Research | Expository | Talks | Teaching | Organization
Welcome to my website.
I’m a fourth-year PhD student of Tom Mrowka at MIT with interests in low-dimensional topology (especially diffeomorphisms and gauge theory in dimension 4) and discrete geometry. My research has been supported by the MIT Dean of Science Fellowship. Previously, I graduated from MIT with a BS in mathematics. You can reach me at my_first_name@mit.edu and can find my CV at this link.
Research:
Designs related through projective and Hopf maps. Discrete & Computational Geometry, 2025.
Formalizes a construction that builds a spherical $t$-design by placing a spherical $t$-design on each projective or Hopf fiber associated to the points of a $\lfloor t/2\rfloor$-design on a quotient projective space or sphere, generalizing work of König, Kuperberg, and Okuda (who was inspired by work of Cohn, Conway, Elkies, and Kumar).
Boundary Dehn twists are often commutators. arXiv:2604.13194, submitted for publication, 2026.
Concretely realizes the boundary Dehn twist as a commutator in the smooth mapping class group rel boundary of all complete intersections, all connected sums thereof, and further broad classes of manifolds, notably showing that boundary Dehn twists known to be nontrivial in the smooth mapping class group rel boundary by work of Baraglia-Konno (see also), Kronheimer-Mrowka, Jianfeng Lin, and Tilton become trivial after abelianization.
Asymptotically optimal $t$-design curves on $S^3$. arXiv:2408.04044, submitted for publication, 2024.
Solves the problem posed by Ehler and Gröchenig of proving that there exist asymptotically optimal sequences of $t$-design curves on the 3-sphere.
Asymptotically short generalizations of $t$-design curves. arXiv:2505.03056, submitted for publication, 2025.
Proves existence of approximate and weighted $t$-design curves which have certain desirable properties and which achieve the asymptotically optimal length of spherical $t$-design curves on the $d$-sphere for all odd $d$ in the approximate setting and all $d$ in the weighted setting.
Dynamical stability of translators under mean curvature flow. Posted to MIT SPUR website, 2022.
Joint with Carlos Alvarado, supported by Tang-Kai Lee. Investigates whether certain classes of perturbations of mean curvature flow translators converge to translators under the flow.
Lifting design curves. In preparation, to be submitted for publication, 2026.
Expository:
Instanton Floer homology and applications. In New Structures in Low-Dimensional Topology, Bolyai Society Mathematical Studies 1, Springer, Cham, 2026.
Joint with John Baldwin, Joye Chen, Nathan Geist, Tomasz Mrowka, Ollie Thakar. These notes, which follow a mini-course presented by John Baldwin and Tomasz Mrowka at the meeting New Structures in Low-Dimensional Topology in Budapest, define instanton Floer homology–a powerful 3-manifold invariant–and discuss applications to sutured manifolds and knot theory.
Khovanov skein lasagna modules for the working topologist. In preparation, 2026.
Joint with Enrico Colón, Gage Martin, Mira Wattal. These notes present Khovanov skein lasagna modules and their impacts from the perspective of low-dimensional topology.
Invited talks:
Boundary Dehn twists are often commutators. AMS Special Session on Geometric and Analytic Aspects of Gauge Theory, 2026.
I will give an overview of my work of the same name which proves that boundary Dehn twists can often be concretely represented as commutators in the smooth mapping class group rel boundary, notably showing that they often become trivial after abelianization.
Geometric design of geometric designs. Dean of Science Graduate Fellows Symposium, 2026.
I spoke about my research formalizing geometric constructions of geometric $t$-designs as a part of a symposium in which recipients of the MIT Dean of Science Graduate Fellowship were asked to present their work to members of the MIT community.
Boundary Dehn twists after abelianization. London Low-Dimensional Topology Seminar, 2026.
I gave an overview of my argument (since disseminated in the paper Boundary Dehn twists are often commutators) showing that boundary Dehn twists often become trivial after abelianization.
Contributed talks:
Boundary Dehn twists are often commutators (lightning). OIST TSVP Autumn School on Gauge Theory, Topology, and Physics, 2026.
I gave a lightning-style overview of my work of the same name which proves that boundary Dehn twists can often be concretely represented as commutators in the smooth mapping class group rel boundary, notably showing that they often become trivial after abelianization.
Boundary Dehn twists are often commutators. Graduate Student Topology and Geometry Conference, 2026.
I gave an overview of my work of the same name which proves that boundary Dehn twists can often be concretely represented as commutators in the smooth mapping class group rel boundary, notably showing that they often become trivial after abelianization.
Boundary Dehn twists are often commutators. AMS New England Graduate Student Conference, 2026.
I gave an overview of my work of the same name which proves that boundary Dehn twists can often be concretely represented as commutators in the smooth mapping class group rel boundary, notably showing that they often become trivial after abelianization.
Skein lasagna modules and handle attachments. Scottish Talbot On Algebra and Topology, 2026.
I discussed work of Manolescu and Neithalath describing how 4- and 3-handle attachments respectively don’t change and don’t enlargen the skein lasagna module of a 4-manifold and establishing an isomorphism between the skein lasagna module of a 2-handlebody and cabled Khovanov-Rozansky homology.
Designs related through projective and Hopf maps. AMS Special Session on Fresh Researchers in Algebra, Combinatorics, and Topology, 2024.
I gave an overview of constructions of $t$-designs on higher-dimensional spheres from those on lower-dimensional spheres and projective spacaes presented in my paper of the same name.
Selected local talks:
Geometrically designing geometric designs. MIT PuMaGraSS, 2025.
I presented work on geometric constructions of spherical $t$-designs and $t$-design curves.
Monopole Floer homology and a refinement of Manolescu. MIT Juvitop, 2024.
I gave an overview of the construction of monopole Floer homology (as in Ch. 22 of Tom and Peter’s book) and of its homotopy refinement (as in work of Manolescu).
In case you’re curious what your local low-dimensional topologists do all day. MIT PuMaGraSS, 2024.
I provided a general introduction to Morse theory and Floer theories.
Teaching:
Organization:
The MIT Geometry & Topology Seminar. MIT, 2026-2027.
Joint with Joye Chen.
We will co-organize the MIT Geometry & Topology Seminar for the 2026-2027 academic year.
The Low-dimensional Cambridge-Organized Student Topology Gathering (the Low COST Gathering). MIT and Harvard, 2027.
Joint with Ollie Thakar.
A meeting on low-dimensional topology which we plan to organize in Fall 2027. Please feel free to e-mail me if you’d be interested in speaking at or attending the event and I’ll be in contact when it is becoming a reality (though note, as is to be expected with the name, we likely won’t have too much funding available for travel or lodging).
The K3@MIT workshop in low-dimensional topology. MIT, 2026.
Joint with Evan Scott and Ollie Thakar.
A workshop which will provide a structured forum for PhD students and early-career mathematicians to work together on problems on the recently released K3 problem list in low-dimensional topology.
The Low-dimensional Princeton-Cambridge Exchange Gathering (the Low PriCE Gathering). MIT and Harvard, 2025.
Joint with Ollie Thakar.
A meeting on low-dimensional topology which we organized. Abstracts from the event can be found here.
Fun:
Coming soon!
This website (including its sub-pages) has been loaded — total times since its creation at the end of 2025.
Many thanks to my good friend Torque, who helped greatly to set up the site!
