Evolutionary and integrodifferential equations, ordinary differential equations, asymptotic behavior

Calculus of variations and weak lower semicontinuity; Partial differential equations - existence of weak solutions; Applications in continuum mechanics - models of solids and their mathematical analysis

Mathematics education – conceptual understanding in learning mathematics (based on APOS theory), using technology in mathematics education (GeoGebra, Maple, etc.), preparing prospective mathematics teachers.

Partial differential equations - existence, regularity and stability theory; continuum thermodynamics; mathematical analysis and modelling of flows and deformation of materials with complicated rheology

numerical linear algebra, high performance matrix computations, parallel algorithms

nonseparable Banach spaces (and related topics in topology and set theory), nonlinear geometry of Banach spaces (most importantly study of ``Lipschitz-free Banach spaces'')

Analytic number theory, distribution of prime numbers, L-functions, multiple Dirichlet series.

numerical solution of partial differential equations with applications in fluid mechanics

Algebraic and combinatorial properties of binary systems (loops, quasigroups, left distributivity).

Boolean functions, finite fields and their applications in cryptography.

Symplectic and contact topology, low-dimensional topology, dynamical systems.

Geometric Function Theory, Mappings of finite distortion, properties of the Jacobian, Functions of several real variables, weak differentiability, approximation, Calculus of Variations, Function Spaces

Combinatorics of words, formalization of mathematics.

General topology (including uniform spaces and topological groups)

Mathematical physics, homological and homotopical methods in string theory and quantum field theory, higher algebraic and geometric stuctures and their applications in physics, BV quantization of gauge theories, generalized geometry

Number theory, universal quadratic forms, class numbers, generalized continued fractions, semifields.

Banach spaces - geometric and topological structures, quantitative versions of their properties, measures of weak non-compactness. Operator algebras and Jordan structures, mainly from the point of view of Banach space theory. Classes of nonseparable Banach spaces and related classes of compact spaces. Descriptive topology and compact convex sets.

Partial differential equations. Far from equilibrium open systems.

Finite element method and its application to the simulation of incompressible flows and to the solution of convection-dominated problems

Universal algebra and algorithms, constraint satisfaction problems.

Mathematical logic and proof complexity in particular.

Theory of sympletic Dirac operators. Hodge theory for elliptic complexes on Hilbert fibre bundles over compact manifolds. Application of Lie groups representation theory in differential geometry.

Numerical solution of partial differential equations, fluid dynamics, chaotic dynamics

Mathematical analysis, hypercomplex analysis. Applications of representation theory of Lie groups and superalgebras. Constructions of Gelfand-Tsetlin bases for polynomial solutions of invariant differential equations.

Analysis of nonlinear partial differential equations, in particular those describing mechanical, thermal and chemical processes in fluids, solids and mixtures. Thermodynamics and mechanics of non-Newtonian fluids.

Integrable partial differential equations: long-time asymptotic analysis of initial value problems with step-like initial data (modified Korteweg - de Vries equation, Camassa - Holm equation, nonlinear Schrödinger equation, Korteweg - de Vries equation, etc). Direct and inverse scattering transforms for non-decreasing and increasing potentials. Riemann - Hilbert problems and asymptotic methods for oscillatory Riemann-Hilbert problems. Further interests: orthogonal polynomials, Painlevé equations, random matrices.

My research focuses on the noncommutative geometry of quantum groups and their quantum homogeneous spaces, and in particular quantum flag manifolds. The approach used involves a mixture of Hopf algebras, monoidal categories, Lie theory, complex and Kähler geometry, C*-algebras, and unbounded operator theory.

Discrete and computational geometry, algebraic and topological combinatorics.

Function spaces, symmetrisation, rearrangement-invariant spaces, Orlicz spaces, Lorentz spaces, embeddings, compact embeddings, optimality, logarithmic Sobolev inequalities, infinite-dimensional analysis on Gaussian spaces, trace theorems, transfer of regularity from data to solutions in differential equations, optimal partnership of function spaces, interpolation theory, approximation theory, boundedness and compactness of operators, measure of noncompactness, supremum operators, integral operators, discretisation, weighted inequalities, elementary topics from analysis, elementary inequalities and estimates, recreational mathematics, history of mathematics, popularisation of mathematics, translation of books.

Real analysis (convex functions, generalized convexity), fractal geometry (fractal curvatures), integral geometry (curvatures for singular sets), other random topics (Tukey depth, stochastic processes)

Mathematical analysis of partial differential equations, particularly equations of mathematical fluid mechanics and thermodynamics. Existence of a solution, regularity, qualitative properties of solutions.

Partial differential equations, dynamical systems. Non-standard analysis, game theory.

Ring theory.

Continuum Theory, construction of spaces with given property. local properties of continua, homogeneity of continua.

(Geometric) measure theory, convex geometry, integral geometry (in particular, curvature measures of sets with singularities and their integral-geometric relations), stochastic geometry.

Mathematics Education - geometry education, uses of technology, teachers' professional vision

PDE, in particular hyperbolic systems of conservaion laws, numerical analysis of finite volume method, popularization of mathematics

Lattice theory, module theory.

Commutative algebra, homological algebra, derived categories.

Banach function spaces, Sobolev-type spaces, linear and multilinear multipliers, singular integral operators, maximal functions, weighted inequalities

Differential and algebraic geometry, Lie groups and algebras (classical, affine, super) and their representation theory, Homogeneous spaces (flag manifolds, (locally) symmetric spaces, reductive spaces), Homogeneous vector bundles and their equivariant homomorphisms, Finite reflection groups and their geometry, Homotopy and (co)homology theories (equivariant spectra), Quantum groups and non-commutative geometry (quantum homogeneous spaces.)

Invariant differential operators on manifolds with a fixed geometric structure. Generalized Cartan geometries, in particular, parabolic geometries. Clifford analysis in one or several variables. Applications of representation theory in analysis. Exact complexes of invariant differential operators.

Mathematical modelling and numerical computations in terrestrial and planetary geophysics (tidally-induced deformation, dissipation and transport processes in the interiors of icy moons Europa and Enceladus, numerical modelling of evolution of ice sheets on Earth and on Mars); Thermodynamics and mechanics of continua (constitutive theory for complex materials, thermodynamics and mechanics of continua on surfaces, thermodynamic modelling of boundary and interface conditions); Theory of multi-component materials (heterogeneous catalysis, porous media flow, partial melting and melt transport)

Integral representation of convex sets; Choquet theory; Banach spaces and algebras; operator spaces and their geometrical and topological properties.

Non-associative algebraic structures, self-distributivity and algebraic invariants of knots, universal algebra, automated theorem proving.

Set theoretic aspect of module theory.

Representation theory of associative algebras, homological algebra, connections to homotopy theory and algebraic geometry.

Development and analysis of methods for the solution of problems of numerical linear algebra and matrix computations, and their application in approximation of functions, numerical optimization and high-performance computing.

Lattice representation theory, applications of algebra in computer science.

solution of large sparse systems of linear equations by direct and preconditioned iterative methods

I am interested in the behavior of complex materials, both fluids, and solids that are dissipating energy. For example non-Newtonian fluids with complicated rheology such as viscoelastic fluids or shape memory alloys undergoing a martensitic transformation. I perform numerical simulations of these models using the finite element method with applications in related areas.

General topology, continuum theory, Polish spaces, Borel reductions, topological dynamical systems

Descriptive set theory. Real and harmonic analysis.

Structural ring theory. Modules and Abelian groups. Abelian categories.