Seminars in Geometric Topology

Bulent Tosun: Contact Surgeries and Symplectic Fillability

9th October, 2023

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, extending an earlier work of the speaker with Conway and Etnyre, we will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (n) surgery along a Legendrian knot to yield a weakly fillable contact manifold, for some integer n > 0. When specialized to knots in the three sphere with its standard tight structure, this result can be effectively used to find many examples of fillable surgeries along with various obstructions and surprising topological applications. For example, we prove that a knot admitting lens space surgery must have slice genus equal to its 4-dimensional clasp number.

Oscar Randal-Williams: Homeomorphisms of Euclidean Space

6th October, 2023

The topological group of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, closely related to understanding the difference between diffeomorphisms and homeomorphisms of all d-dimensional manifolds (except when d=4). Over the last few years a great deal of progress has been made in understanding the algebraic topology of this group. I will report on some of the methods involved, and an emerging conjectural picture.

Arik Wilbert: Real Springer fibers and odd arc algebras

9th November, 2021

Arc algebras were introduced by Khovanov in a successful attempt to lift the quantum 𝔰𝔩₂ Reshetikhin-Turaev invariant for tangles to a homological invariant. When restricted to knots and links, Khovanov’s homology theory categorifies the Jones polynomial. Ozsváth-Rasmussen-Szabó discovered a different categorification of the Jones polynomial called odd Khovanov homology. Recently, Naisse-Putyra were able to extend odd Khovanov homology to tangles using so-called odd arc algebras which were originally constructed by Naisse-Vaz. The goal of this talk is to discuss a geometric approach to understanding odd arc algebras and odd Khovanov homology using Springer fibers over the real numbers.

Stephan Tillmann: On the space of properly convex projective structures

20th May, 2021

This talk will be in two parts. I will outline joint work with Daryl Cooper concerning the space of holonomies of properly convex real projective structures on manifolds whose fundamental group satisfies a few natural properties. This generalizes previous work by Benoist for closed manifolds. A key example, computed with Joan Porti, is used to illustrate the main results.

Marcy Robertson: Expansions, completions and automorphisms of welded tangled foams

22nd April, 2021

Welded tangles are knotted surfaces in ℝ⁴. Bar-Natan and Dancso described a class of welded tangles which have 'foamed vertices' where one allows surfaces to merge and split. The resulting welded tangled foams carry an algebraic structure, similar to the planar algebras of Jones, called a circuit algebra. In joint work with Dancso and Halacheva we provide a one-to-one correspondence between circuit algebras and a form of rigid tensor category called 'wheeled props'. This is a higher-dimensional version of the well-known algebraic classification of planar algebras as certain pivotal categories.

This classification allows us to connect these 'welded tangled foams' to the Kashiwara-Vergne conjecture in Lie theory. In work in progress, we show that the group of homotopy automorphisms of the (rational completion of) the wheeled prop of welded foams is isomorphic to the group of symmetries KV, which acts on the solutions to the Kashiwara-Vergne conjecture. Moreover, we explain how this approach illuminates the close relationship between the group KV and the pro-unipotent Grothendieck–Teichmueller group.