Department of Mathematics - UC Santa Barbara

Geometry, Topology, and Physics Seminar, Spring 2007

Organizers: Sergei Gukov and Dave Morrison.
Meets 3:30 - 5:00 p.m. Fridays in South Hall 6635.

Various topics relating geometry, topology, and physics.

Other Quarters: [ Spring, 2017; Wnter, 2017; Fall, 2016; Spring, 2016; Winter, 2016; Fall, 2015; Spring, 2015; Winter, 2014; Fall, 2013; Fall, 2012; Fall, 2011; Winter, 2011; Spring, 2010; Winter, 2010; Fall, 2009; Spring, 2009; Winter, 2009; Fall, 2008; Spring, 2008; Winter, 2008; Fall, 2007; Spring, 2007; Winter, 2007; Fall, 2006 ]

Apr. 6

Dave Morrison (UCSB)

Organizational meeting.

Apr. 13

Sergei Gukov (UCSB)

Perturbative gauge theory and arithmetic topology

Audio [ mp3, wma ]; Lecture notes.

Apr. 20

Xiaodong Cao (MSRI and Cornell Univeristy)

Cross Curvature Flow on Locally Homogenous Three-manifolds

(joint meeting with Differential Geometry Seminar)

Recently, Chow and Hamilton introduced the cross curvature flow on three-manifolds, which is a weakly parabolic partial differential equation system when the sectional curvatures have a definite sign. They also conjectured the long time existence and convergence of cross curvature flow on closed three-manifolds with negative sectional curvature. In this talk, we will study the cross curvature flow on locally homogenous three-manifolds. We will describe the long time behavior of the cross curvature flow for each case. This is a joint work with Yilong Ni and Laurent Saloff-Coste.

Audio [ mp3, wma ]; Lecture notes.

Apr. 27

No meeting

May 4

No meeting

May 11

Mike Anderson (SUNY, Stony Brook)

Is Anti deSitter spacetime dynamically stable?


A discussion of the wide open question: is AdS spacetime dynamically stable? This is basically a hyperbolic PDE problem, a bit analogous to Christdoulou-Klainerman theorem on stability of Minkowski spacetime.

Audio [ mp3, wma ]; Lecture notes.

May 18

No meeting.

May 21

John Lott (MSRI and University of Michigan)

Dimensional reduction and long-time behavior of Ricci flow

South Hall 4607, 3:30 p.m.

(Differential Geometry Seminar; note unusual day and location)

Lecture notes.

May 25

Paolo Cascini (UCSB)

Kähler-Ricci flow

H. D. Cao introduced the Kähler-Ricci flow for canonical metrics on manifolds with definite first Chern class. In particular he obtained a new proof of Calabi's conjecture on the existence of Kähler-Einstein metrics on manifolds with c1 < 0. More in general, the Kähler-Ricci flow is expected to provide a deeper understanding of the geometry of the underlying manifold. We will survey on some of its property and applications.

Audio, part 1 [ mp3, wma ], audio, part 2 [ mp3, wma ]; Lecture notes.

June 1

James McKernan (UCSB)

The Sarkisov program


The conjectural output of the minimal model program is either a minimal model or a Mori fibre space. Unfortunately the output in neither case is unique. Kawamata has recently shown that any two minimal models are connected by a sequence of flops. The Sarkisov program aims to factorise any birational map between two Mori fibre spaces into a sequence of elementary links. In the case of surfaces, an elementary transformation of P1-bundles is an example of such a link, and the Sarkisov program provides a natural framework to prove that the birational automorphism group of P2 is generated by a Cremona transformation and PGL(3).
We describe recent work with Christopher Hacon where we extend the Sarkisov program to all dimensions.