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Series: Geometry Topology Seminar

We discuss necessary and sufficient conditions of a subset X of the sphere S^n to be the image of the unit normal vector field (or Gauss map) of a closed orientable hypersurface immersed in Euclidean space R^{n+1}.

Series: Geometry Topology Seminar

The study of Legendrian and transversal knots has been an essential part of contact topology for quite some time now, but until recently their study in overtwisted contact structures has been virtually ignored. In the past few years that has changed. I will review what is know about such knots and discuss recent work on the "geography" and "botany" problem.

Series: Geometry Topology Seminar

We introduce two related sets of topological objects in the 3-sphere, namely a set of two-component exchangable links termed "iterated doubling pairs", and a see of associated branched surfaces called "Matsuda branched surfaces". Together these two sets possess a rich internal structure, and allow us to present two theorems that provide a new characterization of topological isotopy of braids, as well as a new characterization of transversal isotopy of braids in the 3-sphere endowed with the standard contact structure. This is joint work with Doug Lafountain, and builds upon previous seminal work of Hiroshi Matsuda.

Series: Geometry Topology Seminar

I will describe some results concerning factorizations ofdiffeomorphisms of compact surfaces with boundary. In particular, Iwill describe a refinement of the well-known \emph{right-veering}property, and discuss some applications to the problem ofcharacterization of geometric properties of contact structures interms of monodromies of supporting open book decompositions.

Series: Geometry Topology Seminar

In the talk, I will gently introduce the Lauda-Khovanov
2-category, categorifying the
idempotent form of the quantum sl(2). Then I will define a complex,
whose Euler characteristic
is the quantum Casimir. Finally, I will show that this complex
naturally belongs to the center
of the 2-category.
The talk is based on the joint work with Aaron Lauda and Mikhail Khovanov.

Series: Geometry Topology Seminar

A noncompact smooth manifold X has a real algebraic structure if and only if X is tame at infinity, i.e. X is the interior of a compact manifold with boundary. Different algebraic structures on X can
be detected by the topology of an algebraic compactification
with normal crossings at infinity. The resulting filtration of the
homology of X is analogous to Deligne's weight filtration for
nonsingular complex algebraic varieties.

Series: Geometry Topology Seminar

For every quantum group one can define two invariants of 3-manifolds:the WRT invariant and the Hennings invariant. We will show that theseinvariants are equivalentfor quantum sl_2 when restricted to the rational homology 3-spheres.This relation can be used to solve the integrality problem of the WRT invariant.We will also show that the Hennings invariant produces integral TQFTsin a more natural way than the WRT invariant.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

I will discuss a conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. I will present examples, as well as computational challenges.

Series: Geometry Topology Seminar

In 1979 Valiant gave algebraic analogs to algorithmic complexity problem such as $P \not = NP$. His central conjecture concerns the determinantal complexity of the permanents. In my lecture I shall propose geometric and algebraic methods to attack this problem and other lower bound problems based on the elusive functions approach by Raz. In particular I shall give new algorithms to get lower bounds for determinantal complexity of polynomials over $Q$, $R$ and $C$.