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My research is mainly on low-dimensional topology, in particular, open book decompositions in the study of 3-manifolds, knot theory, and contact geometry. This implies studying mapping class groups of compact surfaces with boundary. My thesis is now available here. My papers can be seen below, with the link on the title leading to the arXiv version.
I also collaborate with Peter Feller, Lukas Lewark, Isacco Nonino, and Tanushree Shah.
Monodromies of surfaces in 3-manifolds, right-veeringness, and primeness of links (joint with Peter Feller and Lukas Lewark).
We introduce the notion of partial monodromy of an incompressible surface in a 3-manifold as a map defined on (a subset of) the arc set of the surface. We prove a composition formula of these partial monodromies under Murasugi sums, analogous to Gabai's result for fiber surfaces. The idea is as follows. For a Murasugi sum of (M_1, S_1) and (M_2,S_2), given a product disk we can intersect it with the individual manifolds to obtain collections of product disks in the summands, see the figure on the right for a schematic picture.
Using this, we prove an analog of the primeness criterion from Homogeneous braids are visually prime (see below), which we use to establish primeness of a large class of links (similarly to the previous paper, this class contains arborescent links). This allows us to prove Cromwell's conjecture for so-called alternative links, a class which strictly contains all previous results.
Along the way, we prove that strongly quasipositive surfaces are right-veering, and our proof allows us to recover Honda, Kazez, and Matic's characterization of tight contact structures in terms of right-veering open books, both in the classical and the open book case.
Homogeneous braids are visually prime (joint with Peter Feller and Lukas Lewark). Published by Journal of Topology, you can see the published version here.
When is it possible to tell from a link diagram whether the link represented is prime? More than 30 years ago, Cromwell proved that this happens for positive braids, and conjectured that it is also true for diagrams that produce a minimal genus Seifert surface when Seifert's algorithm is applied.
We resolve Cromwell's conjecture for all braid closures. We also show primeness of a large class of fibered links, which we call trees of open books. This class contains fibered arborescent links.
We are currently trying to generalise the result to other families. The main tool is a criterion for primeness under Murasugi sums. (Update: the paper is now out! See above).
I gave a talk at K-OS about this paper, here are the slides. A recording of the talk is also available here.
Detecting right-veering diffeomorphisms. To appear in Algebraic&Geometric Topology.
The right-veering property is important for distinguishing tight contact structures form overtwisted ones. However it is often difficult to detect. My project concerns a combinatorial way of showing whether an open book is right-veering.
I am currently trying to tie this result to work of Baldwin, Ni, and Sivek that relates the right-veering property to an invariant in Knot Floer homology.
On binding sums of contact manifolds.
The binding sum is an operation on contact manifolds similar to the connect sum. However, its behaviour is not as well understood as the connect sum case. I would like to see when certain properties (such as tightness and fillability) are preserved under this operation.
I can provide an explicit computation of the vanishing of the contact class for an infinite family of binding sums whose summands are Stein fillable, showing that many properties of contact manifolds are not necessarily preserved under the binding sum. This can be seen in my thesis here (update: a preprint -with a slightly more general result- is now also available here).
I took part in the project Humanity's Last Exam, which created a dataset of questions to serve as an academic benchmark for Large Language Models (LLMs), since lately the models tend to achieve over 90% on previous benchmarks. The dataset consists of 2,700 questions across over a hundred subjects. This dataset proved significantly more challenging for the models than existing benchmarks.
Consider a once-punctured torus. Take the product of the (positive) Dehn twists along the standard homology generators, and finally take its 9th power (see the image on the right). What is the fractional Dehn twist coefficient of the resulting mapping class? If you can answer this question, congratulations! You can do better than existing AI (although this might be obsolete by the time you read it).
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