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Modern Mathematical Physics:
Gravity, Supersymmetry and Strings

• Construction of renormalization group flows on curved manifolds via the holographic duality. Studies of phase diagrams using the obtained holographic RG flows.
  
  Calculation and study of thermal correlators corresponding to quantum KdV charges in 2d CFT. Construction of the full KdV partition function in the case of free bosons.
  
  Construction of holographic RG flows with several effective charges. Consideration of these RG flows in terms of brane intersections in a relevant supergravity theory. Studies of the RG flows in the framework of the generalized Sachdev-Ye-Kitaev model.
  
  Development of a group-theoretical approach for the twistor description of massless particles with a continuous spin. The comparison of this approach with the Penrose twistor program.
  
  Construction of projectors for the irreducible representations of the multidimensional Poincare group (for an arbitrary type of symmetry) based on the results from the representation theory of the Brauer algebra and the methods of the R-matrices (the solutions of the Yang-Baxter equation, which are constructed in terms of the Brauer algebra generators).
  
  Study of the systems with partially broken supersymmetry, with an arbitrarily high number of spontaneously broken supersymmetries, in particular, systems of many N=1, d=3 scalar and vector multiplets, as well as their analogues in higher dimensions.
  
  Construction of non-symmetric eigenfunctions of the deformed Macdonald–Ruijsenaars systems in terms of the representation theory of the Ding–Iohara algebra and, in the explicit form, calculation of eigenvalues for these eigenfunctions. Construction of quantum Lax pairs for the deformed Calogero–Moser systems (rational, trigonometric, elliptic) by means of the Dunkl operators. Construction of symmetries of the elliptic Gaudin model by means of the quantum spectral curve. Generalization of Manin matrices.
  
  Construction of monotonic lagrangian tori of non-standard type in toric and pseudotoric Fano manifolds in the framework of Mirror Symmetry. Construction of examples of non-standard lagrangian tori which have non-trivial Maslov classes and, therefore, do not admit hamiltonian deformations to the minimal ones.
  
  Construction of the trigonometric and hyperbolic Calogero models with extended supersymmetry.
  
  
• Continuing the study of the quantum structure of N=(1,0), N=(1,1) and N=(2,0) supersymmetric gauge theories in 6 dimensions by the harmonic superspace methods, constructing the superfield invariants and effective action of these theories, further revealing of their relationships with the AdS/CFT correspondence. Analogous superfield analysis of N=(1,0) and N=(1,1) gauge theories with higher-derivatives. Study of the quantum superfield geometry of N=2, 5D super Yang-Mills theory, finding out the relation of the relevant effective action with the D4 brane action.
  
  Investigations of multiparticle Calogero-type systems with extended Poincare and superconformal supersymmetries, construction of their various SU(m|n) deformations on the basis of the superfield gauging of matrix models. Construction of quantum versions of the hyperbolic and trigonometric supersymmetric Calogero-Sutherland models, analysis of their possible integrability. Building new mechanics models with extended supersymmetry on curved spaces, analysis of their quantum properties, as well as the issues of their integrability and relationship with the matrix models of string theory. Study of the question of possible uses of the models constructed in nuclear physics and high-energy particle physics.
  
  Generalization, to the complex, quaternionic and projective spaces, of the known superintegrable oscillator-like systems allowing the inclusion of the constant magnetic/instanton external field, and further supersymmetrization of these generalized systems. Construction and study of the superintegrable versions of the oscillator models with extra Calogero-like potentials on complex/quaternion projective spaces, in interaction with the external constant magnetic/instanton fields, "weak" N=4 and N=8 supersymmetrization of such systems, finding out their superfield formulations. Construction of hyper-Kahler and quaternionic analogs of the Smorodinsky-Winternitz and Rosokhatius systems, as well as of their "weak" N=4 and N=8 supersymmetric extensions, analysis of their symmetries and finding their classical and quantum-mechanical solutions. Generalization to the Calogero-type systems.
  
  Construction of twistor formulations of particles and superparticles with a continuous spin (helicity), as well as their quantization in the component and superfield approaches.
  
  Further investigations of the properties of topological solitons in classical and quantum field theory in flat and curved space-time. Analogous analysis of the black hole solutions, as well as the localized field-configuration solutions in various versions of the gravitation theory coupled to the matter fields, including non-abelian gauge fields.
  
  Analysis of the quasi-classical limits of the three-point functions in the Liouville theory and its superextensions. Study of the light and heavy asymptotic limits in these theories. Clearing up the properties of the fusion matrix based upon the analysis of these limits, as well as of the relationship between the boundary three-point function and the fusion matrix. Study of the boundary three-point function in the heavy limit and its computing proceeding from the boundary Liouville theory defined on the solutions with three boundary singularities. Exhibiting the information about the monodromy of the solutions of the equations of Goin and Painleve VI by means of using the relationship of the conformal blocks with the solutions of these equations in the heavy asymptotic limits.
  
  
• Study of algebra-geometric structures related with the full symmetric Toda system based on the representation theory, inverse scattering method and other modern methods of studying integrable systems. Explanation of the full Toda system's integrability in terms of the Lie-algebraic approach, search for the reason for the existence of large (commutative and noncommutative) families of integrals of this system. Search for a general principle, behind the Bruhat order emerging in the phase portrait of this system, as well as search for its analogs in the infinite-dimensional limits of the system (the KdV system) and providing a complete description of the integrands in the case of degenerate orbits of the Toda system, in particular on RP(n).
  
  Study of stationary (black holes, black hole systems) and cosmological solutions (inflation, dark energy) in Einstein and modified gravitations of the Horndeski type and others. Study of the prospects for the application of the Palatiny formalism, which is characterized by a smaller number of singularities, in the construction of realistic cosmological models.
  
  Investigation of a subclass of the Stephani models with ideal gas and a matter-radiation mixture. Generalization of the model to the case with nonzero cosmological constant and calculation of observable parameters. Calculation of the probability for a black hole formation in the early universe at the dust stage from the growing density contrasts of the scalar inflaton field.
  
  Investigation of the vacuum energy in the boundary vicinity for CFTs. Computation of the entanglement entropy and pursuing the relation between the entropy and the geometry of the manifold and its boundary.
  
  The aim of the forthcoming research is to obtain new constraints on the parameters of black holes and neutron stars from the observational data acquired in 2019 by the Event Horizon Telescope and other observational facilities, as well as restrictions on the alternative theories of gravity.
  
  Investigation of the cosmological perturbations in covariant formulation of teleparallel gravity. Derivation of equations for scalar perturbations within this approach and obtaining the spectrum of scalar perturbations during inflation.
  
  

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