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 involved with the study of geometric invariants of moduli spaces, such as their cohomologies and their derived categories. 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 in positive characteristic.
Publications
- Projectivity of the moduli of equidimensional branchvarieties. (with D. Halpern-Leistner, T. Jones and R. Ramkumar).
Duke Mathematical Journal (to appear). - Topology of Gm-actions and applications to the moduli of Higgs bundles. (with S. Zhang).
Mathematical Research Letters (to appear). - Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic. (with S. Zhang).
Michigan Mathematical Journal (to appear). - Relative étale slices and cohomology of moduli spaces. (with M. A. de Cataldo and A. Ibáñez Núñez).
Annales de l’Institut Fourier (to appear). - Moduli of objects in finite length abelian categories. (with E. Lennen and S. Makarova).
Moduli. 2025. - The moduli stack of principal rho-sheaves and Gieseker-Harder-Narasimhan filtrations. (with T. L. Gomez and A. Zamora).
Mathematische Zeitschrift. 2024. - On the quasicompactness of the moduli stack of logarithmic G-connections over a curve.
Israel Journal of Mathematics. 2024. - Moduli spaces of sheaves via affine Grassmannians.(with D. Halpern-Leistner and T. Jones).
Journal fur die reine und angewandte Mathematik (Crelle’s journal). 2024. - Geometry of the logarithmic Hodge moduli space. (with M. A. de Cataldo, and joint Appendix with S. Zhang).
Journal of the London Mathematical Society. 2024. - On automorphisms of semistable G-bundles with decorations.
Advances in Geometry. 2023. - An inequality of pairs of Hölder conjugate numbers. (with A. Stan).
Journal of Mathematical Inequalities. 2016.
Preprints
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- Quantum operations on the ring of symmetric functions. (with D. Halpern-Leistner).
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- Stability and disconnected groups. (with A. Ibáñez Núñez).
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- Moduli of non-pure sheaves and contractions of Hilbert schemes. (with S. Makarova).
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- There are no exotic compact moduli of sheaves on a curve. (with D. Weissmann and X. Zhang).
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- Hitchin fibrations are Ngo fibrations. (with M. de Cataldo, R. Fringuelli and M. Mauri).
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- Distinguishing algebraic spaces from schemes. (with D. Weissmann and X. Zhang).
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- Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves. (with D. Bejleri, J. Foster, G. Inchiostro, S. Makarova and J. Zhao).
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- Semistable Non Abelian Hodge theorem in positive characteristic. (with S. Zhang).
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- The structure of the moduli of gauged maps from a smooth curve. (with D. Halpern-Leistner)
Other writing
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- Curves in schemes of representations.(Appendix to the paper “Constructible Subcategories and Unbounded Representation Type” by K. Schlegel).
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- A guide to moduli theory beyond GIT. (with T. Gomez and A. Zamora).
Contemporary Mathematics AMS 803, Moduli Spaces and Vector Bundles—New Trends. 2024.
- A guide to moduli theory beyond GIT. (with T. Gomez and A. Zamora).