About my research
My research interests are in low-dimensional topology. Most of my work uses Khovanov homology to examine the existence and uniqueness of surfaces in the 4-ball. Recently, my work has been focused on generalizations of Khovanov homology (e.g., to other 4-manifolds) and their applications to similar existence and uniqueness questions.
I would like to learn more about applied fields like topological data analysis and data science, although I am always curious to know more about 4-manifolds, knotted surfaces, and anything you're working on!
In preparation
- Almost concordance in S1xS3 with Arunima Ray and Rob Schneiderman
- Finding nontrivial Khovanov homology classes
Published papers and preprints
Seifert surfaces in the 4-ball
with Kyle Hayden, Seungwon Kim, Maggie Miller, and JungHwan ParkPreprint (2022)
We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in the 3-sphere that do not become isotopic when their interiors are pushed into the 4-ball. We give examples where the surfaces are not topologically isotopic in the 4-ball, as well as examples that are topologically but not smoothly isotopic. These latter surfaces are distinguished by their associated cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.
Article in Quanta Magazine
Article in AMS: Math in the Media
Article in Quanta Magazine
Article in AMS: Math in the Media
Khovanov homology and uniqueness of surfaces in the 4-ball
DissertationBryn Mawr College 2022
In this dissertation, we formally define the cobordism maps on Khovanov homology induced by surfaces in the 4-ball and discuss its recent applications toward distinguishing surfaces in the 4-ball up to boundary-preserving isotopy. We describe recent techniques for distinguishing such maps, including a SageMath program KhNoDe which distinguishes Khovanov homology classes.
This paper shows that the cobordism maps on Khovanov homology can distinguish between exotically knotted smooth surfaces in the 4-ball that are isotopic through ambient homeomorphisms but not ambient diffeomorphisms. Our proof uses a new approach to distinguishing the cobordism maps on Khovanov homology, and we highlight additional applications of this approach.
In this paper, we define an invariant of the smooth boundary-preserving isotopy class of a surfaces in the 4-ball, coming from the Khovanov homology of the boundary link. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for 9_46; is not hindered by detecting connected sums with knotted 2-spheres.
In this paper, we study multiplication operators on the weighted Banach spaces of an infinite tree. We characterize the bounded and the compact operators, as well as determine the operator norm. In addition, we determine the spectrum of the bounded multiplication operators and characterize the isometries. Finally, we study the multiplication operators between the weighted Banach spaces and the Lipschitz space by characterizing the bounded and the compact operators, determining estimates on the operator norm, and showing there are no isometries.
