My main research interests are in Mathematical Physics in the sense of Mathematics inspired by ideas that come from Theoretical Physics. More precisely, I am interested in algebraic and geometric structures which come from quantum field theory, statistical mechanics and the theory of integrable systems.

Academic Journal Papers, refereed

[1] M. Kontsevich, A. Odesskii, Multiplication kernels, Lett. Math. Phys. 111 (2021), no. 6, Paper No. 152, 59 pp.,

[2] M. Kontsevich, A. Odesskii, p-Determinants and monodromy of differential operators, Selecta Math. (N.S.) 28 (2022), no. 3, Paper No. 52,

[3] A. Odesskii, V. Sokolov, Nonabelian elliptic Poisson structures on projective spaces, J. Geom. Phys. 169 (2021), Paper No. 104330, 15 pp.,

[4] A. Odesskii, Poisson structures on loop spaces of CP^n and an r-matrix associated with the universal elliptic curve, Journal of Geometry and Physics Volume 140, June 2019, Pages 152-160,

[5] A. Odesskii, B. Feigin, “Flat deformations of algebras and functional equations”, Journal of Combinatorial Algebra, Швейцария. 2019. Vol. 3. No. 3. P. 215-236,

[6] A. Odesskii, “Poisson structures on loop spaces of CP^n and an r-matrix associated with the universal elliptic curve”, Journal of Geometry and Physics, Volume 140, June 2019, pp 152-160,

[7] A. Odesskii, “K-projectors”, Journal of Algebra, Volume 511, 1 October 2018, pp 1-15,

[8] A. V. Odesskii, “Integrable structures of dispersionless systems and differential geometry”, Theoretical and Mathematical Physics, May 2017, Volume 191, Issue 2, pp 692-709,

[9] M. Babela, A. Odesskii, “A family of integrable evolution equations of third order”, J. Nonlinear Math. Phys. 24 (2017), no. 1, 73-78

[10] A. Odesskii, “A simple construction of integrable Whitham type hierarchies”, Geometric methods in physics, 195-208, Trends Math., Birkhauser/Springer, Cham, 2014

[11] V. E. Zakharov, A. V. Odesskii, A. V., M. Cisternino, M. Onorato, “Five-wave classical scattering matrix and integrable equations”, Theoret. and Math. Phys. 180 (2014), no. 1, 759-764

[12] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Parameter-dependent associative Yang-Baxter equations and Poisson brackets”, Int. J. Geom. Methods Mod. Phys. 11 (2014), no. 9

[13] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Double Poisson brackets on free associative algebras”, Noncommutative birational geometry, representations and combinatorics, 225-239, Contemp. Math., 592, Amer. Math. Soc., Providence, RI, 2013

[14] A. Odesskii, V. Sokolov, “Non-homogeneous systems of hydrodynamic type possessing Lax representations”, Comm. Math. Phys. 324 (2013), no. 1, 47-62

[15] A. Odesskii, T. Wolf, “Compatible quadratic Poisson brackets related to a family of elliptic curves”, J. Geom. And Physics (63), 2013, 107-117

[16] A. V. Odesskii, V. N. Rubtsov, V. V. Sokolov, “Bi-Hamiltonian ordinary differential equations with matrix variables”, Teoret. Mat. Fiz., 171:1 (2012), 26-32

[17] E.V. Ferapontov, Alexander Odesskii, and N.M. Stoilov, Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions, Journal of Mathematical Physics 52, 2011

[18] A. Odesskii, V. Sokolov, Integrable (2+1)-dimensional systems of hydrodynamic type, Theoretical and Mathematical Physics, 163:2 (2010), 179-221

[19] A. Odesskii, V. Sokolov, Integrable pseudopotentials related to generalized hypergeometric functions, Selecta Mathematica (2010) 16:145.

[20] E. V. Ferapontov, A. V. Odesskii, Integrable Lagrangians and modular forms, Journal of Geometry and Physics 60 (2010) 896-906.

[21] A. Odesskii, V. Sokolov, Classification of integrable hydrodynamic chains, Journal of Physics A: Math. Theor. 43(2010) 434027 (15pp.).

[22] A. Odesskii, V. Sokolov, Integrable elliptic pseudopotentials, Theoretical and Mathematical Physics, 161(1): 1338-1350 (2009).

[23] Odesskii, Alexander; Rubtsov, Vladimir, Integrable systems associated with elliptic algebras. Quantum groups, 81-105, IRMA Lect. Math. Theor. Phys., 12, Eur. Math. Soc., Zurich, 2008.

