Title: “The geometry of eigenvalue extremal problems”
Abstract: When we choose a metric on a manifold we determine the spectrum of
the Laplace operator. Thus an eigenvalue may be considered as a functional
on the space of metrics. For example the first eigenvalue would be the fundamental
vibrational frequency. In some cases the normalized eigenvalues are bounded
independent of the metric. In such cases it makes sense to attempt to find
critical points in the space of metrics. In this talk we will survey two cases in
which progress has been made focusing primarily on the case of surfaces with
Abstract - Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory.
We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of those results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.
I will give introduction to sofic groups and discuss a possible strategy towards finding a non-sofic group. I will show that if the Higman group were sofic, there would be a map from Z/pZ to itself, locally like an exponential map, satisfying a rather strong recurrence property. The approach to (non)-soficity is based on the study of sofic representations of amenable subgroups of a sofic group. This is joint work with Harald Helfgott.
Abstract: The original construction uses the theory of pseudo-holomorphic curves. In this lecture, I will describe an explicit combinatorial algorithm for computing knot Floer homology in terms of grid diagrams. In this lecture, I will describe joint work with Ciprian Manolescu, Sucharit Sarkar, Zoltan Szabo, and Dylan Thurston.
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: Bordered Floer homology is an invariant for three-manifolds with boundary, defined in collaboration with Robert Lipshitz and Dylan Thurston. The invariant associates a DG algebra to a parameterized surface, and a module over that algebra to a three-manifold with boundary. I will explain how methods from bordered Floer homology can be used to give a tidy description of knot Floer homology. This is joint work with Zoltan Szabo.