Harvard Differential Geometry Seminar
Fall 2017
Organizer: T. Collins
The seminar meets on Tuesdays from 4:15 to 5:15 in SC 507, except as noted below.
Schedule
September 12, 2017
Jordan Keller (Black Hole Initiative)
- Linear Stability of Higher Dimensional Schwarzschild Black Holes
[abstract]
The Schwarzschild-Tangherlini black holes are higher-dimensional generalizations of the Schwarzschild spacetimes, comprising a static, spherically symmetric family of black hole solutions to higher-dimensional vacuum gravity. The physical relevance of such solutions is intimately related to their stability under gravitational perturbations. This talk will address results on the linear stability of the Schwarzschild-Tangherlini black holes, part of ongoing joint work with Pei-Ken Hung and Mu-Tao Wang
September 19, 2017
Peter Smillie (Harvard)
- Weingarten foliations of three dimensional Lorentzian spaceforms
[abstract]
We study the existence of foliations of three dimensional Lorentzian spaceforms by families of spacelike Weingarten surfaces and the induced flows on Teichmuller space.
September 26, 2017
No Seminar -
[abstract]
October 3, 2017
Valentino Tosatti (Northwestern)
- Special Kahler geometry and collapsing
[abstract]
Special Kahler geometry was first discovered in the context of N=2 supersymmetric 4D gauge theories, and it also plays a prominent role in mirror symmetry. A key observation of Donagi-Witten is that the base of every algebraic integrable system admits a special Kahler metric, while the total space admits a hyperkahler metric. Z. Lu proved that special Kahler manifolds are never complete unless they are flat.
I will explain how special Kahler geometry arises in the collapsing of hyperkahler metrics on the total space of an algebraic integrable system (i.e. a projective irreducible holomorphic symplectic manifold with a holomorphic Lagrangian torus fibration), when the volume of the torus fibers shrinks to zero. Furthermore, the global collapsed Gromov-Hausdorff limit is equal to the metric completion of the special Kahler metric, and has singularities in real codimension 2, as predicted by the SYZ conjecture. This solves a conjecture of Kontsevich-Soibelman and Gross-Wilson in this setting.
October 10, 2017
No Seminar
-
[abstract]
October 17, 2017
Gang Liu (Northwestern)
- Recent progress of Yau's uniformization conjecture
[abstract]
Yau’s uniformization conjecture states that a complete noncompact Kahler manifold with positive bisectional curvature is biholomorphic to the complex Euclidean space.
We confirm this conjecture when the manifold has maximal volume growth.
October 24, 2017
Yevgeniy Liokumovich (MIT)
- Heegaard splittings isotopic to an index 1 minimal surface
[abstract]
Let $M$ be a $3$-manifold that has no one-sided embedded surfaces and
$\Sigma$ a strongly irreducible Heegaard surface in $M$.
We show that for a generic Riemannian metric on $M$ surface $\Sigma$
is isotopic to a minimal surface of index $1$, confirming a conjecture
of Pitts and Rubinstein. This is a joint work with Daniel Ketover and
Antoine Song.
October 31, 2017
Bill Goldman (Brown University and Maryland University)
- Dynamics on moduli spaces of flat connections
[abstract]
We describe dynamical systems arising from the classification of locally homogeneous
geometric structures are flat connections on topological manifolds.
These arise from natural mapping class group actions on character varieties. The mapping class group action is a discrete version of a continuous object, namely the extension of the Teichmueller flow to a universal character variety over the tangent bundle of Teichmuller space. We give several examples of this construction and discuss joint work with Giovanni Forni on a mixing property of this suspended flow. Finally we discuss specific examples of these dynamics for surfaces with rank two fundamental group (Euler characteristic -1).
November 7, 2017
Julius Ross (UIC)
- The Monge-Ampere Equation and the Hele-Shaw flow
[abstract]
I shall describe duality between solutions of the Dirichlet problem to the complex Monge-Amp\`ere Equation and a flow coming from fluid dynamics called the Hele-Shaw flow. Using this, we can give examples of boundary data for which solutions to this Dirichlet problem exhibit previously unknown phenomena, and use analytic results from complex geometry to deduce new statements about this classical flow. This work is joint with David Witt Nystr\"om.
November 14, 2017
No Seminar
-
[abstract]
November 21, 2017
Siu-Cheong Lau (Boston University)
- Moduli theory of Lagrangian immersions and mirror symmetry
[abstract]
In this talk, we consider the moduli space of Lagrangian immersions coming from a pair-of-pants decomposition and use it to construct mirror symmetry. It is a joint work with Cheol-Hyun Cho and Hansol Hong.
** Special Seminar ** November 22, 2017, 4:00-5:15 SC 507
Ryosuke Takahashi (Tohoku)
- A new parabolic approach to the Kahler-Einstein problem
[abstract]
We introduce the ``Ding-Mabuchi flow'', a new parabolic flow which is designed to deform a given Kahler metric to a Kahler-Einstein one, and fit Donaldson's new GIT picture. We provide three results about the Ding-Mabuchi flow:
(1) Long-time existence and smooth convergence on canonically polarized manifolds
(2) Long-time existence and strong convergence on anti-canonically polarized manifolds with Kahler-Einstein metrics, and discrete automorphism group
(3) Relation between the Ding-Mabuchi flow and optimal destabilizer on Ding-unstable toric Fano manifolds
In the canonically polarized case, the large time behavior of the Ding-Mabuchi flow is similar to that of the Kahler-Ricci flow. On the other hand, the Ding-Mabuchi flow on Ding-unstable Fano manifolds has a different aspect. We observe that it encodes the optimal destabilizing information about Kahler/algebro-geometric structure in the case when the manifold is toric. This talk is based on the joint work with T.Collins (Harvard Univ) and T. Hisamoto (Nagoya Univ).
November 28, 2017
Gabor Székelyhidi (Notre Dame)
- TBA
[abstract]
** Special Seminar ** December 6, 2017, 4:15-5:15 SC 507
Eveline Legendre (Toulouse)
- TBA
[abstract]
Schedule for past years:
Fall 2016