Research Summary

I am an algebraic geometer who uses tools from derived categories to solve problems of interest to algebraists, geometers, and number theorists. Some of my recent work studies the derived geometry of birational maps, investigates the ties between rational points and rational parameterizations and the structure of the derived category, and develops derived approaches to noncommutative algebraic geometry.

My research has received support from the Simons Foundation and National Science Foundation. It has also benefited from a membership at the Institute for Advanced Study. My ORCID number is ORCID iD iconorcid.org/0000-0001-5819-0159.

For a more information on me, see my CV.

Papers

All the following papers are available on the ArXiv. However, please note that there may be differences between the final published version, the ArXiv version, and the current version of a paper. Copies of any article listed as in preparation are available by request.

0. Derived categories of sheaves on quasi-projectives schemes. Thesis. 2008.

1. Meet homological mirror symmetry. Modular forms and string duality, 191–224, Fields Inst. Commun., 54, Amer. Math. Soc., Providence, RI, 2008.

2. Derived categories of sheaves on singular schemes with an application to reconstruction. Adv. Math. 227 (2011), no. 2, 895–919.

3. Sheaves on local Calabi-Yau varieties.

4. (with David Favero) Hochschild dimensions of tilting objects. Int. Mat. Res. Not. 2012 (2012), no. 11, 2607-2645.

5. Equivalences of derived categories of sheaves on quasi-projective schemes.

6. (with David Favero and Ludmil Katzarkov) Orlov spectra: bounds and gaps. Invent. Math. 189 (2012), no. 2, 359-430.

7. (with David Favero and Ludmil Katzarkov) A category of kernels for equivariant factorizations and its implications for Hodge theory. Pub. Math. l'IHES. 120 (2014), no. 1, 1-111.

8. (with David Favero and Ludmil Katzarkov) A category of kernels for equivariant factorizations, II. J. de Math. Pure et App. 102 (2014), no. 4., 702-757.

9. (with David Favero, Colin Diemer, Ludmil Katzarkov, and Gabriel Kerr) The Mori program and non-Fano toric Homological Mirror Symmetry. Trans. Amer. Math. Soc. 367 (2015) 8933-8974.

10. (with David Favero, Dragos Deliu, M. Umut Isik, and Ludmil Katzarkov) Resolutions in factorization categories. Adv. Math. 295 (2016), 195-249.

11. (with David Favero, Dragos Deliu, M. Umut Isik, and Ludmil Katzarkov) Homological Projective Duality via variation of Geometric Invariant Theory quotients. JEMS. Volume 19 Issue 14 (2017) 1127-1158.

12. Wall crossing for derived categories of moduli of sheaves on rational surfaces. Alg. Geo. 4 (3) (2017) 263-280.

13. (with David Favero, Dragos Deliu, M. Umut Isik, and Ludmil Katzarkov) On the derived categories of degree d hypersurface fibrations. Math. Ann. 371 (2018), no. 1-2, 337--370.

14. (with David Favero and Ludmil Katzarkov) Variation of Geometric Invariant Theory quotients and derived categories. Crelle. Issue 746 (2019) 235-304.

15. (with Alexander Duncan and Patrick McFaddin). The toric Frobenius morphism and a conjecture of Orlov. Eur. J. Math. 5 (2019), no. 3, 640-645.

16. (with Alexander Duncan and Patrick McFaddin). On derived categories of arithmetic toric varieties. Annals of K-Theory 4 (2019), no. 2, 211-242.

17. (with Blake Farman). Kernels for noncommutative projective schemes. To appear in J. Noncomm. Geo.

18. (with Alexander Duncan and Patrick McFaddin) Derived categories of centrally symmetric toric Fano varieties. Submitted.

19. (with Colin Diemer and David Favero). Kernels from compactifications. Submitted.

20. (with Nitin Chidambaram, David Favero, Patrick McFaddin, and Robert Vandermolen) Kernels for Grassmann flops. Submitted.

21. (with Alexander Duncan, Alicia Lamarche, and Patrick McFaddin) Brauer invisibility. Preprint.

22. (with Alexander Duncan, Alicia Lamarche, and Patrick McFaddin) Detecting rationality via the derived category. Preprint.

23. (with Alicia Lamarche) On derived categories and rational points for a class of toric Fano varieties.

Books

1. Superschool on derived categories and D-branes. Edmonton, Canada, July 1723, 2016. Lectures from the PIMS Superschool. Edited with Charles Doran, David Favero and Eric Sharpe. Springer Proceedings in Mathematics & Statistics, 240. Springer, Cham, 2018.