Mathematics

Pre-Calabi-Yau structures and moduli of representations

Wai-kit Yeung (Indiana University)

Abstract: Pre-Calabi-Yau structures are certain structures on associative algebras introduced by Kontsevich and Vlassopoulos. This incorporates as special cases many other algebraic structures of diverse origins. Elementary examples include double Poisson algebras introduced by Van den Bergh, as well as infinitesimal bialgebras studied by Aguiar. Other examples also arise from symplectic topology as well as from string topology, whose relation with topological conformal field theory can be formulated in terms of pre-Calabi-Yau structures. In this talk, we will define pre-Calabi-Yau structures, and study it in the context of noncommutative algebraic geometry. In particular, we show that Calabi-Yau structures, introduced by Ginzburg and Kontsevich-Vlassopoulos, can be viewed as noncommutative analogue of symplectic structures. Pushing this analogy, one can show that pre-Calabi-Yau structures are noncommutative analogue of Poisson structures. As a result, we indicate how a pre-Calabi-Yau structure on an algebra induces a (shifted) Poisson structure on the moduli space of representations of that algebra.

Beyond semitoric

Susan Tolman (Illinois)

Abstract: A compact four dimensional completely integrable system $f \colon M \to \mathbb{R}^2$ is semitoric if it has only non-degenerate singularities, without hyperbolic blocks, and one of the components of $f$ generates a circle action. Semitoric systems have been well studied and have many nice properties; for example, the fibers $f^{-1}(x)$ are connected. Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitations. For example, there are blowups of $S^2 \times S^2$ with Hamiltonian circle actions that cannot be extended to semitoric system. We show that, by allowing certain degenerate singularities, we can expand the class of semitonic systems but still prove that $f^{-1}(x)$ is connected. We hope that this class will be large enough to include not only all compact four manifolds with Hamiltonian circle actions, but more generally all complexity one spaces. Based on joint work with D. Sepe.

Integrable billiards and symplectic embedding problems

Daniele Sepe (Universidade Federal Fluminense, Brazil)

Abstract: A driving question in symplectic topology is to determine whether one symplectic manifold embeds symplectically into another of equal dimension. In this talk, we shall consider a family of symplectic manifolds, called Lagrangian products, that have come to the fore in recent years because of their connection with billiards through the work of Artstein-Avidan, Karasev and Ostrover. We shall illustrate some recent results concerning embeddings of sufficiently symmetric Lagrangian products in any dimension. The proof relies on integrable billiards and is inspired by an idea of Ramos. Time permitting, we shall discuss some ongoing developments and open questions. This is joint work with V. G. B. Ramos.

The cohomology of Courant algebroids and their Characteristic Classes

Miquel Cuenca (IMPA)

Abstract: Courant algebroids originated over 20 years ago motivated by constrained mechanics but now play an important role in Poisson geometry and related areas. Courant algebroids have an associated cohomology, which is hard to describe concretely. Building on work of Keller and Waldmann, I will show an explicit description of the complex of a Courant algebroid where the differential satisfies a Cartan-type formula. This leads to a new viewpoint on connections and representations of Courant algebroids and allows us to define new invariants as secondary charcateristic classes, analogous to what Crainic and Fernandes did for Lie algebroids. This is joint work with R. Mehta.

Non-linear Maslov index on lens spaces

Yael Karshon (Toronto)

Abstract: Let L be a lens space with its standard contact structure. We construct a "non-linear Maslov index", which associates an integer to any contact isotopy of L starting at the identity, and that has certain properties that allows us to use it to prove rigidity properties of L as a contact manifold. This is joint work with Gustavo Granja, Milena Pabiniak, and Sheila (Margherita) Sandon, and it follows earlier work of Givental and Theret that applied to real and complex projective spaces.