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Modern Mathematical Physics:
Strings and Gravity, Supersymmetry, Integrability

• Construction of supergravity backgrounds which describe intersecting D-(M-)-brane solutions with Lifshitz-like asymptotics. Analysis of the obtained solutions in the framework of the gauge/gravity duality. Investigation of correlation functions between a Wilson loop and local photons in the obtained backgrounds using the holographic approach.
  
  Construction and analysis of solutions without a horizon (boson stars) with Lifshitz-like asymptotics. Construction of Kerr-Vaidya/ Kerr-Newman-Vaidya solutions with AdS asymptotics for D=5 supergravity model. Study of local operators (two-point correlation functions) for the constructed metrics via holography.
  
  Study of systems with one-half spontaneously broken supersymmetry and construction of the component action for the Born-Infeld theory with N=4, d=4 supersymmetry, with N=2 vector multiplet as Goldstone superfield.
  
  Construction of the supersymmetric extentions of the mechanics, including the Pais-Uhlenbeck oscillator with specific frequencies, which are invariant with respect to the l-conformal Galilei group or its deformations.
  
  Calculation of the monodromy matrix elements action onto universal Bethe vectors in the quantum integrable models associated with supersymmetric extension of the Yangian double. Application of these formulas for obtaining a supersymmetric analog of the Reshetikhin's formula for the scalar products of the Bethe vectors in the supersymmetric models.
  
  Study of the universal Bethe vectors for the quantum integrable models associated with gl(4) and gl(2,2) spin chains. Study of the representations of connected Yangians with the aim to obtain the correlation functions in quantum integrable models.
  
  In the framework of special Bohr-Sommerfeld geometry, applications to the case when the phase space is not simply connected will be studied. Investigation of a relation between special Bohr-Sommerfeld geometry and the Hitchin integrable systems in the simplest case of riemannian surfaces of genus > 1.
  
  Construction of moduli spaces of special Bohr-Sommerfeld cycles in the algebraic case. It will be shown that in the algebraic case these moduli spaces are finite dimensional.
  
  
• The N = (1; 1); 6D SYM superfield invariants including the candidate counterterms found earlier from the pure symmetry considerations will be reproduced from the full-fledged quantum perturbation theory in N = (1; 1) and N = (1; 0) harmonic superspaces. The background field method for N = (1; 1); 6D SYM theory will be developed in full generality.
  
  Construction and study of the SU(2|1) and SU(2|2) invariant extensions of the Calogero-Moser models as deformations of N = 4 supersymmetric extensions.
  
  The models of N = 4 supersymmetric quantum mechanics with "long multiplets", as well as their SU(2|1) deformations, will be thoroughly studied, their bosonic target geometry will be revealed and quantum spectrum will be found for a few particular cases.
  
  The study of 6D supergravity models in the off-shell N = (1; 0) and on-shell N = (1; 1) 6D harmonic superspaces will be started. The constraints of N = (1; 1) 6D supergravity will be solved in terms of N = (1; 0) superfields.
  
  Construction of superfield action for the many-particle systems with N=4 deformed supersymmetry. Superfield formulation of sigma model with Wess-Zumino term possessing the N=4 deformed supersymmetry and describing the interaction of the spinning particle with an external gauge background.
  
  Construction of models for spinning particles and superparticles using momentum twistors. Obtaining of transition amplitudes from the BFV-BRST path integral.
  
  Definition of geometries described by N=4 supersymmetric quantum mechanical sigma models with a variety of dynamical, semi-dynamical and gauge supermultiplets. Determination of the type of supercharges for different geometries.
  
  Construction of new hairy black holes linked to charged scalar clouds in the Kerr-Newman space-time. Investigation of the near BPS spinning self-gravitating Skyrmions. Construction of spinning black holes with the BPS Skyrme hair.
  
  
• Observations of bright stars near the Galactic Center will be used for obtaining constraints on modification of the Newton gravity law in the weak field approximation. The bounds on a graviton mass will be obtained also proceeding from a potential reconstruction at the Galactic Center.
  
  Realistic inflation scenarios consistent with observational data from Plank-2015 will be developed by making use of the classical Yang-Mills fields non-minimally coupled to gravity.
  
  Description of the scalar-tensor perturbations in modified theories of gravity without the Einstein frame; in particular, in theories with an arbitrary function of the torsion scalar.
  
  The stability of modified gravity theories with higher derivatives will be studied regarding different perturbations.
  
  The explicit compact formulae are expected to be derived for the forces exerted on the material medium by the electromagnetic field, taking advantage of the different forms for the relevant energy-momentum tensor.
  
  Derivation of the vacuum energy of quantized fields in the presence of crossed cosmic strings and extraction of its finite part depending on the strings' mutual position.
  
  Study of the vacuum fluctuations on the lattice background formed by multi-dimensional delta functions.
  
  Construction of integrable scalar cosmologies with a cubic integral of motion.
  
  

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