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Series: Geometry Topology Seminar

We will start by counting lattice points in a polytope and showhow this produces many familiar objects in mathematics.For example if one scales the polytope, the number of lattice points givesrise to the Ehrhart polynomials, including binomals and other well knownfunctions.Things get more interesting once we take a weighted sum over the latticepoints instead of just counting them. I will explain how toextend Ehrhart's theory in this case and discuss an application to knottheory. We will derive a new state sum for the colored HOMFLYpolynomial using q-Ehrhart polynomials, following my recent preprint Arxiv1501.00123.

Series: Geometry Topology Seminar

How is the homological torsion of a hyperbolic 3-manifold related to its geometry? In this talk, I will explain some techniques to address this general question. In particular, I will discuss in detail the case of arithmetic manifolds, where the situation is presumably easier to understand.

Series: Geometry Topology Seminar

In this talk we will begin by discussing the problem of understanding the topology of the space of Riemannian metrics of positive scalar curvature on a smooth manifold. Recently much progress has occurred in this topic. We will then look at an application of the theory of operads to this problem in the case when the underlying manifold is an n-sphere. In the case when n>2, this space is a homotopy commutative, homotopy associative H-space. In particular, we show that it admits an action of the little n-disks operad. Via theorems of Stasheff, Boardman, Vogt and May, this allows us to demonstrate that the path component of this space containing the round metric, is weakly homotopy equivalent to an n-fold loop space.

Series: Geometry Topology Seminar

I will discuss Eliashberg and Thurston's theorem that C^2 taut foliations can be approximated by tight contact structures. I will try to explain the importance of their work and why it is useful to weaken their smoothness assumption. This work is joint with Rachel Roberts.

Series: Geometry Topology Seminar

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

Series: Geometry Topology Seminar

In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss theconstruction of this combinatorial invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3 gives back a stabilized version of knot Floer homology.

Series: Geometry Topology Seminar

Among n-dimensional regions with fixed volume, which one hasthe least boundary? This question is known as an isoperimetricproblem; its nature depends on what is meant by a "region". I willdiscuss variations of an isoperimetric problem known as thegeneralized Cartan-Hadamard conjecture: If Ω is a region in acomplete, simply connected n-manifold with curvature bounded above byκ ≤ 0, then does it have the least boundary when the curvature equalsκ and Ω is round? This conjecture was proven when n = 2 by Weil andBol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke. Injoint work with Benoit Kloeckner, we generalize Croke's result to mostof the case κ < 0, and we establish a theorem for κ > 0. It was originally inspired by the problem of finding the optimal shape of aplanet to maximize gravity at a single point, such as the place wherethe Little Prince stands on his own small planet.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Recently, a "symplectic duality" between D-modules on certainpairs of algebraic symplectic manifolds was discovered, generalizingclassic work of Beilinson-Ginzburg-Soergel in geometric representationtheory. I will discuss how such dual spaces (some known and some new) arisenaturally in supersymmetric gauge theory in three dimensions.

Series: Geometry Topology Seminar

Groups, rings, modules, and compact Hausdorff spaces have underlying sets ("forgetting" structure) and admit "free" constructions. Moreover, each type of object is completely characterized by the shadow of this free-forgetful duality cast on the category of sets, and this syntactic encoding provides formulas for direct and inverse limits. After we describe a typical encounter with adjunctions, monads, and their algebras, we introduce a new "homotopy coherent" version of this adjoint duality together with a graphical calculus that is used to define a homotopy coherent algebra in quite general contexts, such as appear in abstract homotopy theory or derived algebraic geometry.