This course has three main parts, i.e., tensor algebra, representation theory of finite groups, and Galois theory.
Prerequisites: Basic Algebra and Linear Algebra.
Books:
1) P. M. Cohn, Algebra, Vol. 1, Wiley
2) I. N. Herstein, Topics in Algebra, Wiley
3) E. Artin, Galois Theory
4) C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras

1. Tensor product. Its universality. Canonical isomorphisms.
2. Coordinates of a tensor and their transformations under change of variables.
3. Tensor algebra. Symmetric and exterior algebras. Symmetric and skew-symmetric tensors.
4. Simple groups. Solvable groups. Sylow's theorems.
5. Representations. Irreducible representations. Schur's lemma.
6. Semisimple algebras and modules.
7. Group algebra. The Maschke theorem.
8. Theory of characters.
9. Algebraic extensions of fields.
10. Galois extensions. Galois correspondence.

In this course we present the basic concepts of differential geometry (metric, curvature, connection, etc.). The main goal of our study is a deeper understanding of the geometrical meaning of all notions and theorems.

1) B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern geometry - methods and applications. Part I, Springer-Verlag, New York, 1992

1. Plane and space curves. Curvature, torsion, Frenet frame. Curves in pseudo-Euclidean spaces.
2. Surfaces in 3-space. Metrics and the second quadratic form. Curvature. `Theorema egregium' of Gauss.
3. Parallel translations. Gauss-Bonnet formulas.
4. Fibrations. Topological connections as parallel translations. Curvature of a topological connection. Frobenius criterion.
5. Vector bundle. Tangent, cotangent, and tensor bundles. Sections.
6. Differential forms on manifolds. Integrals of differential forms. Stokes' theorem.
7. Connections as covariant derivatives. Curvature and torsion tensors.
8. Riemannian manifolds. Symmetries of the curvature tensor. Geodesics. Extremal properties of geodesics.
9. Sard's lemma. Transversality theorem. Applications.

The most important calculus is calculus on manifolds. It explains how to make sense of computations, and to make the computations themselves easy. The main concepts and ideas in this theory are intrinsic, i.e., they are independent of the choice of coordinates. That is why modern math speaks the language of manifolds. The course leads directly to de Rham cohomology and the Stokes formula, which is one of the cornerstones of the theory.

Prerequisites: one-variable calculus (differentiability, derivative, and integration) and linear algebra (vectors and linear forms, linear transformations and determinant).

1) S. Lang, Calculus of Several Variables, Springer, 1987
2) S. Lang, Differential manifolds, Springer, 1985
3) L. Bers, Calculus, Holt, Reinhart and Winston Inc., 1978
4) R. C. Buck, Advanced Calculus, McGraw-Hill Inc., 1978

1. Definition and examples of smooth manifolds.
2. Orientability and orientation.
3. Tangent vectors and tangent space to a manifold at a point. Tangent bundles. Vector fields.
4. Skew-symmetric forms on linear spaces. Wedge product.
5. Differential forms on manifolds. Exterior differential.
6. Smooth maps of manifolds. Diffeomorphisms. The transformation rule under coordinate change for functions, vector fields and differential forms.
7. Integration in . Coordinate change in the integral. Integration of differential forms. Stokes theorem for a cube in .
8. Integration on manifolds.
9. Manifold with boundary. Induced orientation of the boundary.
10. General Stokes theorem. Green's formula, Gauss-Ostrogradskii divergence theorem, Stokes formula for a surface in .
11. Closed and exact forms. The Poincare lemma. De Rham cohomology.

The course is devoted to the theory of functions of one complex variable. This is a necessary part of any serious mathematical education. Complex analysis is an important part of classical and contemporary mathematics and its application to physics and engineering. It is the basis of many other mathematical disciplines.

1) J. B. Conway, Functions of one complex variable, Springer-Verlag, New-York, 1986

1. Complex-valued and holomorphic functions.
2. Cauchy theorem.
3. Integral Cauchy formula.
4. Taylor series and holomorphness test.
5. Laurent series and singular points.
6. Residues and the argument principle.
7. Topological properties of holomorphic functions.
8. Compact families of holomorphic functions.
9. Hurwitz theorem and one-sheeted functions.
10. Analytic continuation.
11. Riemann's theorem.
12. Riemann surfaces and Fuchsian groups.
13. Moduli spaces of complex tori.
14. Analytic functions and algebraic curves.

The Ergodic Theory of Dynamical systems is a branch of the theory of Differential Equations. It was founded as a separate discipline by Poincare, and developed by Birkhoff and Smale. Substancial contributions were made by mathematicians of the Moscow school, e.g. Kolmogorov, Arnold, Alekseev, Anosov, Sinai, and others. The proposed course is an introduction to the subject. It contains a survey of the field. The basic theorems are proved; the ideas of the proofs of more difficult results are explained without technical details.

Prerequisites: introductory course to Ordinary Differential Equations.

1) P. Billingsley. Ergodic theory and information, John Wiley and Sons, Inc., NY, 1965.
2) M. Blank. Ergodic theory of noninvertible transformations, MCCME, Moscow, 2004.
3) B. Hasselblat, A. Katok. Introduction to the modern theory of dynamical systems, Cambridge Univ. Press.
4) I.P. Kornfeld, S.V. Fomin, Ya.G. Sinai. Ergodic theory, M.:Nauka, 1980. 382 pp. (Springer Verlag, 1982)


1. Dynamical systems: trajectories, simple and strange attractors, chaoticity
2. Action on measures, notion of a transfer operator, invariant measures
3. Ergodicity
4. Birkhoff ergodic theorem
5. Mixing
6. Main ergodic constructions:
-- Direct and skew products
-- Poincare and integral maps
-- Natural extension and noninvertibility
7. Entropy: metric and topological approaches
8. Sinai-Bowen-Ruelle and natural/observable measures
9. Direct operator formalism:
-- Banach spaces of signed measures
-- Ionescu-Tulcea and Marinescu ergodic theorem
-- Random perturbations
10. Spectral theory for Koopman and transfer operators
11. Approximation by finite rank operators
12. Multicomponent systems: phase transitions
13. Mathematical background of numerical simulations

We present the basics of number theory emphasizing the striking similarity between the properties of usual integers and those of polynomials over a finite field. This leads to understanding that number theory and algebraic geometry is in fact one and the same domain. The core of number theory being an elementary problem, we stick to concrete examples.

