Differential Geometry Seminar Schedule for Spring 2011

Fridays 3:00 - 3:50pm, SH 6617

4/8 Paul Feehan, Rutgers University ``American-style options, stochastic volatility, and degenerate variational inequalities"

Abstract: Elliptic and parabolic partial differential equations arising in option pricing problems involving the Cox-Ingersoll-Ross or Heston stochastic processes are well-known to be degenerate parabolic. We provide a report on our work on the existence, uniqueness, and regularity questions for stationary and evolutionary variational equalities and inequalities (obstacle problems) involving degenerate elliptic and parabolic differential operators and applications to American-style option pricing problems for the Heston model, as well as stochastic representations for the solutions. This is joint work with Panagiota Daskalopoulos (Department of Mathematics, Columbia University) and Camelia Pop (Department of Mathematics, Rutgers University).

4/22 Changfeng Gui, University of Connecticut, visiting UCI ``Minimal surfaces, Mean curvature solitons and the Allen-Cahn equation"

Abstract: In this talk, I will talk about the connection between the minimal surfaces, mean curvature solitons and the Allen-Cahn equation. Namely, the level sets of stationary and traveling wave solutions of the Allen-Cahn equation resemble minimal surfaces and translating mean curvature surfaces respectively. Some analogous results for stationary Allen-Cahn equation as for minimal surfaces are shown rigorously, as conjectured by De Giorgi. Recently, some progresses are made on the traveling wave solutions as well. In particular, it is shown that any monotone traveling wave solutions must be symmetric in dimension 2. More investigation is needed to understand the deep connection.

4/26, 27, 28 Please attend the Distinguished Lecture by Jeff Cheeger


Lecture 2: LIPSCHITZ MAPS TO $L_1$


Abstract: The lectures will deal with the notion of {\it quantitative differentiation} and its applications. The simplest instance concerns functions $f:[0,1]\to \R$ with $|f'|\leq 1$. The basic assertion, which appears in work of Peter Jones from 1988, can be paraphrased as stating that in a precise quantitative sense, "$f$ is as close as one likes to being linear at most locations and scales". In the first lecture, it will be explained how the above is actually a particular case of something considerably more general. An "axiomatic" formulation is given in an appendix to a joint paper with B. Kleiner and A. Naor. This paper is discussed in the second lecture. The specific quantitative differentiation result concerns Lipschitz maps from the Heisenberg group to $L_1$. It turns out that there is an application to theoretical computer science. In the third lecture we will explain a quantitative differentiation result in riemannian geometry, which leads to curvature estimates for K\"ahler-Einstein manifolds off sets of small volume.

5/5 Please attend the Colloquium by Wolfgang Ziller

Title: Manifolds with positive curvature

Abstract: Over the last 10 years the question of which manifolds do or do not admit positive curvature has been studied extensively under the presence of a large symmetry group. We will review some of the results, discuss a recent new example of positive curvature, and an obstruction as well.

5/6 John Lott, UC Berkeley ``Remarks on optimal transport and geometry"

10/14 Yanir Rubinstein, Stanford University ``"


Differential Geometry Seminar Schedule for Winter 2011

Fridays 3:00 - 3:50pm, SH 6617

1/28 Peng Wu, UCSB ``On gradient steady Ricci solitons"

Abstract: Ricci soliton are natural extension of Einstein manifolds, they are also possible singularity models of the Ricci flow. Recently there are a lot of work on geometry/topology of Ricci solitons and their classifications. In this talk, I will derive an estimate for potential function of complete noncompact gradient steady Ricci solitons. As a consequence, we show there is no nontrivial complete noncompact gradient steady Ricci soliton with bounded potential function or uniformly positive scalar curvature.

2/25 James Isenberg, University of Oregon ``Ricci Flow on NonCompact Surfaces (Plus Comments on Kaluza-Klein Ricci Flow on Surfaces)"

3/4 Dan Knopf, University of Texas, Austin ``Ricci flow through singularities"

Abstract: We construct and describe smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches, without performing an intervening surgery. In the restrictive context of rotational symmetry, our construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities." This is joint work with Sigurd Angenent and Cristina Caputo.

3/11 Yu Zheng, East China Normal University, visiting UCI ``On the curvature preserving under the Ricci flow"

Differential Geometry Seminar Schedule for Fall 2010

Fridays 3:00 - 3:50pm, SH 6617

10/8 Christina Sormani, CUNY Lehman College and Graduate Center ``The Intrinsic Flat Convergence of Riemannian Manifolds"

Abstract: We define a new distance between oriented Riemannian manifolds that we call the intrinsic flat distance based upon Ambrosio-Kirchheim's theory of integral currents on metric spaces. Limits of sequence of manifolds, with a uniform upper bound on their volumes, the volumes of their boundaries and diameters are countably Hm rectifiable metric spaces with an orientation and multiplicity that we call integral current spaces. In general the Gromov-Hausdorff and intrinsic flat limits do not agree. However, we show that they do agree when the sequence of manifolds has nonnegative Ricci curvature and a uniform lower bound on volume and also when the sequence of manifolds has a uniform linear local geometric contractibility function. These results are proven using work of Greene-Petersen, Gromov, Cheeger-Colding and Perelman. We present an example of three manifolds with positive scalar curvature constructed using Gromov-Lawson connected sums attaching two standard 3 spheres with increasingly many tiny wormholes which converge in the Gromov Hausdorff sense to the standard three sphere but in the intrinsic flat sense to the 0 space due to the cancelling orientation of the two spheres. We conjecture this cannot occur if we exclude spaces with interior minimal surfaces. This is joint work with S. Wenger.

11/12 Xianzhe Dai, UCSB ``An Introduction to Orbifolds"

Abstract: Orbifolds are generalizations of smooth manifolds which allow finite quotient singularities. It is a very useful generalization that appears often in math and physics. On the other hand, it does not require significantly new technical tools to deal with. We will start with the basics of the theory and end with the orbifold index theorem.

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