1. K-theory (K-theory of Banach algebras, C*-algebras, monoidal and polynomial algebras, spe­cial normed algebras, bivariant K-theory of Banach algebras and graded categories) [H. Inassaridze, T. Kandelaki, J. Gubeladze, T. Pirashvili, N. Inassaridze, D. Pataraia].

2. Homological and homotopical algebra (Non-abelian derived functors and satellites, co­ho­mo­logy of rings, coalgebras, monoids and categories, non-abelian homology and cohomology of groups and Lie algebras, Hochschild and Leibniz homology, cohomology of semimodules, Koszul algebras and monoid rings, cyclic homology) [H. Inassaridze, A. Pachkoria, T. Pirashvili, N. Inassaridze, T. Datuashvili, M. Jibladze, G. Rakviashvili, E. Khmaladze, G. Donadze, R. Kurdiani].

3. Galois theory of commutative rings and extension theory [G. Janelidze, B. Mesablishvili].

4. Algebraic geometry (toric varieties) [J. Gubeladze].

5. Category theory (Bicategories and internal categories, internal functors, Kan extensions) [G. Janelidze, B. Mesablishvili, D. Zangurashvili, Z. Omiadze, T. Datuashvili].

6. Mathematical logic [D. Pataraia, M. Jibladze].  

1. Homology and homotopy theory of fibrations: algebraic models of fibrations, obstruction theory [N. Berikashvili, T. Kadeishvili, S. Khazhomia, M. Mikiashvili, S. Saneblidze].

2. Homology and homotopy theory of general spaces: shape theory, cohomotopy type functors, latices [S. Khazhomia, L. Mdzinarishvili, Z. Todua].

3. Cobordism theory: self-conjugate and symplectic cobordisms, Adams-Novikov spectral se­quences [M. Bakuradze, G. Pruidze].

4. Applications of topology: Fredholm theories in operator algebras, loop groups techniques in transmission problem [G. Khimshiashvili].

5. Linear control theory: generalized linear dynamical systems [V. Lomadze].

6. Representation theory of Lie groups and algebras [A. Elashvili].

Two-sided estimates for measure of non-compactness, entropy and approximation numbers of a wide class of integral operators in various Banach function spaces [V. Kokilashvili, A. Meskhi].

2. Metric problems of Fourier series and integral operators arising in that theory. Sampling theory and function spaces [L. Ephremidze].

3. The problems of continuity and differentiability of functions of several variables. Application to the boundary properties of derivatives of the allied Poisson integrals on the unit ball of three-dimensional Euclidian space.

The concept of derivative of quaternion functions and its application [O. Dzagnidze].

4. Discontinuous boundary value (Riemann, Riemann-Hilbert) problems for analytic functions in “bad” domains with oscillating conjugation coefficients; Dirichlet and Neumann boundary value problems for certain classes of harmonic functions in the do­mains with non-smooth boundaries, involving the cusps. Mixed boundary value problems for harmonic functions of Smirnov classes.

Problems of Noetherity and index formulas for singular operators with the complex con­ju­ga­tion in the case of piecewise smooth curves with cusps.

 The behavior of derivatives of conformal mapping functions in the neighborhood of an­gular points in case of non-Jordan boundaries [E. Gordadze, G. Khuskivadze, V. Kokilashvili, V. Paatashvili, A. Saginashvili].

5. Ergodic theory: the fundamental inequalities for ergodic maximal functions and ergodic Hil­bert transforms. The estimations for non-increasing rearrangements of the ergodic integral trans­forms. The problem of uniqueness of ergodic maximal functions. Criteria of two-weighted ine­qu­a­li­ties for ergodic integral operators in weighted Banach function spaces. Application to the probability theory [L. Ephremidze].

6. Factorization of matrix-functions. Approximate factorization of spectral measures of station­nary processes [L. Ephremidze].

1. Linear and nonlinear boundary value problems for ordinary differential equations and sys­tems; asymptotic properties of solutions of nonautonomous differential equations; oscillation theory [I. Kiguradze, L. Kokilashvili, R. Koplatadze, G. Kvinikadze, S. Mukhigulashvili, N. Partsvania].

2. Linear and nonlinear hyperbolic and mixed type equations and systems, their general so­lu­tions and integrals; nonlinear variants of initial, boundary and characteristic problems [J. Gvazava, O. Jokhadze, S. Kharibegashvili, T. Kiguradze].

3. Modern methods of investigation of the symmetry in analytic mechanics [R. Sulikashvili].

4. Approximate solutions of functional equations by projective, projective-difference, dif­fe­rence and iteration methods [G. Berikelashvili, I. Bukhnikashvili].

