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 615 Symplectic geometry of character varieties. An Introduction TuTh 11:30am-12:50pm in Physics P 127.

Office hours: TuTh 2:00pm-3:30pm in 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.
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. Supersymmetry and trace formulas I. Compact Lie groups (with C. Choi) arXiv:2112.07942 (2021).
38. Goldman form, flat connections and stable vector bundles Enseign. Math. 68 (2022), no. 3/4, 409–440; arXiv version (2021).
39. Ludwig Dmitrievich Faddeev Biographical Memoirs of Fellows of the Royal Society, published online 24 August 2022 (pdf file).
40. Berezin quantization, conformal welding and the Bott-Virasoro group (with A. Alekseev and S. Shatashvili) arXiv:2212.04822 (2022).
41. Supersymmetry and trace formulas II. Selberg trace formula (with C. Choi) arXiv:2306.13636 (2023).
42. Algebraic de Rham theorem and Baker-Akhiezer function (with I. Krichever) arXiv:2311.04440 (2023).