We report on joint work with O. Röndigs. We construct model structures and Quillen equivalences that capture many notions in Goodwillie's calculus of functors between certain pointed simplicial model categories, such as n-excisive, n-homogeneous homotopy functors, cross effects and derivatives. At the end, I will mention two ongoing projects, one joint with Dwyer/Röndigs and the other with C. Berger.

I will talk about embedded contact homology capacities and some recent applications of it to obstructing symplectic embeddings. I will show how to compute ECH capacities of so-called concave toric domains and I will give some examples of when they give sharp obstructions to symplectic ball packings. In particular, I will discuss the case of the union of a ball with an infinite cylinder. This is joint work with K. Choi, D. Cristofaro-Gardiner, D. Frenkel and M. Hutchings.

Building on Seidel-Solomon’s fundamental work, we define the notion of a g-equivariant Lagrangian brane in a symplectic manifold M if g ⊂ SH 1 (M ) is a sub-Lie algebra of symplectic cohomology of M . This allows us to construct a mirror theory to Bott-Borel-Weil theory on the A-side. We will make our construction completely explicit in the case of sl2 and comment on generalizations to arbitrary semisimple Lie algebras. This is a joint work with James Pascaleff.

Everyone knows that a cat dropped upside downcan turn around and fall on his legs. This ability, which at first glance would seem to contradict the conservation of angular momentum, it is instead a consequence of it and is based on the cat's ability to change shape over the course of the fall. In the first part of the seminar we will discuss the kinematics of a deformable body (the cat, but it could also be a satellite or a robotic arm) from the points of view of differential geometryfollowing R. Montgomery. We will show that the configuration space of a deformable body has the structure of a principal bundle with structure group SO(3) -- the group of rotations of the three-dimensional Euclidean space -- and that the angular momentum defines a connection on this bundle. Finding paths with zero angular momentum thus becomes a problem of parallel transport. In the second part of the seminar we will apply this general theory to a model of cat introduced by Kane and Scher, always followingMontgomery.