About me

It's always about me, isn't it?

Hi! I'm Filippo, 4^th year DPhil (or, for the less pretentious, PhD) student in mathematics at the University of Oxford, supervised by Marc Lackenby. I'm interested in studying the topology of 2- and 3-manifolds from an algorithmic and combinatorial point of view (can one be any more vague than this?).

Email: filippo.baroni@maths.ox.ac.uk ().

CV: here (the list of papers is down here, what even is the point?).

Publications and preprints

Tap on a title to reveal the abstract. Ah, the magic of modern technology!

Uniformly polynomial-time classification of surface homeomorphisms 16/08/24, preprint (arXiv:2402.00231), 2024.

We describe an algorithm which, given two essential curves on a surface \(S\), computes their distance in the curve graph of \(S\), up to multiplicative and additive errors. As an application, we present an algorithm to decide the Nielsen-Thurston type (periodic, reducible, or pseudo-Anosov) of a mapping class of \(S\). The novelty of our algorithms lies in the fact that their running time is polynomial in the size of the input and in the complexity of \(S\) – say, its Euler characteristic. This is in contrast with previously known algorithms, which run in polynomial time in the size of the input for any fixed surface \(S\).

Classification of genus-2 surfaces in \(S^3\) 08/08/24, accepted for publication in Algebraic & Geometric Topology, 2024, arXiv:2309.05387.

We describe an algorithm to decide whether two genus-two surfaces embedded in the 3-sphere are isotopic or not. The algorithm employs well-known techniques in 3-manifolds topology, as well as a new algorithmic solution to a problem on free groups.

Solution of the Hurwitz problem with a length-2 partition 02/08/24 (with Carlo Petronio), in Illinois Journal of Mathematics, 2024, arXiv:2305.06634.

In this note we provide a new partial solution to the Hurwitz existence problem for surface branched covers. Namely, we consider candidate branch data with base surface the sphere and one partition of the degree having length two, and we fully determine which of them are realizable and which are exceptional. The case where the covering surface is also the sphere was solved somewhat recently by Pakovich, and we deal here with the case of positive genus. We show that the only other exceptional candidate data, besides those of Pakovich (five infinite families and one sporadic case), are a well-known very specific infinite family in degree 4 (indexed by the genus of the candidate covering surface, which can attain any value), five sporadic cases (four in genus 1 and one in genus 2), and another infinite family in genus 1 also already known. Since the degree is a composite number for all these exceptional data, our findings provide more evidence for the prime-degree conjecture. Our argument proceeds by induction on the genus and on the number of branching points, so our results logically depend on those of Pakovich, and we do not employ the technology of constellations on which his proof is based.

The Proportionality Principle via Bounded Cohomology, in Bounded Cohomology and Simplicial Volume (pp. 118-131), Cambridge University Press, 2022.