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.

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.

In this talk, we will first survey some known result about curvature behavior at the first finite-singularity time under the Ricci flow. Then we will discuss some recent development in this direction and their applications.

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.

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.