Valentino Tosatti

Columbia University
Department of Mathematics, Room 625
2990 Broadway, New York, NY 10027
Email: tosatti@math.columbia.edu
Fax: 212-854-8962

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I am a Ritt Assistant Professor at Columbia University. Before coming to Columbia I was a graduate student at the Department of Mathematics, Harvard University, where I got my Ph.D. on June 2009. My advisor was Shing-Tung Yau. Before that, I went to college at the Scuola Normale Superiore and at the University of Pisa.

I do research in Differential Geometry, Geometric Analysis, Complex Geometry and Partial Differential Equations. Some specific topics are: Kähler Geometry, Calabi-Yau manifolds, almost-complex, symplectic and Hermitian geometry, geometric flows, complex Monge-Ampère equations.

Here's my CV, updated in May 2012.

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Teaching

The Kähler-Ricci Flow and the Minimal Model Program - MSC Tsinghua University

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Publications and Preprints

• (with B. Weinkove) On the evolution of a Hermitian metric by its Chern-Ricci form
  [arXiv] preprint 2012
  
  
• (with M. Gross and Y. Zhang) Collapsing of abelian fibred Calabi-Yau manifolds
  [arXiv] preprint 2011
  
  
• Calabi-Yau manifolds and their degenerations
  [arXiv] to appear in Ann. N.Y. Acad. Sci. 2012
  
  
• (with B. Weinkove) Plurisubharmonic functions and nef classes on complex manifolds
  [arXiv] [journal] Proc. Amer. Math. Soc. 2012.
  
  
• The K-energy on small deformations of constant scalar curvature Kähler manifolds
  [arXiv] [book] in Advances in Geometric Analysis, Advanced Lectures in Math. 21, International Press, 2012.
  
  
• Kähler-Einstein metrics on Fano surfaces
  [arXiv] [journal] Expo. Math. 30 (2012), no.1, 11-31.
  
  
• (with G. Székelyhidi) Regularity of weak solutions of a complex Monge-Ampère equation
  [arXiv] [journal] Anal. PDE 4 (2011), no.3, 369-378.
  
  
• Degenerations of Calabi-Yau metrics
  [arXiv] [journal] in Geometry and Physics in Cracow, Acta Phys. Polon. B Proc. Suppl. 4 (2011), no.3, 495-505.
  
  
• (with B. Weinkove) The Calabi-Yau equation on the Kodaira-Thurston manifold
  [arXiv] [journal] J. Inst. Math. Jussieu 10 (2011), no.2, 437-447.
  
  
• (with B. Weinkove)The Calabi-Yau equation, symplectic forms and almost complex structures
  [arXiv] [book] in Geometry and Analysis, No. 1, 475-493, Advanced Lectures in Math. 17, International Press, 2011.
  
  
• (with B. Weinkove) The complex Monge-Ampère equation on compact Hermitian manifolds
  [arXiv] [journal] J. Amer. Math. Soc. 23 (2010), no.4, 1187-1195.
  
  
• (with B. Weinkove) Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds
  [arXiv] [journal] Asian J. Math. 14 (2010), no.1, 19-40.
  
  
• Adiabatic limits of Ricci-flat Kähler metrics
  [arXiv] [journal] J. Differential Geom. 84 (2010), no.2, 427-453.
  
  
• Kähler-Ricci flow on stable Fano manifolds
  [arXiv] [journal] J. Reine Angew. Math. 640 (2010), 67-84.
  
  
• Limits of Calabi-Yau metrics when the Kähler class degenerates
  [arXiv] [journal] J. Eur. Math. Soc. (JEMS) 11 (2009), no.4, 755-776.
  
  
• (with B. Weinkove and S.-T. Yau) Taming symplectic forms and the Calabi-Yau equation
  [arXiv] [journal] Proc. London Math. Soc. 97 (2008), no.2, 401-424.
  
  
• A general Schwarz Lemma for almost-Hermitian manifolds
  [arXiv] [journal] Comm. Anal. Geom. 15 (2007), no.5, 1063-1086.
  
  
• (with B. Weinkove) The Calabi Flow with small initial energy
  [arXiv] [journal] Math. Res. Lett. 14 (2007), no.6, 1033-1039.
  
  
• On the Critical Points of the E_k Functionals in Kähler Geometry
  [arXiv] [journal] Proc. Amer. Math. Soc. 135 (2007), no.12, 3985-3988.
  
  
• (with L. Pietronero, E. Tosatti and A. Vespignani) Explaining the uneven distribution of numbers in nature
  [arXiv] [journal] Phys. A, 293 (2001), 297-304.

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Thesis, notes and other material

• Geometry of complex Monge-Ampère equations
  [PDF] 4 June 2009, PhD Thesis, Harvard University, Advisor: Prof. Shing-Tung Yau.

• Extremal Sobolev inequalities and applications
  [PDF] 12 March 2005, Minor Thesis, Harvard University, Advisor: Prof. Ben Weinkove.

• Index theorems for morphisms of vector bundles and foliations
  [PDF] (in Italian) 10 June 2004, Laurea Thesis, University of Pisa, Advisor: Prof. Marco Abate.

• Uniqueness of CPⁿ
  [PDF] 30 October 2007, an exposition of a theorem of Hirzebruch, Kodaira, Yau.

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Links

• Schedule for Columbia's Informal Complex Geometry and PDE Seminar.
• Special Day on Complex Geometry and PDE, April 20, 2012.
• Schedule for Columbia's Informal Kähler Ricci Flow Seminar.
• Schedule for Harvard's Yau's students seminar.

My Erdős number is 3.