Leon A. Takhtajan

Distinguished Professor
Department of Mathematics

Office: Math. Tower 5-111 (631) 632-8287; Fax 632-7631; e-mail: leontak@math.stonybrook.edu

Department Administrator: Anne Duffy, Math 5-116, 632-8290

Research interests: Mathematical physics, in particular applications of quantum field theory to geometry, algebra and analysis.

Teaching schedule: MAT 401 Seminar: Hyperbolic Geometry TuTh 2:00pm-3:20pm in Physics P 129.

Office hours: W 2:00pm-4:00pm in Math Tower 5-111, and by appointment

Former graduate students:

• Joseph Schaefer
• Carlos Florentino
• Rukmini Dey
• Lee-Peng Teo
• Andrew McIntyre
• Joshua Friedman
• Daniel An
• Ki Song
• Joseph Walsh
• Vamsi Pingali
• Claudio Meneses
• Jun Wen

Recent Papers:

1. Quantum field theories on an algebraic curve Lett. Math. Phys. 52 (2000), 79-91.
2. Deformation quantization on Kähler manifolds (with N. Reshetikhin),  Amer. Math. Soc. Transl. (2) 201 (2000), 257-276.
3. Free bosons and tau-functions for compact Riemann surfaces and closed Jordan curves. Current correlation functions Lett. Math. Phys. 56 (2001), 181-228.
4. Generating functional in CFT on Riemann surfaces II: Homological aspects  (with E. Aldrovandi), Commun. Math. Phys. 227 (2002), 303-348; Part I Generating functional in CFT and effective action for two-dimensional quantum gravity on higher genus Riemann surfaces  (with E. Aldrovandi), Commun. Math. Phys. 188 (1997), 29-67.
5. Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler meitrics on M _{0,n} (with P. Zograf), Trans. Amer. Math. Soc. 355 (2003), 1857-1867, Erratum.
6. Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography (with L.-P. Teo), Commun. Math. Phys. 239 (2003), 183-240.
7. Weil-Petersson metric on the universal Teichmuller space I: Curvature properties and Chern forms (with L.-P. Teo), arXiv: math.CV/0312172 (2003).
8. Weil-Petersson metric on the universal Teichmuller space II: Kahler potential and period mapping (with L.-P. Teo), arXiv: math.CV/0406408 (2004).
9. Weil-Petersson geometry of the universal Teichmuller space (with L.-P. Teo), Progress in Math. 237 (2005), 219-227.
10. Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula (with A. McIntyre), Geom. and Funct. Analysis, 16 (2006), 1291-1323.
11. Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces (with L.-P. Teo), Commun. Math. Phys. 268 (2006), 135-197.
12. Weil-Petersson Metric on the Universal Teichmuller Space (with L.-P. Teo) Memoirs of the Amer. Math. Soc. 183 No. 861 (2006), vii + 119 pp.
13. Hamiltonian Methods in the Theory of Solitons (with L.D. Faddeev) Springer "Classics in Mathematics" 2007, xiv + 592 pp, reprint of 1987 original; review in Zentralblatt MATH.
14. Normal matrix models, dbar-problem, and orthogonal polynomials on the complex plane (with A. Its), arXiv:0708.3867 (2007).
15. The first Chern form on moduli of parabolic bundles (with P. Zograf), Math. Annalen 341 (2008), 113-135, Addendum.
16. Quantum Mechanics for Mathematicians Amer. Math. Soc. "Graduate Studies in Mathematics" vol. 95, 2008, xv + 387 pp; Additional Material and Errata; review in Zentralblatt MATH.
17. Quantum Mechanics for Mathematicians Russian translation, 2011, 496 pp.
18. Spectrum of the density matrix of a large block of spins of the XY model in one dimension (with F. Franchini, A. R. Its and V. E. Korepin), Quantum Inf. Process. 10 (2011), 325-341.
19. Hamiltonian Methods in the Theory of Solitons (with L.D. Faddeev) Japanese translation 2012, two volumes.
20. Quantum field theories on algebraic curves. I. Additive bosons Izvestiya RAN: Ser. Mat. 77:2 (2013), 165-196 (in Russian); English translation in Izvestiya: Mathematics 2013, 77:2, 378-406.
21. On Bott-Chern forms and their applications (with V. Pingali), Math. Annalen 360:1-2 (2014), 519-546.
