Projectdetails

Mathematics and theoretical physics (in particular, quantum and string theories) traditionally have a lot of connections and influence each other. This connections work in two sides. Mathematics is the main tool in theoretical physics. On the other side quantum and string theories produce new concepts and problems which become subjects of mathematical research. Most important and interesting results are obtained at the edge between these fields.

The main aim of the Project is to develop interdisciplinary connections between geometry and singularity theory on one side and quantum field theory on the other. It consists of several adjoint themes, connected with quantum field theory: string theory, WDVV equations, Frobenius manifolds, Gromov-Witten invariants, mirror symmetry, integrable systems, Whitham hierarchies, Seiberg-Witten theory, matrix models, quantum topological invariants of knots and 3D-, 4D-manifolds, modern problems of theory of singularity and symplectic geometry.

The goal of the project is to clarify connections between these theories and to use these connections for progress in them.

The project includes, in particular,

- study of topology of moduli spaces of complex and real algebraic curves, corresponding Hurwitz spaces and their applications to quantum invariants of manifolds,

- development of tools for computing Gromov-Witten invariants of complex manifolds based on singularity theory,

- study of various spaces of mappings and their quantum invariats such as Vassiliev invariants of knots, 4-invariants of graphs, finite order invariants of complex mappings,

- mirror symmetry for Fano manifolds and especially 3D- and 4D- Fano manifolds,

- applications of integrable systems KP/Toda/Calogero and Whitham type to study of important problems of mathematical physics: Seiberg-Witten theory, Dijkgraaf-Vafa theory, Landau-Ginzburg models, matrix models etc,

- D-branes theory and associated deformations of open-closed topological field theory, Landau-Ginzburg models with boundaries, noncommutative Frobenius manifolds.

Keywords:

quantum field theory, string theory, WDVV equations, Frobenius manifolds, mirror symmetry, integrable systems, Seiberg-Witten theory, matrix models, quantum topological invariants of knots and 3D-, 4D- manifolds, singularity theory, symplectic geometry

