Abstract: The Gromov non-squeezing theorem in symplectic geometry states that is not possible to embed symplectically a ball into a cylinder of smaller radius, although this can be done with a volume preserving embedding. Hence, the biggest radius of a ball that can be symplectically embedded into a symplectic manifold can be used as a way to measure the "symplectic size'' of the manifold. We call the square of this radius times the number \pi the Gromov width of the symplectic manifold. The Gromov width as a symplectic invariant is extended through the notion of "Symplectic Capacity".
A (countable) group G is homogeneous if whenever g,h are tupples of the same type in G, there is an automorphism of G sending g to h.
We give a characterization of freely-indecomposable torsion-free hyperbolic groups which are homogeneous, in terms of a particular decomposition as a graph of groups - their JSJ decomposition. This is joint work with Chloe Perin.
Abstract: Knot Floer homology is an invariant for knots in the three-sphere defined using methods from symplectic geometry. I will describe a new algebraic formulation of this invariant which leads to a reasonably efficient computation of these invariants. This is joint work with Zoltan Szabo.
Abstract: In this talk, I will introduce diffeological spaces and some (co)homology theories on these spaces. I will also talk on Thom-Mather spaces and their (co)homology in the diffeological context.
In the first talk we gave a brief outline of the contents of the course. In the rest of the semester we will get deeper into some topics. In the coming lecture ( and the next one) we will discuss Kazhdan property T and its connections with expanders and with first cohomology groups. No prior knowledge will be assumed.
The expander Chernoff bound states that random walks over expanders are good samplers, at least for a certain range of parameters. In this talk we will be interested in “Parity Samplers” that have the property that for any test set, about half of the sample sets see the test set an *even* number of times, and we will check whether random walks over expanders are good parity samplers. We will see that:
1. Random walks over expanders fare quite well with the challenge, but, 2. A sparse Random complex does much better.
The "PCP theorem" says that problems in NP are hard in a robust or stable way.
I will give a brief intro to PCPs (and explain the acronym) and then try to outline a proof of the PCP theorem based on "agreement expansion" which is a form of high dimensional expansion.
My aim is to show how high dimensional expansion is inherently present in PCP type questions.