UCSD Differential Geometry Seminar
UCSD Differential Geometry Seminar |
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Thursday November 13th, 2008. *** Note the different time and place from usual ***.
Room and Time: AP&M 7421, 3-4pm.
Speaker: Artem Pulemotov, Cornell University.
Title: The Li-Yau-Hamilton Estimate and the Yang-Mills Heat Equation
Abstract: The talk will focus on two connected subjects. First, we will discuss the Li-Yau-Hamilton estimate for the heat equation on a manifold M with nonempty boundary. Results of this kind play a significant part in proving monotonicity formulas for geometric flows. Second, we will talk about the Yang-Mills heat equation in a vector bundle over M. Our interest is mainly in the long-time existence of solutions.
Wednesday November 19th, 2008.
Room and Time: AP&M 5402, 4-5pm.
Speaker: Yujen Shu, UC Santa Barbara.
Title: CscK metrics on compact complex surfaces
Abstract:
We will introduce the notion of constant scalar curvature
Kahler (cscK) metrics, and investigate the existence of such metrics on
compact complex surfaces. We show that every compact complex surface with
even first Betti number, if it is not in the deformation equivalence class
of CP_2 blown up at one or two points, is deformation equivalent to one
which admits an cscK metric.
Wednesday December 3, 2008.
Room and Time: AP&M 5402, 4-5pm.
Speaker: Mark Haskins, Imperial College London.
Title: Singular special Lagrangian n-folds
Abstract: We discuss recent progress on understanding singular special Lagrangian n-folds. Our focus will be on joint work with N. Kapouleas using gluing methods to construct a wide variety of special Lagrangian cones in every dimension three and greater.Thursday December 4, 2008.
UC IRVINE - UC SAN DIEGO JOINT GEOMETRY SEMINAR.
Room and Time: AP&M 6402, 3-4pm
Speaker: Fred Wilhelm, UC Riverside.
Title: An exotic sphere with positive sectional curvature.
Abstract: I'll discuss joint work with Peter Petersen that
shows that the Gromoll-Meyer exotic 7-sphere admits positive sectional
curvature. I'll discuss the history of the problem and give a coarse
outline of our solution.
Room and Time: AP&M 6402, 5-6pm
Speaker: Jacob Bernstein, MIT.
Title: Helicoid-Like Minimal Disks
Abstract: Colding and Minicozzi have shown that if an embedded minimal
disk in $B_R\subset\Real^3$ has large curvature then in a smaller ball, on a scale
still proportional to $R$, the disk looks roughly like a piece of a helicoid. In
this talk, we will see that near points whose curvature is relatively large the
description can be made more precise. That is, in a neighborhood of such a point (on
a scale $s$ proportional to the inverse of the curvature of the point) the surface
is bi-Lipschitz to a piece of a helicoid. Moreover, the Lipschitz constant goes to
1 as $Rs$ goes to $\infty$ . This follows from Meeks and Rosenberg's result on the
uniqueness of the helicoid of which, time permitting, we will discuss a new proof.
Joint work with C. Breiner.
Wednesday January 14, 2009.
Room and Time: AP&M 5402, 4-5pm.
Speaker: Neil Donaldson, UC Irvine.
Title: Isothermic submanifolds in Euclidean space
Abstract: We give a positive answer to Burstall's question of whether there exists an interesting theory of isothermic submanifolds of dimension>2 in R^n. We relate chains of such manifolds to solutions of a system of PDEs and describe their moduli space. We also describe Christoffel and Darboux/Ribaucour transforms of isothermic chains.
Wednesday January 21, 2009.
Room and Time: AP&M 6402, 3-4pm.
Note change in time and room!
Speaker: Reza Seyyedali, Johns Hopkins University.
Title: Balanced Metrics and Chow Stability of Ruled Manifolds
Abstract: In 1980, I. Morrison proved that slope stability of a vector bundle of rank $2$ over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the statement holds for higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group.
Thursday February 19, 2009.
Room and Time: AP&M 5402, 10.30-11.30am (Note change in time and room!)
Speaker: Valentino Tosatti, Harvard University.
Title: Kahler-Ricci flow and stability
Abstract: I will discuss the relationship between convergence of
the Ricci
flow on a Fano manifold and algebraic stability
of the manifold with the anticanonical polarization. I will show that if
the
curvature remains bounded along the flow then
stability implies convergence of the flow and so in particular existence
of
a Kahler-Einstein metric.
Room and Time: AP&M 7421 3.30-4.30pm (Note change in time and room!)
Speaker: Gabor Szekelyhidi, Columbia University.
Title: Greatest lower bounds on the Ricci curvature of Fano manifolds
Abstract: On Fano manifolds we study the supremum of the possible t
such that there exists a metric in the first Chern class with Ricci
curvature bounded below by t. For the projective plane blown up in one
point we show that this supremum is 6/7.
Saturday, Sunday, February 21-22. SOUTHERN CALIFORNIA
GEOMETRIC
ANALYSIS SEMINAR
Wednesday March 4, 2009.
Room and Time: AP&M 6402, 3-4pm.
Speaker: Burkhard Wilking, Muenster University.
Title: High dimensional Ricci flow
Thursday March 19, 2009.
Room and Time: AP&M 5402, 3-4pm
Speaker: Panagiota Daskalopoulos, Columbia University.