Artikelen

• A. Morozov, D. Krotov (2005). A solvable sector of AdS theory. Journal of High Energy Physics. pp. 062-. ISSN 1029-8479.
• A. Marshakov, L. Chekhov, D. Vasiliev, A. Mironov (2005). Complex Geometry of Matrix Models. Proceedings of the Steklov Institute of Mathematics . pp. 265-306. ISSN 0371-9685.
• A.G. Khovanskii, S.P. Chulkov (2006). The Hilbert polynomial for systems of linear partial differential equations with analytic coefficients. Izvestiya: Mathematics . pp. 153-169. ISSN 1064-5632.
• P.E. Pushkar, E. Ferrand (2006). Morse theory and global coexistence of singularities on wave fronts. Journal of the London Mathematical Society. pp. 527-544. ISSN 0024-6107 .
• I. Luengo, A. Melle-Hernandez, S.M. Gusein-Zade (2006). Integration over the space of non-parametrized arcs and motivic versions of the monodromy zeta-function. Proceedings of the Steklov Mathematical Institute. pp. 71-82. ISSN 0371-9685.
• I. Luengo, A. Melle-Hernandez, S.M. Gusein-Zade (2006). Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points. Michigan Mathematical Journal. pp. 353-359. ISSN 0026-2285.
• O.N. Karpenkov (2006). Mobius energy of graphs. Mathematical Notes. pp. 134-138. ISSN 0025-657X.
• O.N. Karpenkov (2006). Classification of lattice-regular lattice convex polytopes. Functional Analysis and Other Mathematics. pp. 31-54. ISSN 1991-0061.
• Vik.S. Kulikov (2006). On the Alexander polynomials of Hurwitz curves. Izvestiya:Mathematics. pp. 75-94. ISSN 0373-2436.
• V.M. Kharlamov, Vik.S. Kulikov (2006). Surfaces with DIF$\ne$DEF real structures. Izvestiya:Mathematics. pp. 135-174. ISSN 0373-2436.
• A. Mironov (2006). Matrix models vs. matrix integrals. Theoretical and Mathematical Physics. pp. 63-72. ISSN 0564-6162.
• A.F. Costa, S.M. Natanzon, A.M. Porto (2006). Counting the regular coverings of surfaces using the center of a group algebra.. European Journal of Combinatorics. pp. 228-234
• A. Alexandrov, A. Morozov, A. Mironov (2006). Unified description of correlators in non-Gaussian phases of Hermitean matrix model. International Journal of Modern Physics A. pp. 2481-2517. ISSN 0217-751X.
• S.K. Lando (2006). $J$-invariants of ornaments, and framed chord diagrams. Funct. Anal. Appl.. pp. 1-10. ISSN 0016-2663.
• S. Natanzon, A. Alexeevski (2006). Algebra of Hurwitz numbers for seamed surfaces. Russian Math. Surveys. pp. 
• J. Mostovoy, S. Duzhin (2006). A toy theory of Vassiliev invariants. Moscow Mathematical Journal. pp. 85-93. ISSN 1609-4514.
• A.V. Alekseevskii, S.M. Natanzon (2006). Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves. Selecta Mathematica. New Series. pp. 301-377
• O. Karpenkov (2006). On existence and uniqueness conditions of lattice triangles. Russian Math. Surveys. pp. 185-186. ISSN 0042-1316.
• O. Karpenkov (2006). Three examples of three-dimensional continued fractions in the sense of Klein. C.R.Acad. Sci. Paris, Ser.B. pp. 5-7. ISSN 1631-073X.
• O. Karpankov (2006). On examples of difference operators for {0,1}-valued functions over finite sets. Functional Analisys and Other Mathematics. pp. 197-202. ISSN 1991-0061.
• S. Lando (2006). $J$-invariants of ornaments and framed chord diagrams. Funct. Anal. Appl.. pp. 1-10
• F. Delgado, A. Campillo, S.M. Gusein-Zade (2007). On Poincare series of filtrations on equivariant functions of two variables. Moscow Mathematical Journal. pp. 243-255. ISSN 1609-3321.
• S. S. Shadrin, D. Zvonkine (2007). Changes of variables in ELSV-type formulas. Michigan Mathematical Journal. pp. 209-228. ISSN 0026-2285.
• S. S. Shadrin, A. Losev (2007). From Zwiebach invariant to Getzler relation. Communications in Mathematical Physics. pp. 649-679. ISSN 0010-3616.
• S. Shadrin, I. Shneiberg (2007). Belorousski-Pandharipande relation in dGBV algebra. Journal of Geometry and Physics. pp. 597-615. ISSN 0393-0440.
• A. Morozov, A. Mironov (2007). Is strong gravitational radiation predicted by TeV-gravity?. JETP Letters. pp. 9-14. ISSN 0021-3640.
• A. Campillo, F. Delgado, S.M. Gusein-Zade (2007). Multi-index filtrations and generalized Poincare series. Monatshefte fur Mathematik. pp. 193-209. ISSN 0026-9255.
• S.M. S.M.Gusein-Zade, A. Melle-Hernandez, I. Luengo (2007). On the power structure over the Grothendieck ring of varieties and its applications. Proceedings of the Steklov Institute of Mathematics. pp. 53-64. ISSN 0081-5438.
• A. Morozov, A. Alexandrov, A. Mironov (2007). Instantons and merons in matrix models. Physica D. pp. 126-167
• E.A. Gorsky (2007). On the motivic measure on the space of function. Revista Matematica Complutense. pp. 507-521. ISSN 1139-1138.
• E.A. Gorsky (2007). Motivic integrals and functional equations. Algebra and Analysis (St. Petersburg Mathematical Journal). pp. 92-111. ISSN 0234-0852 (1061.
• Vik.S. Kulikov (2007). Hurwitz curves. Uspekhi Mat. Nauk. pp. 3-86. ISSN 0042-1316.
• V Golyshev, J Stienstra (2007). Fuchsian Equations of type DN. Communications in Number Theory and Physics. pp. 323-346
• S. Duzhin, M. Karev (2007). Detecting the orientation of string links by finite type invarants. Funct. Anal. Appl.. pp. 208-216. ISSN 0016-2663.
• Vik.S. Kulikov (2007). Braid monodromy of unibranched plane curve singularity. Vestnik of MSUP. pp. 70-87
• S.K. Lando, M.E. Kazaryan (2007). An algebro-geometric proof of Witten's conjecture. J. Amer. Math. Soc.. pp. 1079-1089. ISSN ISSN 1088-6834.
• D. Zvonkine, S.K. Lando (2007). Counting ramified coverings and intersection theory on spaces of rational functions I. Moscow Mathematical Journal. pp. 85-107. ISSN 1609-4514.
