Tristan C. Collins

Massachusetts Institute of Technology
Department of Mathematics
77 Massachusetts Avenue, Cambridge, MA 02139
Email: tristanc at mit dot edu

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I am the Class of 1948 Career Development Associate Professor in the Mathematics Department at MIT. Formerly I was a Benjamin Peirce Assistant Professor at Harvard University. I completed my Ph.D. under the supervision of D.H. Phong at Columbia University in New York City. Before studying at Columbia, I studied mathematics at the University of British Columbia in Vancouver, B.C., where I worked with Malabika Pramanik.

I do research in geometry and analysis. I am particularly interested in geometric flows, and applications of algebraic geometry to analysis, and vice versa.

My research is partially supported by NSF CAREER grant DMS-1944952.

Here's my CV last updated July 11, 2023.

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Teaching

Spring 2022

• 18.03- Differential Equations

Fall 2020

• 18.03- Differential Equations

Fall 2019

• 18.965- Geometry of Manifolds I

Past Teaching

• Webpages for past courses can be found here

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Geometry and Analysis Seminar for Boston Area Graduate Students

• Fall 2022

• Fall 2021

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Publications and Preprints

• (with J. Lo, Y. Shi, and S.-T. Yau) Stability for Line Bundles and Deformed Hermitian-Yang-Mills Equation on Some Elliptic Surfaces
  [arXiv] preprint 2023
  
  
• (with S. Picard, and S.-T. Yau) The Strominger system in the square of a Kahler class
  [arXiv] preprint 2022
  
  
• (with Y.-S. Lin) Recent progress on SYZ mirror symmetry for some non-compact Calabi-Yau surfaces
  [arXiv] preprint 2022
  
  
• (with Y. Li) Uniqueness of some cylindrical tangent cones to special Lagrangians
  [arXiv] preprint 2022
  
  
• (with Y. Li) Complete Calabi-Yau metrics in the complement of two divisors
  [arXiv] preprint 2022
  
  
• (with D. Jafferis, C. Vafa , K. Xu, and S.-T. Yau) On upper bounds in dimension gaps of CFT's
  [arXiv] preprint 2022
  
  
• (with D.H. Phong) Spinor flows with flux, I: short-time existence and smoothing estimates
  [arXiv] preprint 2021
  
  
• (with S. Gukov, S. Picard, and S.-T. Yau) Special Lagrangian cycles and Calabi-Yau transitions
  [arXiv] preprint 2021
  
  
• (with A. Jacob, and Y.-S. Lin) The Torelli theorem for ALH^* gravitational instantons
  [arXiv] to appear in Forum of Math., Sigma
  
  
• (with S.-T. Yau) Moment maps, nonlinear PDE, and stability in mirror symmetry, I; Geodesics
  Ann. PDE 7 (2021), no. 1, Paper No. 11, 73 pp.
  
  
• (with S. Picard, and S.-T. Yau) Stability of the tangent bundle through conifold transitions
  [arXiv] to appear in Comm. Pure and Appl. Math.
  
  
• (with A. Jacob, and Y.-S. Lin) The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
  [arXiv] preprint 2020
  
  
• (with R. Berman, and D. Persson) Emergent Sasaki-Einstein geometry and AdS/CFT
  [arXiv] Nature Comm., 13, article 365.
  
  
• (with F. Tong, and B. Guo) On the degeneration of asymptotically conical Calabi-Yau metrics
  [arXiv] Math. Ann. 383 (2022), no. 3-4, 867--919.
  
  
• (with Y. Shi) Stability and the deformed Hermitian-Yang-Mills equation
  [arXiv] to appear in Surv. in Diff. Geom.
  
  
• (with J. Chu, and M.-C. Lee) The space of almost calibrated (1, 1) forms on a compact Kahler manifold
  [arXiv] Geom. Topol. 25 (2021), no. 5, 2573--2619.
  
  
• (with S. Picard) The Dirichlet problem for the k-Hessian equation on a complex manifold
  [arXiv] to appear in Amer. J. Math.
  
  
• (with A. Jacob, and Y.-S. Lin) Special Lagrangian submanifolds of log Calabi-Yau manifolds
  [arXiv] Duke Math. J. 170 (2021), no. 7,1291--1375.
  
  
• Concave elliptic equations and generalized Khovanskii-Teissier inequalities
  [arXiv] Pure Appl. Math. Q. 17 (2021), no. 3, 1061--1082.
  
