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Modern Mathematical Physics:
Strings and Gravity, Supersymmetry, Integrability

• Construction of the field theory models, realizing the partial breaking of the global supersymmetries in D=3 and D=4 with the hypermultiplet as the Goldstone superfiled.
  
  Investigation of the determinant formulas for form-factors of the local operators in the quantum integrable models with higher symmetries solvable in the framework of the nested Bethe ansatz.
  
  Study of the basic equations and relations for the relativistic static spherically symmetric stars (SSSS) in the model of minimal dilatonic gravity (MDG), which is locally equivalent to the f(R) theories of gravity and gives an alternative description of the effects of dark matter and dark energy.
  
  Development of new approach based on the symmetric form of the Heun's differential equation yielded by development of the Felix Klein symmetric form of the Fuchsian equations with an arbitrary number N>3 of regular singular points.
  
  Construction of the superconnection framing of the derived category of coherent sheaves on non-compact complex varieties. The technique will be exploited to construct complex moduli spaces of objects on the varieties.
  
  Descripyion of the pairs of unmixed basis in 6 dimensional space with the application to the quantum information theory.
  
  Construction of deformations of lagrangian cycles of Mironov - Panov.
  
  Development of the theory of special Bohr - Sommerfeld cycles. Comparison in these terms of the results of the Kahler and lagrangian approaches in Geometric Quantization.
  
  Longitudinal and transverse gluon propagators in the Landau gauge will be studied in the SU(2) lattice gauge theory at nonzero temperatures near the critical point T_c.
  
  The properties of the thermal Abelian color-magnetic monopoles in the maximally Abelian gauge will be studied in the vicinity of the confinement-deconfinement phase transition in the lattice SU(3) gluodynamics and lattice QCD.
  
  
• Analysis of the loop structure of the Bagger-Lambert-Gustavsson and Aharony-Bergman-Jaffereris-Maldacena models by using the superfield background field method and manifestly covariant superfield proper-time method, and study of the aspects of the AdS(4)/CFT(3) correspondence.
  
  Design and analysis of new models of supersymmetric mechanics, including cases of deformed and quasicomplex supersymmetries, and their application to the description of the motion of spin particles in different background non-Abelian gauge fields and in the theory of the quantum Hall effect.
  
  Further elaboration of the formalism of auxiliary fields in the duality invariant supersymmetric theories and construction of duality-invariant actions for the Born-Infeld theory with partial supersymmetry breaking.
  
  Construction of new twistor formulations of spinning (super)particles and (super)strings and study of their application in higher spin fields theory.
  
  Study of various black-hole configurations and localized particle-like soliton solutions in the higher dimensional (super)gravity and (super)Yang-Mills theories by numerical analysis.
  
  
• Investigation of the Minkowski space-time stability in the framework of the modified gravitation theories with higher derivatives by making use of a new, specially developed, universal method based on the theory of dynamical systems;
  
  Searches for new physics and creation of new theoretical models for interpretation of observational data for the Galactic Center;
  
  Creation of advanced theoretical models for exoplanetary systems with gravitational microlensing and polarization observations;
  
  Development of the heat kernel expansion for manifolds with non-smooth boundaries will be proposed. For domains with boundaries possessing cusps the multiple reflection technique will be extended in order to derive systematically higher order heat kernel coefficients;
  
  Development of a general approach to calculation of the vacuum noncontact friction by applying the Green's function technique in macroscopic electrodynamics and making resort to the Kubo formulas from nonequilibrium statistical mechanics;
  
  Investigation of electronic excitations in graphene will be conducted in the framework of the quantum theory of the Dirac field defined on the 2-dimensional graphene sheet and interacting with the electromagnetic field occupying the whole 3-dimensional space. Study ofthe surface plasmon excitations in this system.
  
  

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