Heegaard Floer theory is a suite of invariants for studying low-dimensional manifolds. In the case of punctured torus, for instance, this theory constructs a particular algebra. And, the invariants associated with three-manifolds having (marked) torus boundary are differential modules over this algebra. This is structurally very satisfying, as it translates topological objects into concrete algebraic…

The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in the three-sphere. In recent years, the realization problem for C, T, O, and I-type spherical manifolds has been solved, leaving the D-type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized…

Grid homology is a version of knot Floer homology in the 3-sphere that is entirely combinatorial and simple to define. Exploiting this, Ozsvath, Szabo, and Thurston defined a combinatorial invariant of transverse links in the 3-sphere using grid homology, which was then used to show that certain knot types are transversely non-simple by Ng, Ozsvath,…

The Gordian distance between two knots is the fewest number of crossing changes necessary to transform one knot into the other. Khovanov homology is a categorification of the Jones polynomial that comes equipped with several spectral sequences. In this talk, we show that the page at which some of these spectral sequences collapse gives a…

The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously…

A strengthening of the Poincare Conjecture asks whether the fundamental group of every closed 3-manifold which is not the 3-sphere admits a nontrivial homomorphism to SU(2). With that as motivation, I'll describe a connection between Stein fillings of a 3-manifold and SU(2) representations of its fundamental group, coming from instanton Floer homology. This connection can…

Given a 3-manifold Y, what are the possible definite intersection forms of smooth 4-manifolds with boundary Y? Donaldson's theorem says that if Y is the 3-sphere, then all such intersection forms are standard integer Euclidean lattices. I will survey some new progress on this problem, for other 3-manifolds, that comes from instanton Floer theory.

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. The proof relies on the involutive Heegaard Floer homology package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

We will talk about Kronheimer and Mrowka’s knot concordance invariant, $s^\sharp$. We compute the invariant for various knots. Our computations reveal some unexpected phenomena, including that $s^\sharp$ differs from Rasmussen's invariant $s$, and that it is not additive under connected sums. We also generalize the definition of $s^\sharp$ to links by giving a new characterization…

We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational…

In the first part of the talk we will discuss famous topological and dynamical questions and conjectures about character varieties and the associated action of the mapping class group. In the second part of the talk we will discuss joint work with F. Palesi and T. Yang about type-preserving representations of the fundamental group of the three-holed projective…

Bordered Floer theory has proven quite useful for studying satellites. In this talk, I'll discuss how to use gradings in bordered Floer theory to study the Floer thickness, 3-genus, and fiberedness of arbitrarily twisted Mazur pattern satellite knots Q_n(K). We'll show that for all but two satellites, Q_n(K) is not Floer homologically thin, we'll give…

A group G is called coherent if every finitely generated subgroup of G is finitely presented.  This is a property enjoyed by the fundamental groups of 3-manifolds, and it is deeply related to the geometry of the group. We show that free-by-free groups satisfying a particular homological criterion are incoherent. This class is large in…

Let M be the 3-manifold obtained by r-surgery on the right handed trefoil knot. Classification of contact structures on such manifolds have been mostly understood for r\geq 1 and r=0. Etnyre-Min-Tosun has an upcoming work on the classification of the tight contact structures for all r. The fillability of contact structures on M is mostly understood if r is not…

By slicing a Heegaard diagram for a knot K in $S^3$, it is possible to retrieve the knot Floer homology of K as a tensor product of bimodules over an $\A_{\infty}$ algebra corresponding to the slice. The first step in this process is to assign an $\mathcal{A}_{\infty}$ algebra to this slice, which can also be…

Dimension four is the lowest dimension where smooth and topological manifolds can differ; any difference between these categories is known as exotica. In particular, a smooth 4-manifold is \emph{exotic} if there is another smooth 4-manifold which is homeomorphic but not diffeomorphic to it. There is a wealth of literature, mostly written between 1983 and 2008,…