UCSD Differential Geometry Seminar

Fall 2011 Schedule

Winter 2012 Schedule

Spring 2012 Schedule

Abstracts

Wed 10/5/11. Ben Weinkove (UCSD). The Kahler-Ricci flow on projective bundles. Abstract: I will discuss the behavior of the Kahler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kahler class, the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base. This is a joint work with J. Song and G. Szekelyhidi.

Wed 11/2/11. Zhenlei Zhang (Capital Normal University). A convergence theorem of the Kahler-Ricci flow to a Kahler-Ricci soliton. Abstract: The aim is to show a convergence theorem of the Kahler-Ricci flow: if the initial metric is sufficiently close to a shrinking Kahler-Ricci soliton with respect to a holomorphic vector field, then the modified Kahler-Ricci flow by this holomorphic vector field will converge to a shrinking Kahler-Ricci soliton.

Wed 12/7/11. Jeff Streets (UC Irvine). Geometric flows in complex geometry. Abstract: I will introduce a new geometric flow on complex, non-Kahler manifolds. I will exhibit Perelman-type functionals for this flow, and some regularity results. Finally I will present an optimal regularity conjecture and discuss its relationship to the long open problem of the classification of Class VII surfaces. This is joint work with G. Tian.

Tue 12/13/11. Feng Luo (Rutgers University). A dilogarithm identity on moduli space of Riemann surfaces. Abstract: Given any closed hyperbolic surface of a fixed genus, we establish an identity involving dilogarithm of lengths of simple closed geodesics in all embedded pairs of pants and one-holed tori in the surface. One may consider this as a counterpart of McShane's identity for closed hyperbolic surfaces. This is a joint work with Ser Peow Tan.

Fri 1/13/12. Frederick Fong (Stanford University). Collapsing Behavior of the Kahler-Ricci flow and its Singularity Analysis. Abstract: In this talk, I will discuss my recent works on the collapsing behavior of the Kahler-Ricci flow. The first work studies the Kahler-Ricci flow on P^1-bundles over Kahler-Einstein manifolds. We proved that if the initial Kahler metric is constructed by the Calabi's Ansatz and is in the suitable Kahler class, the flow must develop Type I singularity and the singularity model is P^1 X C^n. It is an extension of Song-Weinkove's work on Hirzebruch surfaces. The second work discusses the collapsing behavior in a more general setting without any symmetry assumption. We showed that if the limiting Kahler class of the flow is given by a holomorphic submersion and the Ricci curvature is uniformly bounded from above with respect to the initial metric, then the fibers will collapse in an optimal rate, i.e. diam ~(T-t)^{1/2}. It gives a partial affirmative answer to a conjecture stated in Song-Szekelyhidi-Weinkove's work on projective bundles.

Fri 1/13/12. Albert Chau (University of British Columbia). Compact manifolds with nonnegative quadratic orthogonal bisectional curvature. Abstract: In this talk I will discuss nonnegatively curved compact Kahler manifolds and their classification. An overview of past results will be given in the cases of bisectional and orthogonal bisectional curvature. The more recent case of quadratic orthogonal bisectional curvature will then be discussed along with recent results. The talk is based on joint work with L.F. Tam.

Fri 1/13/12. Adam Jacob (Columbia University). The Yang-Mills flow and the Atiyah-Bott formula on compact Kahler manifolds. Abstract: In this talk I will describe the limiting properties Yang-Mills flow on a holomorphic vector bundle E, in the case where the flow does not converge. In particular I will describe how to determine the L^2 limit of the curvature endomorphism along the flow. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. I will then show how to use this result to identify the limiting bundle along the flow, which turns out to be independent of metric and uniquely determined by the isomorphism class of E.

Wed 1/25/12. Nicolaos Kapouleas (Brown University). Gluing constructions for minimal surfaces and self-shrinkers . Abstract: In the first part of the talk I will discuss doubling constructions. In particular I will discuss in some detail a recent doubling construction for an equatorial two-sphere in the round three-sphere, and also potential generalizations for self-shrinkers of the Mean Curvature flow. In the second part of the talk I will briefly discuss the current understanding of desingularization constructions for minimal surfaces and self-shrinkers. In the third and final part I will discuss open uniqueness questions for closed embedded minimal surfaces in the round three-sphere inspired by the above constructions.

Wed 2/8/12. Jim Isenberg (University of Oregon). Asymptotic Behavior of Degenerate Neckpinches in Ricci Flow. Abstract: We discuss the detailed nature of the geometry of rotationally symmetric degenerate neckpinch singularities which develop in the course of Ricci flow.

Wed 2/15/12. Zhiwei Wu (UC Irvine). Equations of KdV type and curve flows in affine space. Abstract: The KdV equation is one of the most important equations in soliton theory. It can be generalized to Gelfand-Dikii hierarchy and there have been a lot of work related to it. In this talk, I will give a geometric interpretation of the equations in Gelfand-Dikii hierarchy as curve flows in R^n. I will also discuss Backlund transformation and Hamiltonian structures for these curve flows.

Wed 2/22/12. Li-Sheng Tseng (UC Irvine). Differential Cohomologies on Symplectic Manifolds. Abstract: In this talk, I will introduce new cohomologies and elliptic operators on symplectic manifolds. Their construction follows from a simple decomposition of the exterior derivative into two first-order symplectic differential operators, which are analogous to the Dolbeault operators in complex geometry. These symplectic cohomologies encode new geometrical invariants especially for non-Kahler symplectic manifolds. This is joint work with S.-T. Yau.

Questions: Contact Ben Weinkove (weinkove@math) or Lei Ni (lni@math).

Email addresses end in ucsd.edu

See the 2010-2011 Differential Geometry Seminar schedule.

See the 2009-2010 Differential Geometry Seminar schedule.

See the 2008-2009 Differential Geometry Seminar schedule.

Return to UCSD Mathematics homepage.