I am a PhD student (expected Spring 2018) and graduate instructor in the Department of Mathematics at Boston College. My advisor is John Baldwin. To contact me, send an email to an address that looks like firstname.lastname@example.org.
I am interested in low-dimensional topology, as well as contact and symplectic geometry. My research has centered around exploring three- and four-dimensional manifolds and the geometric structures they support using knots and surfaces.
Publications and preprints:
Unknotted ribbon disks, band surgeries, and algebraic curves
Available as arXiv:1801.05415.
We study ribbon surfaces in the four-ball up to a natural notion of ribbon-isotopy. We prove that a ribbon disk with unknotted boundary is ribbon-isotopic to the standard ribbon disk with a single local minimum if and only if every regular level set is an unlink and every regular sublevel set is a collection of disks. Our main applications resolve Problems 1.100A and 1.100B from Kirby's list, which concern the Seifert genera of level sets of algebraic curves.
Legendrian ribbons and strongly quasipositive links in an open book.
Submitted, available as arXiv:1710.06036.
We show that a link in an open book can be realized as a strongly quasipositive braid if and only if it bounds a Legendrian ribbon with respect to the associated contact structure. This generalizes a result due to Baader and Ishikawa for links in the three-sphere. We highlight some related techniques for determining whether or not a link is strongly quasipositive, emphasizing applications to fibered links and satellites.
Quasipositive links and Stein surfaces.
Submitted, available as arXiv:1703.10150.
We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. As in the classical case, we see that this generalization of quasipositivity is intimately connected to contact and complex geometry. The main result is an essentially three-dimensional proof that every link in the boundary of a Stein domain which bounds a complex curve in the interior is quasipositive, generalizing a theorem of Boileau-Orevkov.
Minimal braid representatives of quasipositive links.
To appear in Pacific J. Math.
We show that every quasipositive link has a quasipositive minimal braid representative, partially resolving a question posed by Orevkov. These quasipositive minimal braids are used to show that the maximal self-linking number of a quasipositive link is bounded below by the negative of the minimal braid index, with equality if and only if the link is an unlink. This implies that the only amphicheiral quasipositive links are the unlinks, answering a question of Rudolph's.
Positive Knots and Lagrangian Fillability.
with Joshua M. Sabloff. Proc. Amer. Math. Soc. 143 (2015), no. 4, 1813–1821.
This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact three-space and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions.
Topologically Distinct Lagrangian and Symplectic Fillings.
with Chang Cao, Nathaniel Gallup, and Joshua M. Sabloff. Math. Res. Lett. 21 (2014), no. 1, 85–99.
We construct infinitely many Legendrian links in the standard contact three-space with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in the three-sphere that bound topologically distinct pieces of algebraic curves in the unit four-ball, is applied to find contact 3-manifolds with topologically distinct symplectic fillings, and is generalized to higher dimensions.
I am not teaching any courses this semester. In previous semesters, I have been the instructor of record for
Before the above courses, I ran discussion sections, graded exams, and held office hours for the following math courses at Boston College: