Valentino Tosatti
Professor
Department of Mathematics, Room 225
Northwestern University
2033 Sheridan Road, Evanston, IL 60208
Email: tosatti@math.northwestern.edu
Fax: 847-491-8906
I do research in
Differential Geometry, Geometric Analysis, Complex Algebraic 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, transcendental methods in algebraic geometry, dynamics on K3 surfaces.
My research is partially funded by NSF grant DMS-1903147.
My CV is available here.
Selected Teaching
Fall 2019 - Introduction to Differential Geometry
Fall 2019 - Fourier Analysis and Boundary Value Problems
Fall 2018 - Topics in Geometry
Spring 2017 - Introduction to Topology
Winter 2017 - Partial Differential Equations
Fall 2015 - Differential Geometry
Fall 2015 - Algebraic Geometry
Spring 2015 - Cohomology
Winter 2015 - Differentiable Manifolds
Winter 2013 - Differential Geometry
Publications and Preprints
- (with S. Filip) Canonical currents and heights for K3 surfaces
[arXiv] preprint 2021
- (with H.-J. Hein) Smooth asymptotics for collapsing Calabi-Yau metrics
[arXiv] preprint 2021
- (with Y. Li) Diameter bounds for degenerating Calabi-Yau metrics
[arXiv] preprint 2020
- (with T. Collins) Restricted volumes on Kähler manifolds
[arxiv] to appear in Ann. Fac. Sci. Toulouse Math. 2021
- (with S. Filip) Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics
[arxiv] to appear in Amer. J. Math. 2021
- (with S. Kołodziej) Morse-type integrals on non-Kähler manifolds
[arXiv] to appear in Pure Appl. Math. Q. 2021
- (with B. Weinkove) The complex Monge-Ampère equation with a gradient term
[arXiv] to appear in Pure Appl. Math. Q. 2021
- Ricci-flat metrics and dynamics on K3 surfaces
[arXiv] [journal] Boll. Unione Mat. Ital. 14 (2021), no.1, 191-209.
- (with H.-J. Hein) Higher-order estimates for collapsing Calabi-Yau metrics
[arxiv] [journal] Camb. J. Math. 8 (2020), no.4, 683-773.
- (with M. Gross and Y. Zhang) Geometry of twisted Kähler-Einstein metrics and collapsing
[arXiv] [journal] Comm. Math. Phys. 380 (2020), no.3, 1401-1438.
- (with Y. Zhang) Collapsing hyperkähler manifolds
[arxiv] [journal] Ann. Sci. Éc. Norm. Supér. 53 (2020), no.3, 751-786.
- Collapsing Calabi-Yau manifolds
[arXiv] [journal] Surveys in Differential Geometry 23 (2018), 305-337, International Press, 2020.
- (with B. Weinkove) Hermitian metrics, (n-1,n-1) forms and Monge-Ampère equations
[arXiv] [journal] J. Reine Angew. Math. 755 (2019), 67-101.
- (with N. McCleerey) Pluricomplex Green's functions and Fano manifolds
[arxiv] [journal] Épijournal Geom. Algébrique 3 (2019), Art. 9, 15pp.
- (with J. Chu and B. Weinkove) The Monge-Ampère equation for non-integrable almost complex structures
[arxiv] [journal] J. Eur. Math. Soc. (JEMS) 21 (2019), no.7, 1949-1984.
- Orthogonality of divisorial Zariski decompositions for classes with volume zero
[arxiv] [journal] Tohoku Math. J. 71 (2019), no.1, 1-8.
- (with S. Filip) Smooth and rough positive currents
[arxiv] [journal] Ann. Inst. Fourier (Grenoble) 68 (2018), no.7, 2981-2999.
- KAWA lecture notes on the Kähler-Ricci flow
[arxiv] [journal] Ann. Fac. Sci. Toulouse Math. 27 (2018), no.2, 285-376.
- (with B. Weinkove) The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation
[arxiv] [book]
in Nonlinear Analysis in Geometry and Applied Mathematics, Part 2, 147-158, Harvard CMSA Ser. Math. 2, International Press, 2018.
