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Modern Mathematical Physics

• A structure of cotangent bundle over quantum linear groups and its extensions by "spectral variables" will be investigated. The corresponding dynamical R-matrices will be calculated. These results will be applied to the solution of the model of isotropic q-Euler top.
  
  The stochastic "Raise and Peel Model" (RPM) with nontrivial boundary conditions will be investigated by combinatorial methods. Correlation functions (e.g., a stationary distribution of the number of clusters and their sizes) for this model will be calculated. "Minimal" polynomial solutions of the quantum Knizhnik-Zamolodchikov equation will be constructed and applied for a description of the stationary regime of the RPM type models and of the quantum XXZ Heisenberg spin chains.
  
  Method of projections will be applied for gl_N current algebras to obtain and invstigate the universal Bethe equations for nested Bethe ansatz. It will be shown that these equations can be obtained from the condition of the commutativity of the universal transfer matrix and generalized Bethe vectors.
  
  Matrix elements of the local spin operators in the generalized Baxter-Bazhanov-Stroganov model will be calculated on the eigen-vectors of the commuting integrals of motion in this model and obtained by the separation of the variable method. This will allow one to investigate the correlation functions of the local operators in Z_N-symmetric models and in the finite volume.
  
  For classical mechanical systems with compact phase space an appropriate complexification of the Bohr-Sommerfeld condition will be found. In the case of integrable systems this complexified Bohr-Sommerfeld condition will be used for definition of analogs of spectral curves. A new type of integrable systems whose phase spaces are presented by symplectic Fano manifolds will be studied.
  
  
• New models of higher N supersymmetric and superconformal mechanics will be constructed and used for analyzing the structure of supersymmetric gauge theories in D=4 and D=6, including those with higher derivatives.
  
  New superparticle models associated with the higher spin theory will be constructed and studied.
  
  The phenomenon of partial supersymmetry breaking in models formulated in sueprspaces with non-anticommuting Grassmann coordiantes will be explored.
  
  
• Development of the mathematical tools for internal and external description of open systems in the framework of the spectral approach to the quantum field theory and employment of this formalism in calculating vacuum effects.
  
  Development of dimensional reduction in gravitation theory by introducing the geometrical potentials, separation of variables and by making use of the static-cosmological duality.
  
  

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