[24] Odesskii, A. V.; Sokolov, V. V., On (2+1)-dimensional systems of hydrodynamic type possessing a pseudopotential with movable singularities. (Russian) Funktsional. Anal. i Prilozhen. 42 (2008), no. 3, 53–62, 96; translation in Funct. Anal. Appl. 42 (2008), no. 3, 205–212

[25] Odesskii, A. V.; Pavlov, M. V.; Sokolov, V. V., Classification of integrable Vlasov-type equations. (Russian) Teoret. Mat. Fiz. 154 (2008), no. 2, 249–260. (Reviewer: Yuri B. Suris)

[26] Odesskii, Alexander, A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential. Selecta Math. (N.S.) 13 (2008), no. 4, 727–742. (Reviewer: Ian A. B. Strachan)

[27] Odesskii, Alexander; Sokolov, Vladimir, Pairs of compatible associative algebras, classical Yang-Baxter equation and quiver representations. Comm. Math. Phys. 278 (2008), no. 1, 83–99. (Reviewer: Zhong Qi Ma) 17B62 (81R12)

[28] Odesskii, Alexander; Sokolov, Vladimir, Algebraic structures connected with pairs of compatible associative algebras. Int. Math. Res. Not. 2006, Art. ID 43734, 35 pp. (Reviewer: Thierry Dana-Picard)

[29] Odesskii, A. V.; Sokolov, V. V., Integrable matrix equations related to pairs of compatible associative algebras. J. Phys. A 39 (2006), no. 40, 12447–12456. (Reviewer: Stanislav Z. Pakuliak)

[30] Odesskii, A. V.; Sokolov, V. V., Compatible Lie brackets related to elliptic curve, J. Math. Phys. 47 (2006), no. 1, 013506, 14 pp. (Reviewer: Olivier G. Schiffmann)

[31] Odesskii, Alexander, Bihamiltonian elliptic structures. Mosc. Math. J. 4 (2004), no. 4, 941–946, 982.

[32] Odesskii, Alexander, Set-theoretical solutions to the Yang-Baxter relation from factorization of matrix polynomials and $\theta$-functions. Mosc. Math. J. 3 (2003), no. 1, 97–103, 259. (Reviewer: Gigel Militaru)

[33] Odesskii, A. V.; Rubtsov, V. N., Polynomial Poisson algebras with a regular structure of symplectic leaves. (Russian) Teoret. Mat. Fiz. 133 (2002), no. 1, 3–23; translation in Theoret. and Math. Phys. 133 (2002), no. 1, 1321–1337 (Reviewer: Zakaria Giunashvili)

[34] Enriquez, B.; Odesskii, A., Quantization of canonical cones of algebraic curves. Ann. Inst. Fourier (Grenoble) 52 (2002), no. 6, 1629–1663. (Reviewer: Olivier G. Schiffmann)

[35] Belavin, A. A.; Odesskii, A. V.; Usmanov, R. A., New relations in the algebra of the Baxter $Q$-operators. (Russian) Teoret. Mat. Fiz. 130 (2002), no. 3, 383–413; translation in Theoret. and Math. Phys. 130 (2002), no. 3, 323–350 (Reviewer: Andrey V. Tsiganov)

[36] Braden, H. W.; Gorsky, A.; Odesskii, A.; Rubtsov, V., Double-elliptic dynamical systems from generalized Mukai-Sklyanin algebras. Nuclear Phys. B 633 (2002), no. 3, 414–442. (Reviewer: Alexander V. Shapovalov)

[37] Odesskii, A. V., Belavin elliptic R-matrices and exchange algebras. (Russian) Funktsional. Anal. i Prilozhen. 36 (2002), no. 1, 59–74, 96; translation in Funct. Anal. Appl. 36 (2002), no. 1, 49–61 32G34

[38] Feigin, B. L.; Odesskii, A. V., Quantized moduli spaces of the bundles on the elliptic curve and their applications. Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), 123–137, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht, 2001. (Reviewer: Zhenbo Qin)

[39] Feigin, B. L.; Odesskii, A. V., Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras. Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), 109–122, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht, 2001. (Reviewer: Shao-Ming Fei)

[40] Feigin, B. L.; Odesskii, A. V., Vector bundles on an elliptic curve and Sklyanin algebras. Topics in quantum groups and finite-type invariants, 65–84, Amer. Math. Soc. Transl. Ser. 2, 185, Amer. Math. Soc., Providence, RI, 1998. (Reviewer: Michel Van den Bergh)

