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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 and vector supermultiplets as the Goldstone superfields;
  
  We will study basic equations and relationships between relativistic static spherically symmetric stars (SSSS) and other static spherically symmetric objects (SSSO) of different scales - planets, white dwarfs, normal star of solar types, dark spheroids, galaxies and galaxy coupling in the model of minimal dilatonic gravity (MDG) locally equivalent to f(R) gravity theories leading to an alternative description of dark matter and dark energy. We will describe singular manifolds in the phase space of SSSO. We will study new effects of the bifurcation of the phase portrait of the systems under the change of the mass of the dilatonic field. We will develop new effective computation schemes for the listed problems using the parallel programming;
  
  We will develop a new approach based on the symmetrical form of the Hoin differential equation. The approach is based on the generalization of the Klien symmetrical form for the Fuksian equations for any number (N> 3) of regular singular points;
  
  Construction of the universal Bethe vectors for the quantum integrable models associated with supersymmetric extension of the Yangian double. Calculation of the formfactors of the local operators in the supersymmetric quantum integrable models;
  
  For a five-dimensional supergravity model we aim to build new analytic black holes in Lifshitz-like backgrounds, which are able to describe holographically the quark-gluon plasma at strong coupling. We will also construct a Vaidya-Lifshitz solution that represents a thin shell falling at the speed of light and allows one to describe a black hole formation. Using the obtained solutions we will compute and investigate nonlocal operators (entanglement entropy, Wilson loops, etc.) characterizing various stages of the quark-gluon plasma. The impact of the background anisotropy on the QGP thermalization will be studied;
  
  We will construct pseudotoric structures on hyperplane sections of toric varieties. We will study the exotic lagrangian tori arising in this construction;
  
  Special geometry for Bohr - Sommerfeld cycles will be constructed. We will prove that the universal moduli space of special Bohr - Sommerfeld cycles carries a natural universal Kahler structure;
  
  Special Bohr - Sommerfeld geometry can be exploited in the theory of sigma models. We will associate to any embedded Riemann surface a complicated graph in the loop space of the target space.
  
  
• Analysis of loop structure of 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 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;
  
  Analysis of the ultraviolet properties of N=(1,0) and N=(1,1) super Yang-Mills theories in dimension 6 in terms of the harmonic off-shell N=(1,0) and on-shell N=(1,1) superfields. Solving of the defining constraints of 6D N=(1,1) super Yang-Mills theory in terms of harmonic N=(1,0) superfields;
  
  Construction of the superfield description of the new type supermultiplets of supersymmetric quantum mechanics, the so called "long multiplets", for N=2 and N=4 supersymmetries in one dimension, equally as for their SU(2|1) deformations. Construction of new explicit realizations of the conformal supergroup D(2,1;) in the models of N=4 superconformal mechanics.
  
  
• 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 interpretations of observational data for the Galactic Center;
  
  Creation of advanced theoretical models for exo-planetary 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 the higher order heat kernel coefficients;
  
  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 interaction with the electromagnetic field occupying the whole 3-dimensional space. Study of the surface plasmon excitations in this system;
  
  Realistic inflation scenarios consistent with observational data from Plank-2015 will be developed in conformal gravity models with several scalar fields and symmetry violating R ²-term;
  
  It is planned to examine the possibility of constructing the regimes of accelerated expansion in the models of gravity non-minimally coupled to a scalar field with one-loop quantum corrections;
  
  The vacuum friction force between macroscopic bodies will be calculated in the framework of the quantum-statistical theory of the linear response by making use of different versions of the electromagnetic energy-momentum tensor in a medium;
  
  It is expected to accomplish the construction of the uniform asymptotic expansions for hypergeometric function products arising, for instance, in the vacuum energy calculation for quantized fields in the background of the media with variable spherically symmetric refractive index.
  
  

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