• S.K. Lando, M.E. Kazaryan (2007). Thom polynomials for maps of curves with isolated singularities. Proc. Steklov Inst. Math.. pp. 93-106. ISSN 0371-9685.
• S.V. Chmutov, S.K. Lando (2007). Mutant knots and intersection graphs. Algebraic and Geometric Topology. pp. 1579-1598. ISSN 14722747 EISSN:.
• S.M. Natanzon, A.F. Costa (2007). Classification of $\mathbb{Z}_{p^{k}}^{m}$ orientation preserving actions on surfaces. Moscow Math.J.. pp. 419-424
• S.M. Natanzon (2007). Singularities and noncommutative Frobenius manifolds. MIAN, M.. pp. 143-155
• E. Frenkel, A. Losev, N. Nekrasov (2007). Notes on instantons in topological field theory and beyond. Nucl.Phys.Proc.Suppl.. pp. 215-230
• S. Shadrin, A. Losev, I. Shneiberg (2007). Tautological relations in Hodge field theory. Nucl.Phys.. pp. 267-296
• D. Krotov, V. Alexandrov, V. Lysov, A. Losev (2007). Materializing superghosts. JETP Lett. pp. 439-443
• A. Mironov, A. Morozov, T. Tomaras (2007). On n-point Amplitudes in N = 4 SYM. Journal of High Energy Physics. pp. 21-. ISSN 1029-8479.
• A. Mironov, T. Tomaras, A. Morozov (2007). Some Properties of Alday-Maldacena minimum. Physics Letters B. pp. 723-731. ISSN 0370-2693.
• V. Dolotin, A. Morozov, Sh. Shakirov (2007). Higher Nilpotent Analogues of A-infinity structure. Physics Letters B. pp. 71-73. ISSN 0370-2693.
• A. Morozov, A. Mironov, A. Alexandrov (2007). M-theory of matrix models. Theoretical and Mathematical Physics. pp. 153-164. ISSN 0040-5779.
• Vik.S. Alexander modules of irreducible $C$-groups (2008). Alexander modules of irreducible $C$-groups. Izvestiya:Mathematics. pp. 305-344. ISSN 0373-2436.
• Vik.S. Kulikov (2008). On Chisini's Conjecture. II. Izvestiya:Mathematics. pp. 901-913. ISSN 0373-2436.
• A. Morozov, A. Mironov (2008). Radiation Beyond Four Space-Time Dimensions. Theoretical and math. physics. pp.  . ISSN 0564-6162.
• M. Serbyn, Dr. A.N. Morozov (2008). Non-Linear Algebra and Bogolubov's Recursion. Theoretical and Math.physics. pp.  . ISSN 0564-6162.
• Sh. Shakirov, V Dolotin, Dr. A.N. Morozov (2008). A-infinity structure on simplicial complexes. Theoretical and Mathematical Physics. pp.  . ISSN 0564-6162.
• Dhr. S.K. Lando (2008). Combinatorial facets of Hurwitz numbers. Applications of Group Theory to Combinatorics, Koolen, Kwak and Xu, Eds., Taylor & Francis Group. London. pp. 109-132
• G.G. Gusev (2008). Euler characteristic of the bifurcation set for a polynomial of degree 2. Russian Mathematical Surveys. pp. 167-168. ISSN 0042-1316.
• Yu.A. Levitskaya, G.B. Shabat, Yu.Yu. Kochetkov, E.M. Kreines, V.F. Nasretdinova, V.A. Dremov, N.Ya. Amburg, N.M. Adrianov (2008). Catalog of dessins d'enfants with $\le$ 4 edges. Fundamentalnaya i prikladnaya matematika (Journal of Mathematical Sciences). pp.  . ISSN 1560-5159 (1072.
• Vik.S. Kulikov (2008). A remark on the non-rationality problem for generic cubic fourfolds. Mat. Zametki. pp. 61-68. ISSN 0025-567X.
• A. Morozov, Dr. A. Mironov (2008). On the Problem of Radiation Friction Beyond 4 and 6 Dimensions. International Journal of Modern Physics A. pp. 4677-4688. ISSN 0217-751X.
• Dr. A.N. Morozov (2008). Hamiltonian Formalism in the Presence of Higher Derivatives. Theoretical and Mathematical Physics. pp. 208-216. ISSN 0564-6162.
• H. H.Itoyama, Dr. A.N. Morozov (2008). Boundary Ring or a Way to Construct Approximate NG Solutions with Polygon Boundary Conditions. II. Polygons which admit an inscribed circle. PROGRESS OF THEORETICAL PHYSICS. pp. 231-287. ISSN 0033-068X.
• Dr. A.N. Morozov (2008). Alday-Maldacena duality and AdS Plateau problem. International Journal of Modern Physics A. pp. 2118-2134. ISSN 0217-751X.
• Dr. A.N. Morozov (2008). On the Problem of Multiple M2 Branes. Journal of High Energy Physics. pp. 1-76. ISSN 1029-8479.
• Dr. A.N. Morozov (2008). NSR Superstring Measures Revisited. Journal of High Energy Physics. pp. 1-86. ISSN 1029-8479.
• Dr. A.N. Morozov (2008). NSR measures on hyperelliptic locus and non-renormalization of 1,2,3-point functions. Physics Letters B. pp. 116-122. ISSN 0370-2693.
• Dr. A.N. Morozov (2008). From Simplified BLG Action to the First-Quantized M-Theory. Pisma v ZhETF. pp. 759-762. ISSN 0021-3640.
• M. Serbyn, Dr. A.N. Morozov (2008). Non-Linear Algebra and Bogolubov s Recursion. Theoretical and Mathematical Physics. pp. 316-343. ISSN 0564-6162.
• Vik.S. Kulikov (2008). A Remark on the Nonrationality Problem for Generic Cubic Fourfolds. Mathematical Notes. pp. 57-64. ISSN 0025-567X.
• A. Campillo, F. Delgado, Prof. S.M. Gusein-Zade (2008). Title Universal Abelian covers of rational surface singularities and multi-index filtrations. Functional Analysis and Its Applications. pp. 83-88. ISSN 0016-2663.
• S.A. Grishanov, Prof. dr. V.A. Vassiliev (2008). Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds. Topology and its Applications. pp. 1757-1765. ISSN 0166-8641.
• S.M. Natanzon, Prof. dr. A. Alekseev (2008). The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces. Izvestiya Mathematics. pp. 627-646
• A. Morozov, H. Itoyama, Dr. A. Mironov (2008). "Anomaly" in n=infinity Alday-Maldacena Duality for Wavy Circle. Journal of High Energy Physics. pp. 1-23. ISSN 1029-8479.
• A. Morozov, Dr. A. Mironov (2009). Virasoro constraints for Kontsevich-Hurwitz partition function. Journal of High Energy Physics. pp.  . ISSN 1029-8479.
• A. Morozov, H. Itoyama, Dr. A. Mironov (2009). Boundary Ring: a way to construct approximate NG solutions with polygon boundary conditions: I. Z_n-symmetric configurations. Nuclear Physics B. pp. 365-410. ISSN 0550-3213.

