In this talk I will use a long exact sequence (after recalling its construction) relating the Legendrian contact homology of the two ends of a Lagrangian cobordism and the singular homology of the cobordism to deduce some restrictions on the topology of this Lagrangian (when the negative end admits an augmentation). Considering a twisted version of this long exact sequence we will see that Lagrangian cobordisms from a Legendrian sphere admitting an augmentation to itself are smoothly trivial. I will recall all the basic definitions.
Dan Avritzer (Universidade Federal de Minas Gerais)
Friday, 12 December, 2014 - 15:30
Room 0.04 (FC1)
Consider a complete intersection $X=Q_0 \cap Q_1$ of two quadrics in $\mathbb P^5.$ Choose a line $L\subset X$ and a $V_3 \equiv \mathbb P^3$ disjoint from $L.$ Consider the projection $\pi : \mathbb P^5 \longrightarrow M$ with center $L$ and restric $\pi$ to $X,$ this gives a birational map from $X$ to $M$ whose inverse is not defined at the genus two quintic in $M$ given by a quadric and two quartics. In a joint paper with I. Pan and G.
Christian Peskine (Institut de Mathématiques de Jussieu)
Friday, 5 December, 2014 - 15:30
Room 0.04 (FC1)
According to the celebrated Trisecant Lemma, a general projection of a smooth algebraic curve on a plane has only ordinary double points. In other words, the trisecant, the tangents and the stationary bisecants (they shall be explained) do not fill up the space. I intend to explain at length this classical result, hoping in particular to attract the interest of those who have never heard of it. I will then discuss secants and tangents to a smooth algebraic variety and explain how the Trisecant Lemma is a special case of a very natural and general satement.
We present a new method for classifying naturally reductive homogeneous spaces -- i.e. homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion and curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such connections with the geometric structure on the given Riemannian manifold.
In my talk I intend to present a survey of some classical and recent results on algebraic surfaces which analytically arise as quotients of bounded domains by arithmetic groups. These particularly include recent results on fake projective planes and fake quadrics as well as surfaces related to them.
Hyperbolic groups were introduced by Mikhail Gromov in the 80s by considering the geometry of Cayley graphs, viewed as geodesic metric spaces. Several equivalent conditions, due to Gromov and/or Elyahu Rips, are commonly used to characterize hyperbolic geodesic metric spaces. We consider a strengthened version which we call polygon hyperbolicity, and establish equivalent conditions which are variants of the classical alternatives of Gromov. We also characterize those groups whose Cayley graph is a polygon hyperbolic space: they are the finitely generated virtually free groups.
This talk is based on joint work with Weiwei Pan, Daniel Tubbenhauer and with Yasuyoshi Yonezawa. In his seminal work on the categorification of the Jones polynomial, Khovanov introduced a new family of algebras, which he called the arc algebras. He showed that the Grothendieck group of the category of f.d. representations of an arc algebras is isomorphic to a certain space of invariant quantum sl(2) intertwiners.
The topological complexity of a space has been introduced by M. Farber in order to give a measure of the complexity of the motion planning problem in robotics. This invariant is, by its definition, closely related to the Lusternik-Schnirelmann category. By analogy to the notion of weak (LS) category, we define the "weak topological complexity". This invariant is a new lower bound for the topological complexity and turns out to be equal to the weak category of the homotopy cofibre of the diagonal map. This is joint work with J.M. García Calcines.
We use Higgs bundles to study the character variety for representations of a surface group in the non-compact dual of the special orthogonal group. This talk is based on joint work with Steve Bradlow and Oscar Garcia-Prada.