Georgia Institute of Technology College of Sciences

Knot concordance, in both the smooth and topological categories, is a well-studied equivalence relation on knots. We will introduce the smooth, topological, and algebraic concordance groups, and discuss several invariants that help answer questions about the structures of these groups and the relationships between them. We will also discuss the ways facts about knot concordance can be used to construct exotic pairs of 4-manifolds.

We discuss a new problem by Dmitri Burago and Anton Petrunin. Consider a $C^2$-smooth surface embedded in $\RR^3$ with principal curvatures $\leq 1$. Does such a surface necessarily enclose a volume at least that of the unit ball? We address results from two opposite perspectives: On the one hand, we can construct a counterexample by deforming Lagunov's fishbowl (a fattened Bing's house); on the other hand, there are positive results with additional conditions. Many problems remain open.

Given a link in a closed 3-manifold, which Dehn surgery multislopes give rise to 3-manifolds with taut foliations?

Heegaard Floer homology is a modern algebraic package that has seen utility in many areas of low dimensional topology. In this introductory talk, we will give a birds eye view of its construction and properties, and present a highlight reel on the problems it helps solve. As many of our beloved faculty find use for Floer homology in their research, this talk intends to ease the newcomer into the vernacular they may hear in a Seminar talk, and at the same time, convince them that it is a beautiful and rich field of study.

Engel structures are maximally non-integrable rank-two plane fields on four-dimensional manifolds. They are closely related to contact geometry, but their global behavior is still much less understood.

In contact topology, complex tangencies of real hypersurfaces in complex manifolds give a fundamental source of contact structures, often with strong rigidity properties. This motivates the Engel analogue: can a compact four-dimensional submanifold of $\mathbb C^3$ have complex tangencies forming an Engel structure?

The colored Jones polynomial is a quantum knot invariant which can be constructed as a Reshetikhin–Turaev invariant using representations of $U_q(sl_2)$. Khovanov homology categorifies the Jones polynomial and by extension categorifies the representation theory of $sl_2$. Of particular interest is sutured annular Khovanov homology, which admits a structure as an $sl_2$-module. We will discuss a result of Grigsby–Licata–Wehrli that this structure is a representation-theoretic invariant of an annular link.