Differential Geometry Seminar Schedule for
Spring 2008
Fridays 3:00 - 4:00pm, SH 6617
4/4, no meeting, see Distinguished Lectures
5/9, Robert McCann, University of Toronto
``Extremal doubly stochastic measures and optimal transport"
5/23, Rugang Ye, UCSB
``"
5/30, , UCSB
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6/6, , UCSB
``"
Differential Geometry Seminar Schedule for
Winter 2008
Fridays 3:00 - 4:00pm, SH 6617
1/18, Seungsu Hwang, Chung-Ang University and visiting UCSB
``Critical points of the total scalar curvature functional"
Abstract: On a compact n-dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation. In 1987 Besse proposed a conjecture that a solution of the critical point equation is Einstein. Since then, a number of mathematicians have given partial proof of the conjecture and obtained many geometric consequences. However, none has given a complete proof.
The purpose of the talk is to show that a compact 3-dimensional manifold M is isometric to the round 3-sphere $S^3$ if ker $s_g'* \not= 0$ and its second homology vanishes. Note that this Theorem implies that M is Einstein and hence that Conjecture holds on 3--dimensional compact manifold under certain topological conditions.
1/25 Jeffery Case, UCSB
``The Bakry-Emery tensor in Lorentzian geometry"
Abstract: In this talk, we will consider extensions of the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem to the Bakry-Emery-Ricci tensor. We will point out similarities and differences to the setting in Riemannian geometry, and spend the most time on ideas less familiar in Riemannian geometry.
2/1 Colin Hinde, UCLA
``Synthetic Ricci Curvature and Optimal Transportation"
Abstract: We will review the solution of the optimal transportation problem on Riemannian manifolds. The investigation leads to a synthetic curvature condition equivalent to a lower bound on Ricci curvature yet requiring only the structure of a locally compact length space with a Borel measure. Yet, as in the case of Alexandrov Space, the synthetic curvature condition secures additional structure for free. We will discuss some results in this direction as well as localization of the condition, as time permits.
2/8, organization meeting
2/15, Rugang Ye, UCSB
``The entropy functional and the kappa-noncollapsing estimate for
the Ricci flow"
2/22, no meeting
2/29, Guofang Wei, UCSB
``An Introduction to Optimal Transportation"
3/7, Guofang Wei, UCSB
``Space of probability measures with Wasserstein distances"
Differential Geometry Seminar Schedule for
Fall 2007
Fridays 3:00 - 4:00pm, SH 6617
9/28, organization meeting
10/5 Rugang Ye, UCSB
``The Logarithmic Sobolev inequality along the Ricci flow"
10/12 Rugang Ye, UCSB
``The Logarithmic Sobolev inequality along the Ricci flow, continue"
10/19 Yujen Shu, UCSB
``Constant scalar curvature Kahler metrics on ruled surface"
Abstract: We will introduce the notion of constant scalar curvature Kahler
(cscK) metrics, and investigate the existence of such metrics on ruled
surfaces.
10/26 Zhenghan Wang, Microsoft/UCSB
``On Exotic (2+1)-TQFTs"
Abstract:
G. Moore and N. Seiberg conjectured in 1990s that every (2+1)-TQFT could be realized as
a Chern-Simons-Witten theory based on a pair (G,\lamda), where G is a compact (not necessarily connected)
Lie group, and \lambda a cohomology class in H^4(BG;Z). I will discuss examples of TQFTs constructed from Jones' subfactor theory which might not be realized in this way. This is a joint work with Seung-moon Hong of Indiana Univ and Eric Rowell of Texas A&M.
11/2 Zhongmin Shen, IUPUI
``Ricci curvature in Finsler geometry"
11/9 Rugang Ye, UCSB
``Analytic and geometric estimate for Ricci flow with surgeries"
11/16 Peter Petersen, UCLA
``Rigidity of shrinking Ricci solitons"
Abstract: I will survey several recent results on Ricci solitons. The goal is to find suitable conditions that make it possible to classify these objects. Several different conditions on curvature and symmetry allow us to classify these objetcs. Best of all in dimension 3 we can classify all shrinking solitions without further assumptions.
11/23 Thanksgiving
11/30 see Karen Vogtmann's talk
12/7 Andreas Malmendier, UCSB
``Rational elliptic surfaces and the u-plane integral"
Abstract: I will give a brief review of how the Seiberg-Witten families of elliptic curves with N_f=0,1,2,3,4 massive hypermultiplets define certain rational elliptic surfaces, p: Z -> CP^1, with non-stable singularities. I will then explain how the masses of the hypermultiplets are closely related to the homology of this surface: the root lattice T corresponding to the singularity type is embedded into H_2(Z). Oguiso and Shioda classified all possible structures for T and the Mordell-Weil group of p. Roughly, the evaluation of the holomorphic two-form on classes perpendicular to T then gives the masses of the hypermultiplets.
If time permits, I will explain how the Donaldson invariants for CP^2
can be computed by evaluating a certain ratio of functional determinants of a family of elliptic operators parametrized by the rational elliptic surface for N_f =0.
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