Bijdragen aan boeken

• Vik.S. Kulikov (2006). $C$-groups: their realizations and properties. in: Mathematisches Forschungsinstitut Oberwolfach Report No 42/2006. Oberwolfach: . pp. 11-16
• A. Losev, M. Shifman (2007). From Berezin Integral to Batalin-Vilkovissky Formalism: A mathematical Physicist's point of view. in: Felix Berezin life and death of the mastermind of supermathematics. : . pp. 
• S. Chmutov, S. Duzhin, G.L. Naber, S.T. Tsou, J.-P. Francoise (2007). Kontsevich integral. in: Encyclopedia of Mathematical Physics. : . pp. 231-239. ISBN 978-0-1251-2666-3.
• S.M. Gusein-Zade, W. Ebeling, Le Dung Trang, J.-P. Brasselet, J. Damon, M. Oka (2007). Lectures on Monodromy. in: Singularities in Geometry and Topology. : . pp. 234-252 . ISBN 978-981-270-022-3.
• S.M. Gusein-Zade, W. Ebeling, J.-P. Brasselet, M. Oka, J. Damon, Le Dung Trang (2007). Indices of collections of 1-forms on singular varieties. in: Singularities in Geometry and Topology. : . pp. 629-639. ISBN 978-981-270-022-3.
• A. Losev (2007). From Berezin Integral to Batalin-Vilkovisky Formalism: A Mathematical Physicist's Point of View. in: "Felix Berezin: The Life and Death of the Mastermind of Supermathematics, edited by M.Shifman. : . pp. 1-28. ISBN ISBN 978-981-270-532.
• W. Ebeling, S.M. Gusein-Zade, A. Pichon, C. Murolo, D. Trotman, D. Cheniot, N. Dutertre (2007). Chern obstructions for collections of 1-forms on singular varieties. in: Singularity Theory. Singapore: . pp. 557-564. ISBN 978-981-270-410-8.
• I. Luengo, A. Melle-Hernandez, Prof. S.M. Gusein-Zade, V.M. Buchstaber, I.M. Krichever (2008). An equivariant version of the monodromy zeta function. in: American Mathematical Society Translations, Series 2, v.224.. Providence, RI: . pp. 139-146. ISBN 0-8218-4674-4.
• S.M. Natanzon, A.V. Alexeevski, V.M. Buchstaber, Prof. dr. I.M. Krichever (2008). Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups. in: American Mathematical Society Translations, Series 2, v.224.. Providence, RI: . pp. 1-27. ISBN 0-8218-4674-4.