Research

My work aims to understand the geometry of moduli problems arising in algebraic geometry. I am especially fond of the moduli of G-bundles, as well as the moduli of pure sheaves. More recently I have considered problems related to the moduli of curves and stable maps.

I enjoy thinking about constructions of stability conditions, Harder-Narasimhan stratifications, and moduli spaces.

Part of my research is also involved with the study of geometric invariants of moduli spaces, such as their cohomology and their derived category. It is often interesting to try to match up invariants coming from different moduli problems.

I am also interested in geometric incarnations of the Langlands program. Some of the relevant mathematical objects are: G-bundles, Higgs bundles, connections and automorphic forms over function fields.

Publications

1. The moduli stack of principal rho-sheaves and Gieseker-Harder-Narasimhan filtrations. (joint with T. L. Gomez and A. Zamora). Accepted in Math. Z. (arXiv: 2107.03918)
2. On the quasicompactness of the moduli stack of logarithmic G-connections over a curve. To appear in Israel J. Math. (arXiv: 2002.11832)
3. Moduli spaces of sheaves via affine Grassmannians. (joint with D. Halpern-Leistner and T. Jones). J. reine angew Math, 2024.(arXiv: 2107.02172)
4. Geometry of the logarithmic Hodge moduli space. (joint with M. A. de Cataldo, with an Appendix joint with S. Zhang). J. London Math. Soc. Issue 109, 1: e12857, 2024. (arXiv: 2211.06754)
5. On automorphisms of semistable G-bundles with decorations. Advances in Geometry 23 (3), 389-400, 2023. (arXiv: 2202.13360)
6. An inequality of pairs of Hölder conjugate numbers. (joint with A. Stan). J. Math. Ineq. 2016.

Preprints

1. Semistable Non Abelian Hodge theorem in positive characteristic. (joint with S. Zhang) (arXiv: 2310.09923)
2. Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic. (joint with S. Zhang) (arXiv: 2307.16755)
3. Relative étale slices and cohomology of moduli spaces. (joint with M. A. de Cataldo and A. Ibáñez Núñez) (arXiv: 2307.00350)
4. Moduli of objects in finite length abelian categories. (joint with E. Lennen and S. Makarova) (arXiv: 2305.10543)
5. The structure of the moduli of gauged maps from a smooth curve. (joint with D. Halpern-Leistner) (arXiv: 2305.09632)
6. Harder-Narasimhan stratification for the moduli stack of parabolic vector bundles. (arXiv: 2101.08871)
7. Classification of affine normal SL(2) varieties with a dense orbit. (joint with R. Horruitiner) (arXiv: 2012.02815)
8. Reduction theory for connections over the formal punctured disc. (arXiv: 2003.00008)