1. Boundary value problems of classical elasticity: existence, uniqueness, regularity and asym­p­totic properties of solutions of static, oscillation and dynamic boundary value problems [T. Buchukuri, O. Chkadua, R. Duduchava].

2. Mixed, crack and screen type boundary value problems for elliptic systems with constant coefficients: solvability, regularity and asymptotics of solutions [O. Chkadua, R. Duduchava].

3. Boundary value problems for various models of continuous media: electroelasticity, ther­mo­elasticity, couple-stress (moment) elasticity, elastic mixtures, hydrodinamics (Navier-Stokes equa­tion) etc., unilateral, non-classical and ill-posed problems in the classical and the moment theory of elasticity [T. Buchukuri, O. Chkadua, A. Gachechiladze, R. Gachechiladze].

4. Pseudodifferential and singular integral equations on manifolds with and without boundary in various functional spaces. (Hölder, Besov, Bessel potential etc.) [R. Duduchava, D. Kapanadze].

5. One-dimensional singular and convolution operators and Banach algebras generated by such operators. Integral equations with fixed singularities in kernel and its applications to mechanics [R. Duduchava].

6. Approximate solutions of singular integral equations. Approximations of integrals with weak kernel [R. Duduchava].

1. Mixed problems of elasticity with moving boundary conditions, in particular, the problems of crack distribution with constant velocity in a piecewise homogeneous orthotropic semi-infinite plane and with varying velocity in homogeneous media [R. Bantsuri].

2. The contact problems of elasticity and bending of a plate with elastic inclusions of varying ri­gidity [N. Shavlakadze].

3. Some particular contact problems for inhomogeneous domains [O. Shinjikashvili].

4. Dynamical stability for rotary shells similar to cylindrical in case of periodical loads [S. Kukudzhanov].

5. The reverse problems for infinite anisotropic plate with unknown opening [G. Zhorzholiani].

6. The influence of wall permeability on the stability of flows between two rotating cylinders [L. Shapakidze].

7. One class of plane stationary problems of the theory of filtration and the methods for explicit solutions [A. Tsitskishvili].

1. Bound state problems in QFT based on the B-S and quasipotential equations for two and three particles [V. Garsevanishvili, A. Khvedelidze, A. Kvinikhidze].

2. The quark and parton models, QCD (Quantum Chromodynamics) (including nonperturbative approaches) [V. Garsevanishvili, V. Gogokhia, G. Jorjadze, A. Khvedelidze, A. Kvinikhidze, G. Lavrelashvili, B. Magradze, I. Sarishvili, A. Tavkhelidze].

3. Electromagnetic and weak interactions within strongly interacting systems, gauge invariance [V. Garsevanishvili, A. Kvinikhidze].

4. Nontrivial vacuum structure in the gauge theories, fermion number nonconservation and fini­te-temperature phase transitions [G. Lavrelashvili].

5. Quantization of the constrained systems, geometric quantization, gauge theories, gravity [G. Chechelashvili, Z. Giunashvili, G. Jorjadze, A. Khvedelidze, I. Sarishvili, A. Shurgaia].

6. Solitonic solutions, sphalerons, black-hole solutions and their quantum behaviour [G. Lavrelashvili, A. Shurgaia].

7. Chern-Simons theories and (2+1)-dimensional QFT [M. Eliashvili].

8. Low dimensional systems [M. Eliashvili, G. Tsitsishvili].

9. Ground state problem in QFT and Quantum statistics [M. Eliashvili, G. Tsitsishvili].

10. Baby Universe (Wormhole) physics, creation of particles in the tunneling processes [G. Lavrelashvili].

11. Relativistic nuclear physics [V. Garsevanishvili, A. Tavkhelidze].

Department of Theoretical Physics is organizer of regular international workshops (1990, 1991, 1996, 1998-2002 ), devoted to the modern problems of theoretical and mathematical physics.

1. Properties of distributions of sums of independent and conditionally independent random variables [T. Shervashidze, N. Gamkrelidze].

2. Semimartingale backward equations and mean-variance hedging [M. Mania].

3. Semimartingale characterization of generalized derivatives and application to options pricing [M. Mania].

4. Measure-valued solutions of second order stochastic parabolic equations [O. Purtukhia].

5. Robust estimation in filtered parametric statistical models [N. Lazrieva, T. Toronjadze].

6. Stochastic approximation and recursive estimation procedures for statistical models associated with semimartingales [N. Lazrieva, T. Toronjadze].

7. Functional and martingale methods in statistics [E. Khmaladze].

8. Empirical processes and goodness of fit theory [Z. Tsigroshvili].

9. Change point problem [R. Mnatsakanov].

10. Applications of statistics: biology and medicine, linguistics, actuarial and insurance [T. Toronjadze, Z. Tsigroshvili].