Valentino Tosatti

Born c. 1981, in Trieste, Tosatti studied from 2000 at the Scuola Normale Superiore di Pisa and at the University of Pisa, graduating with a laurea in 2004. He then studied at Harvard University, where he graduated with an M.A. in 2005 and a Ph.D. in 2009.^[1] His Ph.D. thesis Geometry of complex Monge-Ampère equations was supervised by Shing-Tung Yau.^[2][3] Tosatti was from 2009 to 2012 a Joseph Fels Ritt Assistant Professor at Columbia University. At Northwestern University, he was an associate professor from 2012 to 2015 and a full professor from 2015 to 2020. From 2020 to 2022, he taught as a professor at McGill University. In 2022 he became a professor at NYU's Courant Institute of Mathematical Sciences.^[1]

Tosatti does research on complex and differential geometry; geometric analysis on complex, Hermitian, and symplectic manifolds; and partial differential equations. He is also interested in the connections of his main research topics with algebraic geometry and dynamical systems.^[1]

In 2017, with Gábor Székelyhidi and Ben Weinkove, Valentino Tosatti proved a conjecture published in 1984 by Paul Gauduchon.^[4][5] Acta Mathematica published their proof that on n-dimensional compact complex manifolds there is always a Gauduchon metric with prescribed volume form. For this purpose, the behavior of a large class of elliptic nonlinear partial differential equations of second order on Hermitian manifolds had to be investigated.^[4]

In 2011, Tosatti received a Blavatnik Award.^[6] In 2012, he was awarded a two-year Sloan Research Fellowship.^[1] In 2018, he received the Caccioppoli Prize. In 2019, he was elected a Fellow of the American Mathematical Society for "contributions in geometric analysis and complex geometry".^[7]

• Tosatti, Valentino; Weinkove, Ben; Yau, Shing-Tung (2008). "Taming symplectic forms and the Calabi-Yau equation". Proceedings of the London Mathematical Society. 97 (2): 401–424. arXiv:math/0703773. doi:10.1112/plms/pdn008.
• Tosatti, Valentino (2010). "Adiabatic limits of Ricci-flat Kähler metrics". Journal of Differential Geometry. 84 (2). arXiv:0905.4718. doi:10.4310/jdg/1274707320.
• Gross, Mark; Tosatti, Valentino; Zhang, Yuguang (2013). "Collapsing of abelian fibered Calabi–Yau manifolds". Duke Mathematical Journal. 162 (3). arXiv:1108.0967. doi:10.1215/00127094-2019703.
• Tosatti, Valentino; Weinkove, Ben (2015). "On the evolution of a Hermitian metric by its Chern-Ricci form". Journal of Differential Geometry. 99. arXiv:1201.0312. doi:10.4310/jdg/1418345539.
• Tosatti, Valentino; Wang, Yu; Weinkove, Ben; Yang, Xiaokui (2015). "${\displaystyle C^{2,\alpha }}$ estimates for nonlinear elliptic equations in complex and almost complex geometry". Calculus of Variations and Partial Differential Equations. 54: 431–453. arXiv:1402.0554. doi:10.1007/s00526-014-0791-0.
• Collins, Tristan C.; Tosatti, Valentino (2015). "Kähler currents and null loci". Inventiones Mathematicae. 202 (3): 1167–1198. arXiv:1304.5216. Bibcode:2015InMat.202.1167C. doi:10.1007/s00222-015-0585-9.
• Tosatti, Valentino; Weinkove, Ben (2016). "The Monge-Ampère equation for (𝑛-1)-plurisubharmonic functions on a compact Kähler manifold". Journal of the American Mathematical Society. 30 (2): 311–346. arXiv:1305.7511. doi:10.1090/jams/875.
• Tosatti, Valentino; Yang, Xiaokui (2017). "An extension of a theorem of Wu–Yau". Journal of Differential Geometry. 107 (3). arXiv:1506.01145. doi:10.4310/jdg/1508551226.
• Tosatti, Valentino; Weinkove, Ben; Yang, Xiaokui (2018). "The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits". American Journal of Mathematics. 140 (3): 653–698. arXiv:1408.0161. doi:10.1353/ajm.2018.0016.
• Tosatti, Valentino; Weinkove, Ben (2019). "Hermitian metrics, ( n -1, n -1) forms and Monge–Ampère equations". Journal für die Reine und Angewandte Mathematik (Crelles Journal) (755): 67–101. arXiv:1310.6326. doi:10.1515/crelle-2017-0017.