Spring 2016 Abstracts
- Hsuan-Yi Liao, Pennsylvania State University
- Algebraic exponential maps
- Exponential maps arise naturally in the contexts of Lie theory and connections on smooth manifolds. We will explain how exponential maps can be understood algebraically, how these maps can be extended to graded manifolds and how this problem leads naturally to Dolgushev-Fedosov resolutions.
- Eckhard Meinrenken, University of Toronto
- Convexity for twisted conjugation
- Let $\kappa$ be an automorphism of a Lie group $G$. The $\kappa$-twisted conjugation action of $G$ is given by $g\cdot a=ga\kappa(g^{-1})$; its orbits are the twisted conjugacy classes. I will define a notion of $\kappa$-twisted group-valued moment maps, and describe a convexity theorem for this setting. As an application, we obtain a convexity theorem for products of twisted conjugacy classes.
- Ana Rita Pires, Fordham University
- Symplectic embeddings and infinite staircases
- McDuff and Schlenk determined when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional ball, and found that if the ellipsoid is close to round, the answer is given by an ``infinite staircase'' determined by the odd index Fibonacci numbers, while if the ellipsoid is sufficiently stretched, all obstructions vanish except for the volume obstruction. Infinite staircases have also been found when embedding ellipsoids into polydisks (Frenkel–Muller) and into the ellipsoid $E(2,3)$ (Cristofaro-Gardiner–Kleinman). In this talk, we will see how the sharpness of ECH capacities for embedding of ellipsoids implies the existence of infinite staircases for these and three other domains. This is joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini.
- Nick Sheridan, Princeton University
- Homological mirror symmetry for Greene-Plesser mirrors
- I will start by explaining what mirror symmetry is about, paying special attention to the 'mirror map' which matches up the family of symplectic forms on one manifold with the family of complex structures on another. I will explain how this works for Batyrev's beautiful toric construction of mirror families from dual reflexive polytopes. Then I will give a template for proving cases of Kontsevich's homological mirror symmetry conjecture, based on a 'versality' result for the Fukaya category, which roughly gives a criterion for the existence of a mirror map. The proof can be completed when the reflexive polytope in Batyrev's construction is a simplex: this special case of the construction is due to Greene and Plesser. The latter result is joint work with Ivan Smith.
- Robert Yuncken, Université Blaise Pascal
- Constructing pseudodifferential operators from groupoids
- Pseudodifferential calculi are a technical tool for proving analytical properties of elliptic differential operators and their cousins. Examples of such analytical properties include smoothness of solutions (hypoellipticity) and Fredholmness. In this talk I will explain how classical pseudodifferential operators can be characterized by a relatively simple algebraic/geometric condition on their Schwartz kernels. This condition is described in terms of Connes' tangent groupoid, which I will also explain. Flipping this around, one can then use this condition as a definition of pseudodifferential operators which applies to more general classes of differential operators.