Embedded Files
For mathematicians
For mathematicians
My research interests lie in low-dimensional geometric topology. I am especially interested in 3- and 4-manifolds and, in particular, concordance and homology cobordism. While the most familiar setting for studying concordance is the (topological or smooth) concordance group of knots in the 3-sphere, I enjoy thinking about concordance in extended settings, including concordance of links, concordance of surfaces and surface links, and concordance in non-simply-connected manifolds. I also like applying tools from concordance to other questions in geometric topology.
For detailed information on my research program, see my research statement.
For non-mathematicians
For non-mathematicians
I primarily study concordance, a notion of equivalence between knots, objects like the one pictured to the left. The knots I study live in spaces called manifolds. The surface of the earth is a 2-dimensional manifold; even though it may look flat from where we're standing, we know it has a more interesting global structure. The knots I study live in 3-dimensional manifolds, which again look "flat" locally but may not be globally. Knot concordance is a 4-dimensional relation, studying surfaces that knots bound in certain 4-dimensional manifolds.
Papers
Papers
We prove that every closed orientable 3-manifold may be obtained via 0-surgery on some link in the 3-sphere and examine the consequences of this fact.
We define concordance invariants of surfaces and surface links in closed orientable 4-manifolds and produce examples of pairwise homotopic but non-concordant positive genus surfaces.
Triple linking and rational homology cobordism, in preparation.
Triple linking and rational homology cobordism, in preparation.
We prove that if a rational homology sphere bounds a rational homology ball with no second homology, then the Freedman-Krushkal torsion triple linking form vanishes on the Lagrangian for the torsion linking pairing.
We study topological concordance modulo local knotting, or almost-concordance, of knots in non-simply-connected 3-manifolds M, partially resolving a conjecture of A. Levine, Celoria, and Friedl-Nagel-Orson-Powell.
Milnor's invariants for knots and links in closed orientable 3-manifolds, to appear in Geometry & Topology.
Milnor's invariants for knots and links in closed orientable 3-manifolds, to appear in Geometry & Topology.
We extend Milnor's link invariants to topological concordance invariants of knots and links in general closed orientable 3-manifolds. These invariants unify and generalize all previous versions of Milnor's invariants in dimension 3.
Undergraduate Research:
Undergraduate Research:
Classical invariants of spiral knots (with Sarah Blackwell, Ashish Das, Sydney Mayer, Luke Moyar, and Faisal Quraishi), submitted.
Classical invariants of spiral knots (with Sarah Blackwell, Ashish Das, Sydney Mayer, Luke Moyar, and Faisal Quraishi), submitted.
This project is a result of the 2024 University of Virginia Geometry & Topology REU. We give a general recursive formula for the Alexander polynomials of spiral knots, which are braid-theoretic generalizations of torus knots.
Sequences of Spiral Knot Determinants (with Seong Ju Kim and Laura Taalman), Journal of Integer Sequences 19 no.1 (2016).
Sequences of Spiral Knot Determinants (with Seong Ju Kim and Laura Taalman), Journal of Integer Sequences 19 no.1 (2016).
My undergraduate thesis. We give related recursive formulas for the determinants of several families of spiral knots.
Page updated
Google Sites
Report abuse