Student Complex Geometry Archives for Fall 2023 to Spring 2024
Holomorphic connections
When: Wed, September 14, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Kwong Sze Hong (UMD) -
Donaldson's proof of Narasimhan-Seshadri Theorem
When: Wed, September 21, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Kwong Sze Hong (UMD) -
Donaldson's proof of Narasimhan-Seshadri Theorem (Part II)
When: Wed, September 28, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Kwong Sze Hong (UMD) -
Santaló's inequality in \C^n by complex interpolation
When: Wed, October 5, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Vlasios Mastrantonis (UMD) -
Abstract: We discuss Cordero-Erausquin's proof of Santaló's inequality for symmetric convex bodies in \C^n, obtained by combining complex interpolation with Berndtsson's generalization of Prékopa's inequality.
Complex Monge–Ampère equation: the first step
When: Wed, October 12, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Prakhar Gupta (UMD) -
Envelope techniques for complex Monge-Ampère equation
When: Wed, October 19, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Prakhar Gupta (UMD) -
Batyrev's construction of mirrors of Calabi-Yau manifolds
When: Wed, October 26, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Yong Cui (UMD) -
Geometry of the space of Kähler metrics
When: Wed, November 2, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Vasanth Pidaparthy (UMD) -
Abstract: We continue the study of the complex Monge-Ampère operator initiated by Prakhar. The homogeneous complex Monge--Ampère equation is the geodesic equation for a natural Riemannian metric on the space of Kähler metrics on a Kähler manifold. In this talk, we will compute the Levi-Civita connection and curvature on this infinite dimensional Riemannian manifold, and show that it is a symmetric space.
ABC of GIT
When: Wed, November 9, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Kwong Sze Hong (UMD) -
Asymptotic of Bergman Kernel I:Positive Case
When: Wed, November 16, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Yu-Chi Hou (UMD) -
Asymptotic of Bergman Kernel II: Indefinite Case
When: Wed, November 30, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Yu-Chi Hou (UMD) -
Abstract: When the line bundle is not positive, another useful tool to study the cohomologies of line bundles is Demailly's holomorphic Morse inequalities. We will extend Bergman kernel to (0,q)-forms valued in a holomorphic line bundle and discuss local holomorphic Morse inequalities due to Berman in his Ph.D thesis. Finally, if time permits, I will mention some works in my master thesis regarding this direction.
Asymptotic of Bergman Kernel II: Indefinite Case
When: Wed, December 7, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Yu-Chi Hou (UMD) -
Abstract: When the line bundle is not positive, another useful tool to study the cohomologies of line bundles is Demailly's holomorphic Morse inequalities. We will extend Bergman kernel to (0,q)-forms valued in a holomorphic line bundle and discuss local holomorphic Morse inequalities due to Berman in his Ph.D thesis. Finally, if time permits, I will mention some works in my master thesis regarding this direction.
Organizational meeting
When: Wed, January 25, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: () -
Calabi-Yau Theorem I
When: Wed, February 8, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Cheuk-Yan Fung (UMD) -
Abstract: The goal of the talk in these 2 weeks is to prove Calabi-Yau Theorem. This week, we will have a quick review on the definition and properties of Kähler manifold. Then we will formulate Calabi-Yau Theorem into a PDE problem and we will show the theorem using the continuity method. We will outline the several key steps of the proof and leave the technical details in next week. The main reference is chapter 1 -3 of An Introduction to Extremal Kähler Metrics written by Gabor Szekelyhidi.
Calabi Yau Theorem Part II
When: Wed, February 15, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Cheuk-Yan Fung () -
Abstract: This week, we continue the proof of Calabi-Yau Theorem. In particular, we will show the remaining proof of openness and closeness from last week and Yau's C^0 estimate. If time permits, we can discuss its application in Kähler-Ricci Flow.
Donaldson’s heat flow approach to DUY theorem
When: Wed, February 22, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Yu-Chi Hou (UMD) -
Abstract: In this talk, we continue discussion on stability for holomorphic vector bundles initiated by Sze-Hong in the last semester. We will discuss Donaldson-Uhlenbeck-Yau theorem/Kobayashi-Hitchin correspondence, which generalizes Narasimhan-Seshadri theorem to higher dimensions. Following Yum-Tong Siu’s note, we will present Donaldson’s heat flow proof to prove that any stable bundle on projective manifolds admits Hermitian-Einstein metric.
Donaldson’s heat flow approach to DUY theorem Part II
When: Wed, March 1, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Yu Chi Hou (UMD) -
Holomorphic discs and the homogeneous Monge-Ampere equation
When: Wed, March 15, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Vasanth Pidaparthy (UMD) -
Pluripotential Theory on compact Kahler manifolds
When: Wed, March 29, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Prakhar Gupta (UMD) -
Abstract: After recalling the pluripotential theory on Open sets in Cn discussed last semester, I'll describe the pluripotential theory on compact Kahler manifolds. I'll set up the problem of solving the complex Monge Ampere equation using the variational method.
Kolodziej's approach to C^0 estimates for complex Monge-Ampère equation
When: Wed, April 5, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Prakhar Gupta (UMD) -
Variations of Hodge structures
When: Wed, April 12, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Kwong Sze Hong (UMD) -
Berkovich spaces -- applications to complex dynamics and non-Archimedean potential theory
When: Wed, April 19, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Drimik Roy Chowdhury (UMD) -
Berkovich spaces II: spectrum of integers and harmonic functions
When: Wed, April 26, 2023 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Drimik Roy Chowdhury (UMD) -