My research lies in the field of low-dimensional topology, particularly knot theory and its applications towards the study of 3- and 4-manifolds. Preprints of much of my work may be found at my page on the math arXiv .

Low-dimensional topology is the branch of topology which studies manifolds of dimension four and lower. Techniques which have yielded much information about manifolds of dimension five and higher often fail for 3- and 4-manifolds, and in fact, many specialized tools have been needed for studying these two particular cases. In some sense, one may consider dimension four as a boundary case between low and high dimensions: there are enough dimensions for the manifold topology to exhibit complex behavior, but not enough space for our usual tools to work. This behavior is exemplified by the following: a closed manifold of dimension three or lower admits exactly one smooth structure; a closed manifold of dimension five or higher admits at most finitely many distinct smooth structures; however, a closed 4-manifold may have infinitely many distinct smooth structures.

A link is the image of a smooth embedding of a disjoint collection of circles into 3-space, considered up to isotopy; a knot is a link with a single component. The study of knots and links is intimately connected with the study of 3-manifolds as seen in the following famous theorem: any closed, connected, orientable 3-manifold can be obtained from the 3-sphere by performing a certain operation ('surgery') on some link. Just as the 3-dimensional relation of isotopy is related to the classification of 3-manifolds, there exist 4-dimensional relations on knots which are relevant to the classification of 4-manifolds; one such relation is (smooth) concordance. The set of concordance classes of knots forms an abelian group called the knot concordance group, denoted by C. Knots representing the trivial class in C are called slice.

Broadly speaking, my research seeks to understand the structure of C, using the following paradigms.

The action of satellite operators. A reasonable approach towards studying any mathematical object is studying functions on it. In the case of knots, there is a natural choice of such functions, namely satellite operations, described below.

Satellite operations are of independent interest beyond knot theory since they can be used to construct interesting examples of 3- and 4-manifolds.

Filtrations of the knot concordance group. It is natural to seek to assess how `close' a knot is to being trivial in C, i.e. concordant to an unknotted circle. This notion was formalized when Cochran-Orr-Teichner introduced the n-solvable filtration of C and showed that the lower levels of the filtration encapsulate the information one can extract from various classical concordance invariants. Studying filtrations gives us a way of understanding the structure of C, a large unwieldy object, in terms of smaller (and hopefully simpler) pieces. There are several other filtrations of knot concordance. In my PhD thesis, I defined several new filtrations of C and established relationships between the various filtrations.

Investigating sliceness of knots. Examining notions closely related to sliceness can yield insights into what it means for a knot to be slice. I have studied a generalization of slice knots, called shake slice knots. Similarly, one can study particular cases of slice knots.

In my work so far I have used tools from geometric and algebraic topology, contact geometry, Heegaard-Floer homology, and other techniques.