Title: Ancient solutions to the Curve shortening flow and Ricci flow on surfaces
Wednesday April 8, 2009.
Room and Time: AP&M 5402, 4-5pm
Speaker: Owen Dearricott, UC Riverside.
Title: Positive curvature on 3-Sasakian 7-manifolds
Abstract: We discuss metrics of positive curvature on 3-Sasakian 7-manifolds including one on a new diffeomorphism type.
Tuesday April 14, 2009.
Room and Time: AP&M 5402, 2.30-3.30pm
Speaker: Dan Knopf, University of Texas, Austin.
Title: Minimally-invasive surgery for Ricci flow singularities
Abstract: If a solution (M,g(t)) of Ricci flow develops a local singularity at a finite time T, there is a proper subset S of M on which the curvature becomes infinite as time approaches T. Existing approaches to Ricci-flow-with-surgery, due to Hamilton and Perelman, require one to modify the solution in a small neighborhood of S by gluing in a highly curved but nonetheless nonsingular solution. This must be done with careful regard to various surgery parameters in order to preserve critical a priori estimates. In case the local singularity is a rotationally-symmetric neckpinch (in any dimension n>2), we can now restart Ricci flow directly from the singular limit g(T), without performing an intervening surgery or making ad hoc choices. We show that any complete smooth forward evolution from g(T) is necessarily compact and has a unique asymptotic profile as it emerges from the singularity, which we describe. (This is joint work with Sigurd Angenent and Cristina Caputo.)
Friday April 24, 2009.
Room and Time: AP&M 6402, 4-5pm
Speaker: Julie Rowlett, UC Santa Barbara.
Title: The Fundamental Gap Conjecture for Triangles
Abstract: The Fundamental Gap Conjecture due to S. T. Yau and M. van de Berg states that for a convex domain in R^n with diameter d, the first two positive eigenvalues of the Dirichlet Laplacian satisfy $$\lambda_2 - \lambda_1 \geq \frac{3 \pi^2}{d^2}.$$ $\lambda_2 - \lambda_1$ is known as the fundamental gap and has been studied by many authors. It is of natural interest to spectral geometers, and moreover, estimates for the fundamental gap have applications in analysis, statistical mechanics, quantum field theory, and numerical methods.
I will discuss joint work with Zhiqin Lu on the fundamental gap when
the domain is a Euclidean triangle. Our first result is a compactness
theorem for the gap function, which shows that the gap function is
unbounded as a triangle collapses to a segment. I will outline our
current work which indicates that the equilateral triangle is a strict
local minimum for the gap function on triangular domains. Finally, I
will discuss how these results combined with numerical methods may be
used to prove the well known conjecture that among all triangular
domains, the fundamental gap is minimized by the equilateral triangle.
Wednesday May 6, 2009.
Room and Time: Room AP&M 5402, 4-5pm
Speaker: Ben Weinkove, UC San Diego.
Title: The Kahler-Ricci flow on Hirzebruch surfaces
Abstract: I will discuss the metric behavior of the
Ka¤hler-Ricci flow on Hirzebruch surfaces assuming that the initial metric
is invariant under a maximal compact subgroup of the automorphism group. I
will describe how, in the sense of Gromov-Hausdorff, the flow either
shrinks to a point, collapses to P^1 or contracts an exceptional divisor.
This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work
with Jian Song.
Thursday May 14, 2009.
Room and Time: AP&M 6402, 3-4pm
Speaker: Brett Kotschwar, MIT.
Title: Backwards-uniqueness for the Ricci flow
Abstract: I will discuss the problem of backwards-uniqueness
or "unique-continuation" for the Ricci flow, and prove that
two complete solutions $g(t)$, $\tilde{g}(t)$ to the Ricci flow
on $[0, T]$ of uniformly bounded curvature that agree at $t=T$
must agree identically on $[0, T]$. A consequence is that the
isometry group of a solution to the Ricci flow
cannot expand in finite time.
Wednesday May 20, 2009.
Room and Time: Room AP&M 5402, 4-5pm
Speaker: Lei Ni, UC San Diego.
Title: Two theorems on positive curved manifolds via Ricci flow.
Room and Time: Room AP&M 5402, 4-5pm
Speaker: Shengli Kong, UC San Diego.
Title: Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations
Abstract: I will discuss a construction of ancient solutions to Ricci flow on spheres and complex projective spaces which generalize Fateev's examples on three spheres. These examples supply counter-examples to some folklore conjectures on ancient solutions of Ricci flow on compact manifolds. This a joint work with Ioannis Bakas and Lei Ni.
Thursday May 28, 2009.
Room and Time: Room AP&M 6402, 4-5pm
Speaker: Neshan Wickramasekera, University of Cambridge.
Title: A general regularity theory for stable codimension 1 integral varifolds
Tuesday June 23, 2009.
Room and Time: Room AP&M 6402, 4-5pm
Speaker: Albert Chau, University of British Columbia.
Title: Lagrangian mean curvature flow for entire Lipschitz graphs
Abstract: In this joint work with Jingyi Chen and Weiyong He, we
prove
existence of longtime smooth solutions to
mean curvature flow of entire Lipschitz Lagrangian graphs. As an
application, we obtain results on entire translating and self-expanding
solutions to to the Lagrangian mean curvature flow.
Questions: Contact Ben Weinkove (weinkove@math) or Lei Ni (lni@math).
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