Georgia Institute of Technology College of Sciences

We show that an embedding of a (small) ball into a contact manifold is contact if and only if it preserves the (modified) shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back (to a closed one-form) of a contact form by a coisotropic embedding of a fixed manifold (of maximal dimension) and of a given homotopy type.

In 1988, Penner conjectured that all pseudo-Anosov mapping classes arise up to finite power from a construction named after him. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. I will sketch the proof (joint work with Hyunshik Shin) that the conjecture is false for most surfaces.

In this lecture series, held jointly (via video conference) with the University of Buffalo and the University of Arkansas, we aim to understand the lecture notes by Vincent Guirardel on geometric small cancellation. The lecture notes can be found here: https://perso.univ-rennes1.fr/vincent.guirardel/papiers/lecture_notes_pcmi.pdf This week we will compete the first of two steps in proving the small cancellation theorem (Lecture 3).

I will discuss a process called a cork twist for relating homeomorphic but not diffeomorphic smooth 4-manifolds. This involves finding a contractible submanifold of a given 4-manifold, removing it, and re-gluing by a diffeomorphism of the boundary. This is a surprisingly simple way of relating non-diffeomorphic manifold that was discovered in the 1990s but has recently been getting a lot of attention.

I will describe new techniques for computing the homology of braid groups with coefficients in certain exponential coefficient systems. An unexpected side of this story (at least to me) is a connection with the cohomology of certain braided Hopf algebras — quantum shuffle algebras and Nichols algebras — which are central to the classification of pointed Hopf algebras and quantum groups. We can apply these tools to get a bound on the growth of the cohomology of Hurwitz moduli spaces of branched covers

A knot is a smooth embedding of S^1 into S^3 or R^3. There is a natural way to "add" two knots, called the connected sum. Under this operation, the set of knots forms a monoid. We will quotient by an equivalence relation called concordance to obtain a group, and discuss what is known about the structure of this group.

Many algebraic results about free groups can be proven by considering a topological model suggested by Whitehead: glue two handlebodies trivially along their boundary to obtain a closed 3-manifold with free fundamental group. The complex of embedded spheres in the manifold gives a combinatorial model for the automorphism group of the free group. We will discuss how Hatcher uses this complex to show that the homology of the automorphism group is (eventually) independent of the rank of the free group.

In this lecture series, held jointly (via video conference) with the University of Buffalo and the University of Arkansas, we aim to understand the lecture notes by Vincent Guirardel on geometric small cancellation. The lecture notes can be found here: https://perso.univ-rennes1.fr/vincent.guirardel/papiers/lecture_notes_pcmi.pdf This week we will finish the section on rotating families (Lecture 3).

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)

A celebrated theorem of Nikolai Ivanov states that the automorphism group of the mapping class group is again the mapping class group. The key ingredient is his theorem that the automorphism group of the complex of curves is the mapping class group. After many similar results were proved, Ivanov made a metaconjecture that any “sufficiently rich object” associated to a surface should have automorphism group the mapping class group.