- Computing heights via limits of Hodge structures, with S. Bloch and R. de Jong
- Heights on curves and limits of Hodge structures, with S. Bloch and R. de Jong
Here is a video of my talk recorded in Oberwolfach explaining the result.
- Separation of periods of quartic surfaces, with P. Lairez,
To appear in Algebra & Number Theory.
- Deep Learning Gauss–Manin Connections, with K. Heal and A. Kulkarni,
Advances in Applied Clifford Algebras 32(24), 2022 (DOI), in a special issue on Machine-Learning Mathematical Structures.
- Effective obstruction to lifting Tate classes from positive characteristic, with E. Costa,
“Arithmetic geometry, number theory, and computation,” Simons Symposia, Springer (DOI).
- An octanomial model for cubic surfaces, with M. Panizzut, B. Sturmfels,
Le Matematiche, 75(2), 517-536, 2020 (DOI).
- On reconstructing subvarieties from their periods, with H. Movasati
Rendiconti del Circolo Matematico di Palermo, II. Series, 70(3), 1441-1457, 2021 (DOI)
- A compactification of the moduli space of multiple-spin curves
- A numerical transcendental method in algebraic geometry, with P. Lairez,
SIAM Journal on Applied Algebra and Geometry, 3(4), 559–584, 2019 (DOI).
- Prym varieties of genus four curves, with N. Bruin,
Transactions of the American Mathematical Society 373 (2020) 149-183 (DOI).
- Certifying reality of projections (extended abstract), with J. Hauenstein, A. Kulkarni, S. Sherman,
Lecture Notes in Computer Science, 10931, 200–208, 2018 (DOI).
- Computing periods of hypersurfaces,
Mathematics of Computation, 88, 2987–3022, 2019 (DOI), software: PeriodSuite.
- Computing images of polynomial maps, with C. Harris and M. Michalek,
Advances in Computational Mathematics 45 (2019) 2845–2865 (DOI).
Noether–Lefschetz divisors are the coefficients of a modular form
Given a pencil of quartics, how many of the smooth fibers contain a curve of given degree and genus? The solutions to these problems for all degree and genus form the coefficients of an explicit modular form. These are expository notes for the proof of Maulik and Pandharipande that rely on a theorem of Borcherds on modularity of Heegner divisors. We also state Borcherds’ theorem and summarize its proof.
Twistor connectivity of the cohomological moduli spaces
This is a detailed exposition of Section 4 in Buskin’s paper. The goal is to show that moduli spaces parametrizing pairs of K3s and a rational Hodge isometry between them are connected by “twistor” paths. In these notes we bypass some calculations of Buskin on the variation of complex structures on products of K3s and reduce it to a tautology.
Finiteness of isogeny classes
We give a detailed exposition of the proof that there are only finitely many isogeny classes of abelian varieties (of fixed dimension, and with good reduction outside of a fixed finite set). Lecture notes for my talk in IRTG College Seminar, Summer 2015, with the goal of understanding Falting’s Theorem.
Line bundles and cohomology of complex tori
A summary of Chapter 1 in Mumford’s book “Abelian Varieties” covering complex compact abelian varities, i.e., complex tori. This is a lecture note for my talk on IRTG College Seminar, Winter 2014/15, with title “Abelian Varieties and Derived Categories”.
Elliptic curves and automorphic forms
An introduction to automorphic forms from an algebraic perspective. Hence, you get to see them as sections of line bundles on modular curves. These are my lecture notes for IRTG College Seminar, Summer 2014, with the general theme of understanding Fermat’s Last Theorem.