Hopkins-Maryland Complex Geometry Seminar

Johns Hopkins University, University of Maryland

DATE: Tuesdays at 4:30pm.

ROOM: Ames 218(JHU), MATH1308(UMD).

ORGANIZED BY:
T. Darvas,
Y. A. Rubinstein,
B. Shiffman,
R. Wentworth,
S. Wolpert,
H. Xu.

COORDINATORS: Y. A. Rubinstein, B. Shiffman.

PREVIOUS YEARS: 2012-2013, 2013-2014, 2014-2015, 2015-2016.

- September 13(UMD)

Ben Weinkove (Northwestern University)

*Title: Monge-Ampere equations on complex and almost complex manifolds*

Abstract: Yau's Theorem on the complex Monge-Ampere equation shows that one can prescribe the volume form of a Kahler metric on a compact Kahler manifold. I will describe extensions of this result to non-Kahler settings. In each case, a Monge-Ampere type equation is used to prescribe the volume form of a special metric on a complex or almost complex manifold. This talk is based on joint works with Tosatti, Szekelyhidi and Chu. - September 27(JHU)

Hao Xu (University of Pittsburgh)

*Title: Asymptotic expansion of Bergman and heat kernels*

Abstract: The asymptotic expansion for the Bergman kernel has important applications in complex analysis. Short-time asymptotic expansion of the heat kernel played an important role in spectral geometry. We will present our work on Feynman diagram formulas for the coefficients in the asymptotic expansion of Bergman and heat kernels on Kahler manifolds and their applications. - November 15(UMD)

Mu-Tao Wang (Columbia University)

*Title: Lagrangian curvature flows in cotangent bundles of spheres*

Abstract: I shall present some new long time existence and convergence theorems of Lagrangian curvature flows in cotangent bundles of spheres with either the canonical metric or the Stenzel (Calabi-Yau) metric. The talk will be based on joint work with Knut Smoczyk and Mao-Pei Tsui, and joint work with Chung-Jun Tsai. - December 2, 10:30 AM (UMD, within MADGUYS) 0112 Chemistry/Biochemistry building

Jake Solomon (Hebrew University, Jerusalem)

*Title: Point-like bounding chains in open Gromov-Witten theory*

Abstract: Over a decade ago, Welschinger defined real enumerative invariants in dimensions 2 and 3. It has remained an open problem to extend these invariants to higher dimensions. I will discuss a solution to this problem in the language of open Gromov-Witten theory. The key idea is that boundary point constraints should be replaced with canonical gauge equivalence classes of Maurer-Cartan elements (bounding chains) in the relevant Fukaya A-infinity algebra. The resulting invariants satisfy an open WDVV equation. All invariants for projective spaces have been calculated. In connection with open WDVV, a relative version of the quantum product appears. Real structures do not play an essential role in our arguments. This is joint work with S. Tukachinsky. - February 21(UMD)

Duong Phong (Columbia)

*Title: Supersymmetric vacua of superstrings and geometric flows*

Abstract: In the mid 1980s, C. Hull and A. Strominger proposed a system of equations for supersymmetric vacua of superstrings, which are generalizations with torsion of the Calabi-Yau condition proposed shortly before by P. Candelas, G. Horowitz, A. Strominger, and E. Witten. As such, they are also of interest from the point of view of non-Kahler geometry and partial differential equations. We introduce a flow, called the Anomaly Flow, whose fixed points would provide solutions of the Hull-Strominger system. We provide criteria for the long-time existence of the flow, and show that it can recapture the celebrated solution found in 2006 by J. Fu and S.T. Yau on toric fibrations over K3 surfaces. This last result may be of particular interest in the theory of non-linear partial differential equations, as the corresponding parabolic scalar equation is not concave. This is joint work with S. Picard and X.W. Zhang. - March 28(JHU)

Xiaofeng Sun (Lehigh)

*Title: Deformation of Fano Manifolds*

Abstract: In this talk we will describe a new necessary and sufficient condition on the existence of KE metrics on all small deformation of a Fano KE manifold with nontrivial automorphism group. We will also describe a canonical extension of pluri-anticanonical forms from a Fano KE manifold to its small deformations which leads to simultaneous embedding of a family of Fano manifolds into projective spaces with effective control. We will also discuss a construction of plurisubharmonic functions on Teichmuller spaces of KE manifolds of general type by using energy of equivariant harmonic maps. - April 18(JHU)

Xiaojun Huang (Rutgers)

*Title: Bergman-Einstein metrics on strongly pseudoconvex domains of C^n.*

Abstract: In this talk, we explain how to combine Fefferman's invariant theory, the Chern-Moser theory, the Cheng-Yau solution of the Fefferman equation, as well as CR extension theory to provide an affirmative solution of a conjecture posed by Cheng, Cheng-Yau more than 30 years ago. The conjecture stated that the Bergman metric of a bounded strongly pseudoconvex domain is Einstein if and only if the domain is holomorphically equivalent to the ball. This is a joint work with M. Xiao. - May 2

Mattias Jonsson (University of Michigan)

*Title: A variational approach to the Yau-Tian-Donaldson conjecture*

Abstract: I will present joint work with Robert Berman and Sebastien Boucksom, on a new proof of a uniform version of the Yau-Tian-Donaldson conjecture for Fano manifolds with finite automorphism group. Our approach does not involve the continuity method or Cheeger-Colding-Tian theory. Instead, the proof is variational and uses pluripotential theory and certain non-Archimedean considerations.

Driving directions to JHU: Park in South Garage (see map) on any level (except the reserved spaces). Take a ticket when entering. The Department will provide a visitor parking pass to use when exiting.

Driving and parking directions to UMD: Park in Paint Branch Drive Visitor Lot (highlighted in yellow in the lower right corner of the second map in the previous link), or in Regents Drive Garage (highlighted in the upper right corner). If you arrive after 4pm you do not need to pay: see the instructions in the previous link.