Publications

• The Hamilton-Jacobi theory for contact Hamiltonian systems, joint with M. de Leon and M. Lainz

Mathematics 2021, 9(16), 1993

The Hamilton-Jacobi formulation of classical (Hamiltonian) mechanics turns the problem of solving n PDEs of second order to 2n first order PDEs. In this paper, we extend this formalism to contact Hamiltonian systems (roughly speaking, contact dynamics deals with mechanical systems in which the energy is dissipated).

Preprints

• Fourier transforms and a filtration on the Lagrangian cobordism group of tori

submitted

This paper has two parts. In the first part, I elucidate some structure on the infinite-dimensional Lagrangian cobordism group of tori. Namely, assuming polarizability of a tropical affine base, I show that the fibered Lagrangian cobordism group admits a natural geometric filtration and bound its length (the bound depends only on the dimension of the torus). In the second half, I construct a Fourier transform between Fukaya categories of dual symplectic tori. I show that, under Abouzaid's homological mirror symmetry result, the symplectic Fourier transform coincides with the well-known Fourier-Mukai transform between the derived categories of dual abelian varieties––see the work of Mukai. I also show that, under homological mirror symmetry, the filtration I construct on the cobordism group matches the Bloch filtration on the Chow groups of the mirror.

• Dimensionality of tropical Chow groups

submitted

In this paper I prove that the existence of non-zero tropical forms (of degree at least two) implies infinite-dimensionality of tropical Chow groups. This is a tropical analog of classical results of Mumford and Roitman in the 60s. I also compute the Chow groups of a Klein bottle and mention some connections to Lagrangian cobordism groups

• Lagrangian cobordisms and K-theory of symplectic bielliptic surfaces

submitted

This paper studies the Lagrangian cobordism group of a closed symplectic 4-manifold. Whereas Sheridan-Smith show that when the canonical bundle is trivial Lagrangian cobordism groups are too large to be computed, I work with an example where the canonical bundle is 2-torsion but not trivial. This allows me to compute the subgroup generated by tropical Lagrangians. I also show this is an interesting subgroup: using Abouzaid's homological mirror symmetry result, I prove it maps isomorphically to the Grothendieck group of the Fukaya category.

Expository

• Cobordism group of tori (available upon request)

This is a summary of the things I was thinking about for most of my first year. It includes a computation of the cobordism group of a two-torus without using homological mirror symmetry and discusses some essential differences for higher-dimensional tori. I also sketch how to define a Bloch-type filtration on the fibered Lagrangian cobordim group of tori. Using a Fourier transform as Beauville one can show that the filtration is finite. (These ideas are further developed in my last paper.)

• A survey on Morse theory and its applications, joint with G. Carfora and R. Gianini.

  Expository project on Morse theory for GLAMs. I contributed the second section on Lagrangian Floer homology, following Auroux's paper.

• A case study in intersection theory

  This is my Part III essay. It is a survey about intersection theory and the Chow rings. I focus on the Chow ring of the Grassmannian and develop its Schubert calculus. I use this to present the solution to two classical problems in intersection theory. A solution using Chern classes is also included.

• Entanglement entropy and holography.

  This is my undergraduate physics dissertation, supervised by Prof. José Edelstein. I looked at (von Neumann) entropy as a measure of entanglement in a physical system. In the last sections, I discussed the Ryu-Takayanagi conjecture, which turns the computation of entanglement entropy from an algebraic problem to a geometric one.