Research Summary

I am an algebraic geometer who uses tools from derived categories to solve problems of interest to algebraic geometers, algebraists, and symplectic geometers. My recent work studies the derived geometry of birational maps, hunts for exceptional collections over general fields, and develops derived approaches to noncommutative algebraic geometry.

My research has received support from the Simons Foundation and National Science Foundation DMS-1501813. 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.

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

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

15. (with Alexander Duncan and Patrick McFaddin). On derived categories of arithmetic toric varieties. To appear in Ann. of K-Theory.

16. (with Blake Farman). Kernels for noncommutative projective schemes.

17. (with Alexander Duncan and Patrick McFaddin). The toric Frobenius morphism and a conjecture of Orlov. To appear in Eur. J. of Math.

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

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

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