  
• (with S.-T. Yau) Moment maps, nonlinear PDE, and stability in mirror symmetry
  [arXiv] preprint 2018
  
  
• (with T. Hisamoto and R. Takahashi) The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics
  [arXiv] J. Differential Geom. 120 (2022), no. 1, 51--95.
  
  
• (with D. Xie, and S.-T. Yau) The deformed Hermitian-Yang-Mills equation in geometry and physics
  [arXiv] Geometry and Physics, Volume I: A Festschrift in honour of Nigel Hitch, Oxford University Press, December, 2018
  
  
• Log-canonical thresholds in real and complex dimension 2
  [arXiv] Ann. Inst. Fourier (Grenoble), 68 (2018), no. 7, 2883--2900
  
  
• (with C. Mooney) Dimension of the minimum set for the real and complex Monge-Ampere equations in critical Sobolev spaces
  [arXiv] Anal. and PDE 10 (2017), no. 8 2013--2041
  
  
• (with V. Tosatti) Restricted volumes on Kahler manifolds
  [arXiv] Ann. Fac. Sci. Toulouse Math. (6) 31 (2022), no. 3, 907--947.
  
  
• (with S. Picard and X. Wu) Concavity of the Lagrangian phase operator and applications
  [arXiv] Calc. Var. and Partial Differential Equations 56 (2017), no. 4, Art. 89
  
  
• (with D. Xie and S.-T. Yau) K stability and stability of chiral ring
  [arXiv] preprint 2016
  
  
• (with G. Székelyhidi) Sasaki-Einstein metrics and K-stability
  [arXiv] Geom. Topol. 23 (2019), no. 3, 1339--1413.
  
  
• (with A. Jacob and S.-T. Yau) (1,1) forms with specified Lagrangian phase.
  [arXiv] Camb. J. Math. 8 (2020), no. 2, 407--452.
  
  
• (with A. Jacob and S.-T. Yau) Poisson metrics on flat vector bundles over non-compact curves
  [arXiv] Comm. Anal. Geom. 27 (2019), no. 3, 529--597
  
  
• (with V. Tosatti) A singular Demailly-Paun theorem
  [arXiv] C. R. Math. Acad. Paris 354 (2016), no. 1, 91--95
  
  
• C^2,α estimates for nonlinear elliptic equations of twisted type
  [arXiv] Calc. Var. Partial Differential Equations 55 (2016), no. 1, Art. 6
  
  
• (with G. Székelyhidi) Convergence of the J-flow on toric manifolds
  [arXiv] J. Differential Geom. 107 (2017), no. 1, 47--81
  
  
• (with V. Tosatti) Kahler currents and null loci
  [arXiv] Inventiones Math. 202 (2015), no. 3, 1167--1198
  
  
• (with G. Székelyhidi) The twisted Kahler-Ricci flow
  [arXiv] J. Reine Angew. Math. 716 (2016), 179--205
  
  
• (with G. Székelyhidi) K-Semistability for irregular Sasakian manifolds
  [arXiv] J. Differential Geom. 109 (2018), no. 1, 81--109
  
  
• (with A. Jacob) On the convergence of the Sasaki-Ricci flow
  [arXiv] to appear in Contemp. Math.
  
  
• Stability and convergence of the Sasaki-Ricci flow
  [arXiv] J. Reine Angew. Math. 716 (2016), 1--27
  
  
• (with V. Tosatti) An extension theorem for Kahler currents
  [arXiv] Ann. Fac. Sci.Toulouse Math. (6) 23 (2014), no. 4, 893-905
  
  
• Uniform Sobolev inequality along the Sasaki-Ricci flow
  [arXiv][Journal] J. Geom. Anal. 24 (2014), no. 3, 1323-1336
  
  
• (with A. Jacob) Remarks on the Yang-Mills flow on a compact Kahler manifold
  [arXiv] Univ. Iagel. Acta Math. no. 51 (2013), 17-43.
  
  
• The transverse entropy functional and the Sasaki-Ricci flow
  [arXiv] Trans. Amer. Math. Soc. 365 (2013), no. 3, 1277-1303
  
  
• (with A. Greenleaf and M. Pramanik) A multidimensional resolution of singularities with applications to analysis
  [arXiv] American Journal of Mathematics, 135 (2013), 1179-1252.
  
  

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