[41] Feigin, B. L.; Odesskii, A. V., Coordinate ring of the quantum Grassmannian and intertwiners for the representations of Sklyanin algebras. Topics in quantum groups and finite-type invariants, 55–64, Amer. Math. Soc. Transl. Ser. 2, 185, Amer. Math. Soc., Providence, RI, 1998. (Reviewer: Michel Van den Bergh)

[42] Feigin, Boris; Jimbo, Michio; Miwa, Tetsuji; Odesskii, Alexandr; Pugai, Yaroslav, Algebra of screening operators for the deformed $W\sb n$ algebra. Comm. Math. Phys. 191 (1998), no. 3, 501–541. (Reviewer: Jun’ichi Shiraishi)

[43] Odesskii, A. V.; Feigin, B. L., Elliptic deformations of current algebras and their representations by difference operators. (Russian) Funktsional. Anal. i Prilozhen. 31 (1997), no. 3, 57–70, 96; translation in Funct. Anal. Appl. 31 (1997), no. 3, 193–203 (1998) (Reviewer: V. Leksin)

[44] Feigin, Boris; Odesskii, Alexander, A family of elliptic algebras. Internat. Math. Res. Notices 1997, no. 11, 531–539. (Reviewer: Michel Van den Bergh)

[45] Odesskii, A. V.; Feigin, B. L., Sklyanin’s elliptic algebras. The case of a point of finite order. (Russian) Funktsional. Anal. i Prilozhen. 29 (1995), no. 2, 9–21, 95; translation in Funct. Anal. Appl. 29 (1995), no. 2, 81–90

[46] Odesskii, A. V.; Feigin, B. L., Constructions of elliptic Sklyanin algebras and of quantum $R$-matrices. (Russian) Funktsional. Anal. i Prilozhen. 27 (1993), no. 1, 37–45; translation in Funct. Anal. Appl. 27 (1993), no. 1, 31–38 (Reviewer: Tomasz Brzeziiski)

[47] Odesski, Aleksandr V., Rational degeneration of elliptic quadratic algebras. Infinite analysis, Part A, B (Kyoto, 1991), 773–779, Adv. Ser. Math. Phys., 16, World Sci. Publ., River Edge, NJ, 1992. (Reviewer: Michel Van den Bergh)

[48] Odesskii, A. V.; Feigin, B. L., Sklyanin’s elliptic algebras. (Russian) Funktsional. Anal. i Prilozhen. 23 (1989), no. 3, 45–54, 96; translation in Funct. Anal. Appl. 23 (1989), no. 3, 207–214 (1990) (Reviewer: S. Paul Smith)

[49] Odesskii, A. V., An analogue of the Sklyanin algebra. (Russian) Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 78–79

Review Articles, refereed

[50] Odesskii, A. V., Elliptic algebras. (Russian) Uspekhi Mat. Nauk 57 (2002), no. 6(348), 87–122; translation in Russian Math. Surveys 57 (2002), no. 6, 1127–1162 (Reviewer: Dmitriy A. Rumynin)

Conference contributions, refereed

[51] A.V.Odesskii and B.L.Feigin, Quantized moduli spaces of the bundles on the elliptic curve and their applications. Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), 123–137, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht , 2001. (Reviewer: Zhenbo Qin), math.QA/9812059

[52] A.V.Odesskii and B.L.Feigin, Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras. Integrable structures of exactly solvable two-dimensional models of quantum field theory ( Kiev , 2000), 109–122, NATO Sci. Ser. II Math. Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht , 2001. (Reviewer: Shao-Ming Fei), math.QA/9912037

• MATH 3P09 Partial Differential Equations
• MATH 2P08 Ordinary Differential Equations
• MATH 3P08 Advanced Differential Equations

Mathematicians, Gold Prospectors, and AI

Imagine gold prospectors who have spent their lives searching for gold by hand along a beautiful river. They love their work, have developed remarkable skills, and take pride in their discoveries.  And suddenly, heavy mining machinery arrives.  The prospectors may love the river, but the gold does not care who discovers it.

I think something similar may be happening to mathematics.  Will AI agents eventually become the main creators of new mathematical theories, leaving human mathematicians primarily to study and understand what AI has discovered?  And would that really be the end of mathematics?

I have written a short essay on anthropocentrism, artificial intelligence, and the possible future of mathematical research.