Abstract: We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces.

Title: Invariants of Random Knots.
Abstract:
Random curves in space and how they are knotted give an insight into the behavior of "typical" knots and links, and are expected to introduce the probabilistic method into the mathematical study of knots. They have been studied by biologists and physicists in the context of the structure of random polymers. There have been many results obtained via computational experiment, but few explicit computations.

Abstract: We use a geometric idea to give an analytic estimate for the word-length in the pure braid group of S^2. This yields that the L^1-norm (and hence each L^p-norm, including L^2) on the group of area-preserving diffeomorphisms of S^2 is unbounded. This solves an open question arising from the work of Shnirelman and Eliashberg-Ratiu. Joint work in progress with Michael Brandenbursky.

Abstract:

Abstract: The unit circle viewed as a Riemannian manifold has diameter (not 2 but rather) π, illustrating the difference between intrinsic and ambient distance. Gromov proceeded to erase the difference by pointing out that when a Riemannian manifold is embedded in L∞, the intrinsic and the ambient distances coincide in a way that is as counterintuitive as it is fruitful. Witness the results of his 1983 Filling paper.

Abstract: The classical theorem of Van Kampen and Flores states that the k-dimensional skeleton of (2k+2)-dimensional simplex cannot be embedded into R2k.
We present a version of this theorem for chain maps and as an application we prove a qualitative topological Helly-type theorem.
If we define the Helly number of a finite family of sets to be one if all sets in the family have a point in common and as the largest size of inclusion-minimal subfamily with empty intersection otherwise, the theorem can be stated as follows:

**Note the special location**
Abstract:

Abstract:
In this pair of talks I will discuss how to obtain fixed-point expressions
for open Gromov-Witten invariants. The talks will be self-contained,
and the second talk will only require a small part of the first talk,
which we will review.
The Atiyah-Bott localization formula has become a valuable tool for
computation of symplectic invariants given in terms of integrals on
the moduli spaces of closed stable maps. In contrast, the moduli spaces
of open stable maps have boundary which must be taken into account

Abstract: One of the first applications of model categories was Quillen homology. Building on the notion of Beck modules, one defines the cotangent complex of an associative or commutative (dg)-algebras as the derived functor of its abelianization. The latter is a module over the original algebra, and its homology groups are called the (Andre'-)Quillen homology. The caveat of this approach is that the cotangent complex is not defined as a functor on the category of all algebras.

Abstract: An elastic energy functional of a Riemannian manifold  is a function that measures the distance of an embedding u:→ℝd from being isometric. In many applications, the manifold in consideration is actually a limit of other manifolds, that is,  is a limit of n in some sense. Assuming that we have an elastic energy functional for each n, can we obtain an energy functional of  which is a limit of the functionals of n?

Abstract: For a toric Fano manifold $X$ denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds with $rk(Pic(X)) \leq 3$ which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}=\left \{ E(z) \vert z \in Crit(X) \right \}$ is a full strongly exceptional collection satisfying the M-aligned property.

Abstract: The contact mapping class group of a contact manifold V is the set of contact isotopy classes of its contactomorphisms. When V is the 2n-dimensional (n at least 2) Am Milnor fiber times the circle, with a natural contact structure, we show that the full braid group Bm+1 on m+1strands embeds into the contact mapping class group of V. We deduce that when n=2, the subgroup Pm+1 of pure braids is mapped to the part of the contact mapping class group consisting of smoothly trivial classes. This solves the contact isotopy problem for V.

Abstract: In the paper "Formal noncommutative symplectic geometry'', Maxim Kontsevich introduced three versions of cochain complexes GCCom, GCLie and GCAs "assembled from'' graphs with some additional structures. The graph complex GCCom (resp. GCLie, GCAs) is related to the operad Com (resp. Lie, As) governing commutative (resp. Lie, associative) algebras. Although the graphs complexes GCCom, GCLie and GCAs (and their generalizations) are easy to define, it is hard to get very much information about their cohomology spaces.

Abstract: I will review a beautiful construction of an infinite collection of monotone Lagrangian tori in the projective plane (and other del Pezzo surfaces) due to Renato Vianna. These tori are obtained from a single one by a procedure called mutation, and I will talk about the wall-crossing formula which relates this geometric procedure to algebraic mutation known from cluster algebra. A proof of the wall-crossing formula is work in progress.