Seminars in Differential Topology

Michael Entov: Kähler-type and Tame Embeddings of Balls into Symplectic Manifolds

5th May, 2023

A symplectic embedding of a disjoint union of domains into a symplectic manifold M is said to be of Kähler type (respectively tame) if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M which is compatible with (respectively tamed by) the symplectic form. I'll discuss when Kähler-type embeddings of disjoint unions of balls into a closed symplectic manifold exist and when two such embeddings can be mapped into each other by a symplectomorphism. If time permits, I'll also discuss the existence of tame embeddings of balls, polydisks and parallelepipeds into tori and K3 surfaces.

Thibaut Mazuir: Higher algebra of A_∞-algebras in Morse theory

28th January, 2022

In this short talk, I will introduce the notion of n-morphisms between two A_∞-algebras. These higher morphisms are such that 0-morphisms correspond to standard A_∞-morphisms and 1-morphisms correspond to A_∞-homotopies. Their combinatorics are then encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra. Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A_∞-algebras naturally arises in the context of Morse theory, using moduli spaces of perturbed Morse gradient trees.

Tashi Walde: Higher Segal spaces via Higher Excision

20th May, 2021

Starting from the classical Segal spaces, Dyckerhoff and Kapranov introduced a hierarchy of what they call higher Segal structures. While the first new level (2-Segal spaces) has been well studied in recent years, not much is known about the higher levels and the hierarchy as a whole.

In this talk I explain how this hierarchy can be understood conceptually in close analogy to the manifold calculus of Goodwillie and Weiss. I describe a natural 'discrete manifold calculus' on the simplex category and on the cyclic category, for which the polynomial functors are precisely the higher Segal objects. Furthermore, this perspective yields intrinsic categorical characterizations of higher Segal objects in the spirit of higher excision.