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.
In this talk I will present a complete set of Legendrian Kirby moves relating diagrams of the same contact 3–manifold, yielding a contact analogue of Kirby's theorem. Our approach refines Avdek's ribbon moves by using Gervais' presentation of mapping class groups using only nonseparating curves, together with explicit constructions of Legendrian representatives. This is a joint work with with Marc Kegel and Eric Stenhede.
Over the last 40 years, a considerable amount of progress has been made in studying the diffeomorphism group of the ball in all dimensions. It is now known that the space of diffeomorphisms of the n-ball that restrict to the identity on the boundary is not contractible for all dimensions n greater than or equal to four whereas in dimensions three and below, it is contractible. Over twenty years ago, Eliashberg asked the same question for the space of contactomorphisms/symplectomorphisms of the ball.
A contact 3-manifold is planar if it is compatible with an open book whose page genus is 0. Wendl's theorem that any symplectic filling of a planar contact 3-manifold comes as a Lefschetz fibration filling a given planar open book translates questions about symplectic fillings of planar contact 3-manifolds into group theoretical questions about Dehn twist factorizations in the planar mapping class groups. I will outline work with Matt Day studying the nilpotent levels and truncations of the planar mapping class groups.
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.
In his 1956 paper "On manifolds homeomorphic to the 7-sphere'', John Milnor constructed some examples of manifolds that are homeomorphic, but not diffeomorphic, to the standard unit sphere. They are now called exotic 7-spheres. This example established that the differential structure of a manifold can carry information not given by its topological structure.