22. On real projective connections, V.I. Smirnov's approach, and black hole type solutions of the Liouville equation Teoret. Mat. Fiz. 181:1 (2014), 206-217 (in Russian); English translation in Theor. and Math. Physics 181:1 (2014), 1307-1316.
23. The spectral theory of a functional-difference operator in conformal field theory (with L. Faddeev), Izvestiya RAN: Ser. Mat. 79:2 (2015), 181-204 (in Russian); English translation in Izvestiya: Mathematics 2015, 79:2, 388-410.
24. Explicit computation of the Chern character forms Geom Dedicata 181:1 (2016), 223-237.
25. Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves (with A. Laptev and L. Schimmer), Geometric and Functional Analysis (GAFA), 26:1 (2016), 288-305.
26. Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces (with J. Park and L.-P. Teo), Adv in Math 305 (2017), 856-894.
27. Remarkable lives an legacy of Sofia Kovalevskaya and Emmy Noether ICTS Public Lecture Bangalore, India (2017).
28. Ludwig Faddeev (1934-2017): His work and legacy Newsletter of the European Mathematical Society, June 2017, issue 4, 35-41 (with I. Aref’eva and M. Semenov-Tian-Shansky); Ludwig D. Faddeev (March 23, 1934 - February 26, 2017) IAMP News Bulletin, October 2017 (with N. Reshetikhin and M. Semenov-Tian-Shansky); Ludwig Dmitrievich Faddeev (obituary) Uspekhi Mat. Nauk 72:6 (2017) 191-196 (in Russian); English translation in Russian Math. Surveys 72:6 (2017), 1157-1163 (with I.Ya. Aref’eva, V.E. Zakharov, V.V. Kozlov, I.M. Krichever, V.P. Maslov, S.P. Novikov, A.M. Polyakov, N.Yu. Reshetikhin, M.A. Semenov-Tian-Shansky, E.K. Sklyanin, F.A. Smirnov, and S.L. Shatashvili).
29. Scientific heritage of L. D. Faddeev. Survey of papers Uspekhi Mat. Nauk 72:6 (2017) 3-112 (in Russian); English translation in Russian Math. Surveys 72:6 (2017) 977-1081 (with A. Yu. Alekseev, I. Ya. Aref’eva, M. A. Semenov-Tian-Shansky, E. K. Sklyanin, F. A. Smirnov, and S. L. Shatashvili).
30. Unravelling nonlinearity - Exactly Interview with BhAvanA magazine, vol. 1, issue 3 (2017).
31. Local index theorem for orbifold Riemann surfaces (with P. Zograf), Lett. Math. Phys. 109:5 (2019), 1119-1143. Erratum and updated arXiv file.
32. Weyl asymptotics for perturbed functional difference operators (with A. Laptev and L. Schimmer), J. Math. Phys. 60, 103505 (2019).
33. On Kawai theorem for orbifold Riemann surfaces Math. Annalen, 375:3 (2019), 923-947.
34. Etudes of the resolvent Uspekhi Mat. Nauk, 75:1(451) (2020), 155-194 (in Russian); English translation: Russian Math. Surveys, 75:1 (2020), 147-186.
35. Trace formulas for the modified Mathieu equation in " Partial Differential Equations, Spectral Theory and Mathematical Physics, The Ari Laptev Anniversary Volume" (Eds. P. Exner et al), Published by EMS Press; 427-442, 2021.
36. Logarithmic connections, WZNW action, and moduli of parabolic bundles on the sphere (with C. Meneses) Commun. Math. Phys., 387:2 (2021), 649-680.
37. Goldman form, flat connections and stable vector bundles Enseign. Math. 68 (2022), no. 3/4, 409–440; arXiv version (2021).
38. Ludwig Dmitrievich Faddeev Biographical Memoirs of Fellows of the Royal Society, published online 24 August 2022 (pdf file).
39. Supersymmetry and trace formulas II. Selberg trace formula (with C. Choi) arXiv:2306.13636 (2023).
40. Berezin quantization, conformal welding and the Bott-Virasoro group (with A. Alekseev and S. Shatashvili) Ann. Henri Poincaré 25, 35–64 (2024).
41. Supersymmetry and trace formulas I. Compact Lie groups (with C. Choi) JHEP 06 (2024) 026.
42. Algebraic de Rham theorem and Baker-Akhiezer function (with I. Krichever) Izvestiya RAN: Ser. Mat. 88:3 (2024), 101-110 (in Russian); English translation: Izvestiya: Mathematics 2024, 88:3, 506-514.