

摘    要：This report reviews the basic theories of optimal transport problems, focusing on the modern framework and new developments of optimal transport problems, introducing Kantorovich theory and Brenier theory, and discussing the basic properties of Wasserstein space based on optimal transmission theory. Finally, as an application, we also give two examples– we prove the isoperimetric inequality and Sobolev inequality to show how the tool of optimal transprot could be applied.

第 73 周 复几何线上讨论班

时   间:  北京时间1月4日上午9:00-10:00

地   点:  腾讯会议（会议号: 220 055 232无密码)

报告人:  徐  飞

题    目：Chern‘s study on S^6

摘    要：In this talk we will first introduce some property about the exceptional group G_2 and the structure equation associated to G_2. Then we will give the Lebrun's proof of the non-existence of the orthogonal complex structure on S^6. Next, we will show Chern's work on S^6 that R.Bryant have compiled. Finally, we will introduce that Chern's work can actually be extended and apply to a much wider class of almost-symplectic 6-manifolds.

第 72 周 复几何线上讨论班

时   间:  北京时间12月29日上午9:00-10:00

地   点:  腾讯会议（会议号: 466 082 323无密码)

报告人:  曹诗禹

题    目：Balanced embedding and Stability Condition

摘    要：In this talk we will first recall the stability conditions in the contexts of GIT and moment map. And then the speaker will present the several proofs of the famous theorem stating that for a projective manifold the balanced condition is equivalent to the Chow stability by Zhang, Luo, Wang, Phong- Sturm, respectively. Moreover, the speaker will also present the definition of differential geometric and algebraic geometric Futaki invariant and state some of their applications in the study of extremal metrics on Kähler manifolds.

第 71 周 复几何线上讨论班

时   间:  北京时间12月14日上午9:00-10:00

地   点:  腾讯会议（会议号: 875 046 026无密码)

报告人:  肖   邦

题    目：Hilbert's Theorem, via moving frames

摘    要：报告 William D. Dunbar 的一篇文章。1901年，D.Hilbert 证明了双曲平面不能等距浸入到3维欧氏空间。此文给出一种新的证明方法，主要用到活动标架。

第 70 周 复几何线上讨论班

时   间:  北京时间12月7日上午9:00-10:00

地   点:  腾讯会议（会议号: 390 007 806无密码)

报告人:  智艳辉

题    目：affine geometry and frobenius algebra

摘    要：in this paper，it shows that there is a Frobenius algebra structure, if and only if the sectional K-curvature vanishes. And it also show that there exists a Frobenius manifold structure on real 2-torus by using Bryant’s construction of compact Hessian manifold.

第 69 周 复几何线上讨论班

时   间:  北京时间11月30日上午9:00-10:00

地   点:  腾讯会议（会议号: 553 033 867无密码)

报告人:  夏天澄

题    目：The deformation of the holomorphic vector bundles from the transition functions

摘    要：In this talk, we discuss the infinitesimal deformation of the holomorphic structures of the vector bundle E_0 on X from the point of view of the transition functions. A homomorphism h(t) from E_0's smooth section to E_t's is defined and the associated Maurer-Cartan equation are derived. At last, we consider the case when transition functions vary continuously with t.

第 68 周 复几何线上讨论班

时   间:  北京时间11月23日上午9:00-10:00

地   点:  腾讯会议（会议号: 869 007 316无密码)

报告人:  陈如通

题   目: Hironaka's example: the limiting fiber of projective manifolds

摘   要: In this report, we will talk about Hironaka's construction of non-Kähler manifold: a 3-dimensional Moishezon manifold as the limiting fiber of projective manifolds, then we will introduce some related work subsequent to it.

第 67 周 复几何线上讨论班

时   间:  北京时间11月16日上午9:00-10:00

地   点:  腾讯会议（会议号: 614 082 387无密码)

报告人:  曹诗禹

题   目：On Chow Stability and Balanced Embeddings

摘   要：In this talk, the speaker will report the recent paper written by Fong with the same title which gave a new proof to the folklore theorem that Chow stability implies the existence of balanced embedding. We will first recall some prerequisites of moment map and GIT. And then the speaker will report the main result of the paper. If time permits, we will also report some applications of stability and balanced embeddings.

第 66 周 复几何线上讨论班

时   间:  北京时间11月9日上午9:00-10:00

地   点:  腾讯会议（会议号: 931 095 004无密码)

报告人:  周   雷

题   目：Equivariant Cohomology of a Symplectic Manifold acted by a Compact Lie Group

摘   要：Kirwan (1984) used the moment map to define a real valued function on a symplectic manifold, then she applied the ideas of Morse theory to this function and revealed the connection between the equivariant cohomology and the ordinary cohomology of the symplectic manifold. In this report I will mainly talk about this process.

第 65 周 复几何线上讨论班

时   间:  北京时间11月2日上午9:00-10:00

地   点:  腾讯会议（会议号: 210 052 558无密码)

报告人:  胡孟戈

题   目：Atiyah-Bott localization, equivariant index theorem and Â-vanishing theorem

摘   要：In this presentation, we would first review the Atiyah-Bott localization in equivariant cohomology. Similarly, as a generalized cohomology theory, equivariant K-theory would be introduced later and there is also a localization formula for index. As an application, we prove the Atiyah-Hirzebruch Â-vanishing theorem which is one of the sources that lead to elliptic genus and various rigidity theorems.

第 64 周 复几何线上讨论班

时   间:  北京时间10月26日上午9:00-10:00

地   点:  腾讯会议（会议号: 580 092 812无密码)

报告人:  魏定畅

题    目: Spin^c Dirac operator and Lichnerowicz formula

摘    要: We will review the definition of the spin^c Dirac operator on complex manifolds and prove the related Lichnerowicz formula. If time permitted, we will also explain the Lichnerowicz formula and Atiyah-Singer index theorem for the modified Dirac operator.