Georgia Institute of Technology College of Sciences

In the early 80's, Freedman discovered that the Whitney trick could be performed in 4-dimensions which quickly led to a complete classification of closed, simply connected topological 4-manifolds. With gauge theory, Donaldson showed that 4-manifolds differ greatly from their higher dimensional counterparts which uncovered the stark differences between topological and smooth results in dimension 4. In this introductory talk, we will give a brief overview this classification and why dimension 4 is so unique.

The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston.

This talk will be an introduction to the theory of surfaces, some tools we use to study surfaces, and some uses of surfaces in "real life". In particular, we will discuss the mapping class group and the curve complex. This talk will be aimed at an audience with a minimal background in low-dimensional topology. 

The fine curve graph of a surface S was introduced by Bowden–Hensel–Webb in 2019 to study the diffeomorphism group of S. We consider a variant of this graph, called the fine 1-curve graph, whose vertices are essential simple closed curves and edges connect curves that intersect in at most one point. Building on the works of Long–Margalit–Pham–Verberne–Yao and Le Roux–Wolff, we show that the automorphism group of the fine 1-curve graph is isomorphic to the homeomorphism group of S. This is joint work with Katherine W. Booth and Daniel Minahan.

We will start by introducing the Teichmüller space of a surface, which parametrizes the possible conformal structures it supports. By defining this space analytically, we can equip it with the Lp metrics, of which the Teichmüller and Weil-Petersson metrics are special cases. We will discuss the incompleteness of the Lp metrics on Teichmüller space and what we know about their completions.

We show that in Cartan-Hadamard manifolds M^n, n≥ 3, closed infinitesimally convex hypersurfaces S bound convex flat regions, if curvature of M^n vanishes on tangent planes of S. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in M^3 with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985.

Fintushel and Stern’s knot surgery constructions has been a central source of exotic 4-manifolds since its introduction in 1997. In the simply connected setting, it is known that there are also embedded corks in knot-surgered manifolds whose twists undo the knot surgery. This has been known abstractly since the construction was first given, but the explicit corks and embeddings have remained elusive.

I will discuss a mixture of results and conjectures related to the Khovanov homology and Knot Floer homology for ribbon knots. We will explore relationships with fusion numbers (a measure of complexity on ribbon disks) and particular families of symmetric unions (ribbon knots given by particular diagrams). This is joint work with Nathan Dunfield, Sherry Gong, Tom Hockenhull, and Marco Marengon.

Locally conformal symplectic (LCS) geometry is a variant of symplectic geometry in which the symplectic form is locally only defined up to positive scale. For example, for the symplectization R x Y of a contact manifold Y, translation in the R direction are symplectomorphisms up to scale, and hence the quotient (R/Z) x Y is naturally an LCS manifold. The importation of symplectic techniques into LCS geometry is somewhat subtle because of this ambiguity of scale.