- Regularity of envelopes in Kähler classes
[arxiv] [journal] Math. Res. Lett. 25 (2018), no.1, 281-289.
- Nakamaye's Theorem on complex manifolds
[arxiv] [book] in Algebraic Geometry: Salt Lake City 2015. Part 1, 633-655, Proc. Sympos. Pure Math. 97.1, American Mathematical Society, 2018.
- (with B. Weinkove and X. Yang) The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
[arxiv] [journal] Amer. J. Math. 140 (2018), no.3, 653-698.
- (with Y. Zhang) Finite time collapsing of the Kähler-Ricci flow on threefolds
[arxiv] [journal] Ann. Sc. Norm. Super. Pisa Cl. Sci. 18 (2018), no.1, 105-118.
- (with J. Chu and B. Weinkove) C^{1,1} regularity for degenerate complex Monge-Ampère equations and geodesics rays
[arxiv] [journal] Comm. Partial Differential Equations 43 (2018), no.2, 292-312.
- (with G. Székelyhidi and B. Weinkove) Gauduchon metrics with prescribed volume form
[arxiv] [journal] Acta Math. 219 (2017), no.1, 181-211.
- (with X. Yang) An extension of a theorem of Wu-Yau
[arXiv] [journal] J. Differential Geom. 107 (2017), no.3, 573-579.
- (with J. Chu and B. Weinkove) On the C^{1,1} regularity of geodesics in the space of Kähler metrics
[arxiv] [journal] Ann. PDE 3 (2017), no.2, 3:15.
- Uniqueness of CP^{n}
[arxiv] [journal] Expo. Math. 35 (2017), no.1, 1-12.
- (with B. Weinkove) The Monge-Ampère equation for (n-1)-plurisubharmonic functions on a compact Kähler manifold
[arXiv] [journal] J. Amer. Math. Soc. 30 (2017), no.2, 311-346.
- (with S. Fang, B. Weinkove and T. Zheng) Inoue surfaces and the Chern-Ricci flow
[arxiv] [journal] J. Funct. Anal. 271 (2016), no.11, 3162-3185.
- The Calabi-Yau Theorem and Kähler currents
[arxiv] [journal] Adv. Theor. Math. Phys. 20 (2016), no.2, 381-404.
- (with M. Gross and Y. Zhang) Gromov-Hausdorff collapsing of Calabi-Yau manifolds
[arXiv] [journal] Comm. Anal. Geom. 24 (2016), no.1, 93-113.
- (with T. Collins) A singular Demailly-Păun theorem
[arxiv] [journal] C. R. Math. Acad. Sci. Paris 354 (2016), no.1, 91-95.
- (with H.-J. Hein) Remarks on the collapsing of torus fibered Calabi-Yau manifolds
[arxiv] [journal] Bull. Lond. Math. Soc. 47 (2015), no.6, 1021-1027.
- (with T. Collins) Kähler currents and null loci
[arXiv] [journal] Invent. Math. 202 (2015), no.3, 1167-1198.
- (with Y. Zhang) Infinite time singularities of the Kähler-Ricci flow
[arxiv] [journal] Geom. Topol. 19 (2015), no.5, 2925-2948.
- Families of Calabi-Yau manifolds and canonical singularities
[arxiv] [journal] Int. Math. Res. Not. IMRN (2015) no.20, 10586-10594.
- (with Y. Wang, B. Weinkove and X. Yang) C^{2,α} estimates for nonlinear elliptic equations in complex and almost complex geometry
[arxiv] [journal] Calc. Var. Partial Differential Equations 54 (2015), no.1, 431-453.
- Non-Kähler Calabi-Yau manifolds
[arxiv] [book] Contemp. Math. 644 (2015), 261-277.
- (with B. Weinkove and X. Yang) Collapsing of the Chern-Ricci flow on elliptic surfaces
[arXiv] [journal] Math. Ann. 362 (2015), no.3-4, 1223-1271.
- (with B. Weinkove) On the evolution of a Hermitian metric by its Chern-Ricci form
[arXiv] [journal] J. Differential Geom. 99 (2015), no.1, 125-163.
