Long time existence of minimizing movement solutions of Calabi flow

Speaker: 

Jeff Streets

Institution: 

UCI

Time: 

Tuesday, February 12, 2013 - 4:00pm

Location: 

RH 306

In 1982 Calabi proposed studying gradient flow of the L^2 norm
of the scalar curvature (now called Calabi flow) as a tool for finding
canonical metrics within a given Kahler class. The main motivating
conjecture behind this flow (due to Calabi-Chen) asserts the smooth long
time existence of this flow with arbitrary initial data. By exploiting
aspects of the Mabuchi-Semmes-Donaldson metric on the space of Kahler
metrics I will construct a kind of weak solution to this flow, known as a
minimizing movement, which exists for all time.

HF=ECH via open book decompositions

Speaker: 

Ko Honda

Institution: 

University of Southern California

Time: 

Tuesday, January 22, 2013 - 4:00pm

Location: 

RH 306

Floer homology theories have had an enormous impact on low-dimensional topology over the last 2-3 decades.  The goal of this talk is to introduce two Floer homology theories -- Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings) -- and to sketch a proof of the equivalence of the two.  This is joint work with Vincent Colin and Paolo Ghiggini.

The talk will be accessible to beginning graduate students.

Holomorphic functions on certain Kahler manifolds

Speaker: 

Ovidiu Munteanu

Institution: 

University of Connecticut

Time: 

Tuesday, February 26, 2013 - 4:00pm

Location: 

RH 306

We first survey some development regarding the study of holomorphic functions on manifolds. We insist mostly on Liouville theorems or, more generally, dimension estimates for the space of polynomially growing holomorphic functions. Then we present some recent joint work with Jiaping Wang on this topic. Our work is motivated by the study of Ricci solitons in the theory of Ricci flow. However, the most general results we have do not require any knowledge of curvature.

Positive curvature in Sasaki geometry

Speaker: 

Weiyong He

Institution: 

University of Oregon

Time: 

Tuesday, January 15, 2013 - 4:00pm to 5:00pm

Location: 

RH 306

This is based on joint work with Song Sun.
As an analogue of Frankel conjecture (Mori, Siu-Yau theorem) in Kahler geometry, we
classify compact Sasaki manifolds with positive curvature by deforming metrics.
Roughly speaking, such Sasaki structure is a standard Sasaki structure on (odd
dimensional) spheres. Our theorem gives a new proof of Frankel conjecture as a
special case. We have also similar results as in Kahler setting for nonnegative
curvature.

An open mirror theorem for toric varieties

Speaker: 

Siu-Cheong Lau

Institution: 

Harvard University

Time: 

Tuesday, January 8, 2013 - 4:00pm

Location: 

RH 306

Mirror map is a central object in the study of mirror symmetry. They are obtained in hypergeometric series by solving Picard-Fuchs equations. In this talk, I will explain a geometric meaning of mirror maps for toric varieties in terms of counting of holomorphic discs bounded by Lagrangian submanifolds. It is motivated by the study of SYZ mirror symmetry. This is a joint work with K. Chan, N.-C. Leung and H.-H. Tseng.

Symplectic harmonic forms and the Federer-Fleming deformation theorem

Speaker: 

Yi Lin

Institution: 

Georgia Southern University

Time: 

Tuesday, March 5, 2013 - 4:00pm

Location: 

RH 306

Symplectic harmonic theory was initiated by Ehresmann and Libermann in 1940's, and was rediscoverd by Brylinski in late 1980's. More recently, Bahramgiri showed in his MIT thesis that symplectic harmonic representatives of Thom classes exhibited some interesting global feature of symplectic geometry. In this talk, we discuss a new approach to symplectic Harmonic theory via geometric measure theory. The new method allows us to establish a fundamental property on symplectic harmonic forms, which is a non-trivial generalization of Bahramgiri's result, and enables us to provide a complete solution to an open question asked by V. Guillemin concerning symplectic harmonic representatives of Thom classes.  This talk is based on a very recent work of the speaker.

J-holomorphic curves in a nef class

Speaker: 

Tian-Jun Li

Institution: 

University of Minnesota

Time: 

Tuesday, January 29, 2013 - 4:00pm

Location: 

RH 306

We investigate properties of reducible J-holomorphic subvarieties in 4-manifolds. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is J-nef.

For a spherical class, it has particularly strong consequences: for any tamed J, each irreducible component is a smooth rational curve. We also completely classify configurations of maximal dimension. To prove these results we treat subvarieties as weighted graphs and introduce several combinatorial moves. This is a joint work with Weiyi Zhang.

Rational analogs of projective planes

Speaker: 

Zhixu Su

Institution: 

UC Irvine

Time: 

Tuesday, November 6, 2012 - 4:00pm

Location: 

RH 306

There does not exist closed manifold along the line of projective planes
above the dimension of octonions due to the inexistence of hopf invariant
1 map in higher dimensions. I investigated the existence dimension of such
manifold in the rational sense, such that the rational cohomology is rank
one in dimension 0, 2k and 4k and is zero otherwise. Applying rational
surgery, the problem can be reduced to finding possible Pontryagin classes
satisfying the Hirzebruch signature formula and a set of congruence relations
determined by the Riemann-Roch integrality conditions, which is eventually
equivalent to solving a system of Diophantine equations. After a negative
answer in dimension 24, the first existence dimension of such manifold is 32.

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