Math:

Seidel's exact triangle and sections of 4-dimensional Lefschetz fibrations. (Work in progress - joint with T. Perutz)

We are working towards a formula that gives some lowerbound on the number of pseudo-holomorphic sections of a Lefschetz fibration over the disk, while keeping track of their relative homology class. To achieve this, we developed a particular local coefficient system and gave a fully explicit and geometric proof of the exactness of Seidel's triangle in Lagrangian and Fixed Point Floer homology.

On the left, a decomposition of a pseudo-holomorphic section over the disk into a sequence of sections over annuli which are easier to count. On the right, the moduli space that shows a certain Massey product is chain-homotopic to the identity

Fixed point Floer cohomology of disjoint Dehn twists on a w⁺-monotone manifold with rational symplectic form

Journal of Symplectic Geometry 2024 vol 22 issue #3. arXiv.

I gave an explicit description of the Floer cohomology of a family of Dehn twists about disjoint Lagrangian spheres in a weakly⁺-monotone rational symplectic manifold. This is a generalization of a classic result by P. Seidel from 1996 and it is based on a neck-stretching argument and some delicate reasoning with an energy filtration for \(CF(\tau_V)\) which show that certain "bad trajectories" do not count towards the differentials for \(CF(\tau_V)\), proving that the chain complex can be naturally identified with the one for Morse relative cohomology of \((M,V)\).

A "bad trajectory" that goes through the Lagrangian \(V\). These are the obstruction for having \(HF(\tau_V)\cong H(M,V)\)

Stable classification of four-manifolds with fundamental group D_2n

This is my master thesis project, where I (almost) completely classified four-manifold with prescribed fundamental group up to stable diffeomorphism. As a corollary I got some restriction on the divisibility of the signature of such manifolds under some additional assumptions. A. Debray communicated me he was able to figure out the classification in the missing case.

One of the many Atiyah-Hirzebruch spectral sequences I studied in my master thesis.

Math Education:

Developing Ethical Reasoning Skills as a Mathematics Student (joint with Jennifer Austin, Milica Cudina and Tristan Pace

PRIMUS, April, 1-15. T&F link.

We will share three mini-lessons for guiding mathematics majors to develop ethical reasoning skills. These activities have been tested, evaluated, and revised in a 1-hour seminar for entry-level mathematics majors over two recent offerings of this course. In the first activity, students collaboratively develop a set of Ethical Guidelines for Mathematics Majors. Then, in the second activity, students apply these guidelines to analyze vignettes posing an ethical challenge linked to being an undergraduate math major. Finally, in the third activity, students reflect on their experiences through the lens of the collaboratively developed Ethical Guidelines for Mathematics Majors. We hope our paper will be relevant for college and university mathematics instructors who wish to support the development of their students' ethical reasoning skills.