- (with Y. Zhang) Triviality of fibered Calabi-Yau manifolds without singular fibers
[arXiv] [journal] Math. Res. Lett. 21 (2014), no.4, 905-918.
- (with T. Collins) An extension theorem for Kähler currents with analytic singularities
[arXiv] [journal] Ann. Fac. Sci. Toulouse Math. 23 (2014), no.4, 893-905.
- (with B. Weinkove) The Chern-Ricci flow on complex surfaces
[arXiv] [journal] Compos. Math. 149 (2013), no.12, 2101-2138.
- (with M. Gross and Y. Zhang) Collapsing of abelian fibred Calabi-Yau manifolds
[arXiv] [journal] Duke Math J. 162 (2013), no.3, 517-551.
- Blowup behavior of the Kähler-Ricci flow on Fano manifolds
[arXiv] [journal] Univ. Iagel. Acta Math. 50 (2012), 117-126.
- Calabi-Yau manifolds and their degenerations
[arXiv] [journal] Ann. N.Y. Acad. Sci. 1260 (2012), 8-13.
- (with B. Weinkove) Plurisubharmonic functions and nef classes on complex manifolds
[arXiv] [journal] Proc. Amer. Math. Soc. 140 (2012), no.11, 4003-4010.
- The K-energy on small deformations of constant scalar curvature Kähler manifolds
[arXiv] [book]
in Advances in Geometric Analysis, 139-150, 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.
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.
- A finiteness theorem for polarized manifolds
(with Y. Zhang) [PDF] [book] 28 July 2015, a short note, appendix to this paper, published in Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds & Picard-Fuchs Equations, 551-592, Advanced Lectures in Math. 42, International Press, 2018.
- Metric limits of Calabi-Yau manifolds
[journal] Notices Amer. Math. Soc. 65 (2018), no.4, 413-414.
- Extension of semi-flat forms across singular fibers
[PDF] 27 March 2018, a short note, appendix to this paper by Cao, Guenancia and Păun.
Editorial boards
Links
Other events that I recently co-organized
- Conference on Calabi-Yau and Geometry, Sapienza University of Rome, May 29 - June 1, 2019.
- Nonlinear PDEs in real and complex geometry, AIM, August 13 - 17, 2018.
- Workshop on degenerations of Calabi-Yau manifolds, IHP, Paris, May 14 - 16, 2018.
- Young Geometric Analysts' Forum 2018, Tsinghua Sanya International Mathematics Forum, January 29 - February 2, 2018
- Summer Northwestern Analysis Program SNAP 2017, July 24 - August 11, 2017
- Analytic Methods in Algebraic Geometry Day, March 18, 2017
- Young Geometric Analysts' Forum 2017, Tsinghua Sanya International Mathematics Forum, January 9 - 13, 2017
- The complex Monge-Ampère equation, AIM, August 15 - 19, 2016.
- Moduli spaces and singularities in algebraic and Riemannian geometry, Stony Brook, August 17 - November 20, 2015
- Collapsing Calabi-Yau Manifolds, Stony Brook, August 31 - September 4, 2015
- Special Session on Geometric Partial Differential Equations, AMS Meeting Chicago, October 2-4, 2015
- Emphasis Year in Geometric Analysis at Northwestern University, 2014-15
- Special Day on Eigenfunctions of the Laplacian on manifolds, October 25, 2014
- Special Day on Complex Geometry and Analysis on real analytic Riemannian manifolds, February 21, 2015
- Special Day on Analytic Methods in Algebraic Geometry, April 11, 2015
- Workshop on Ricci curvature, May 28-31, 2015
- Summer School in Geometric Analysis, July 6-12, 2015
- 73rd Midwest PDE Seminar, May 10-11, 2014
- Summer School in Complex Geometry, Rutgers University, August 12-18, 2013
- Great Lakes Geometry Conference 2013, April 20-21, 2013
- Special Session on Complex Geometry and Partial Differential Equations, AMS Meeting Akron, OH, October 20-21, 2012
My Erdős number is 3.