1) K. Ireland and M. Rosen, A classical introduction to modern number theory. Springer, 1982.

1. What is a number theory problem?
2. Rings of residues, finite fields
3. Integers and polynomials
4. Quadratic fields
5. Global fields
6. Prime decomposition and class group
7. Sign and units
8. Zeta-functions
9. Arithmetic of curves

The emphasis of the course is on the interconnections of modern topology with other branches of mathematics and on concrete topological spaces (manifolds, vector bundles) rather than the most general abstract categories.

1. Chain complexes, cycles, boundaries, and homology groups. Polyhedra, triangulations, and simplicial homology groups of topological spaces. Betti numbers and Euler characteristic. Homology groups of classical surfaces: sphere, torus, Klein bottle, projective plane.

2. Cell spaces or CW-complexes. Cell chains, cycles, boundaries, and incidence coefficients. Cell homology groups and their coincidence with simplicial homology groups (without proof). Examples: multidimensional spheres, tori and projective spaces.

3. The long exact sequence associated with a short sequence of chain complexes. Relative homology groups. The exact sequence of a pair of topological spaces. Mayer--Vietoris exact sequence. Computing the homology groups of some topological spaces.

4. Main topological constructions: product, quotient space, cone, wedge, suspension, join, loop space and their homology groups.

5. Homotopy. Classification of mappings from a circle to itself and their degrees or rotation numbers. The index of an isolated singular point of a plane vector field. Poincaré index theorem: the sum of the indices of a vector field on a surface is equal to its Euler characteristic. Brushing a sphere. Brouwer fixed point theorem.

6. Singular homology groups of topological spaces. Homology groups of a point. Homotopy invariance of singular homology groups. Exact sequences of pairs and triples. Homology groups of spheres. The coincidence of singular, cell, and simplicial homology groups.

7. Intersection of submanifolds and cycles. Homological interpretation of the index of a vector field. Lefschetz fixed point theorem and its applications.

8. Cohomology groups and Poincaré duality. De Rham cohomology groups. Multiplication of cocycles and its applications.

Algebraic Geometry: start-up course

Algebraic geometry studies geometric loci defined by polynomial equations, for example the complex plane curve f(x,y)=0. It plays an important role at both elementary and sophisticated levels in many areas of mathematics and theoretical physics, and provides the most visual and elegant tools to express all aspects of the interaction between different branches of mathematical knowledge. The course gives the flavor of the subject by presenting examples and applications of the ideas of algebraic geometry, as well as a first discussion of its technical apparatus.

Prerequisites: basic linear and multilinear algebra (tensor products, polylinear maps), basic ideas of commutative algebra (polynomial rings and their ideals). Some experience in geometry and topology (projective spaces, metric and topological spaces, simplicial complexes and homology groups) is desirable but not essential.

1) C. H. Clemens, A scrapbook of complex curve theory, Plenum Press, 1980
2) M. Reid, Undergraduate algebraic geometry, CUP, 1988
3) I. R. Shafarevich, Basic algebraic geometry, Vol 1, Springer 1994
4) J. G. Semple and L. Roth, Introduction to algebraic geometry, OUP, 1986

1. Projective spaces.
2. Projective conics and PGL(2)
3. Geometry of projective quadrics. Spaces of quadrics
4. Grassmannians.
5. Examples of projective maps: Pluecker, Segre, Veronese.
6. Integer ring extensions, polynomial ideals, affine algebraic geometry and Hilbert's theorems.
7. Algebraic varieties, Zarisky topology, schemes, geometry of ring homomorphisms.
8. Irreducible varieties. Dimension.
9. Plane projective algebraic curves: point multiplicities, intersection numbers, Bezout's theorem.
10. Plane projective algebraic curves: singularities, duality, Pluecker formulas.
11. Rational curves. Veronese curve. Cubic curves.
12. Curves on surfaces. The 27 lines on a smooth cubic surface.
13. Vector bundles and their section sheaves. Vector bundles on the projective line.
14. Linear systems and invertible sheaves, the Picard group, line bundles on affine and projective spaces.
15. Tangent, cotangent, normal and conormal bundles. The Euler exact sequence.
16. Singularities and tangent cone. Blow up.
17. Complex projective curves: canonical class, genus, Serre duality and Riemann-Roch theorem.
18. If time allows: Ponselet's porism; quadrics through a canonical curve; Klebsh and Luroth problems, and so on.

Group symmetry is one of the most fundamental geometrical properties of the real world. It suffices to say that three of the four fundamental types of physical interactions (namely, electromagnetic, weak, and strong) are controlled by symmetries of the first three special unitary groups: U(1), SU(2), and SU(3), respectively. The representation theory of these groups and of some related structures will be considered in the proposed course.

1) W. Fulton, J. Harris. Representation Theory. A first course, Springer-Verlag, 1991

1. Representations of finite groups:
---Schur lemma
---complete reducibility
---1-st and 2-d Burnside theorems
---characters and equivalence

2. Representations of the special unitary group:
---Peter-Weyl theorem
---Weyl formula for characters
---representations of SU(2)
---the hydrogen atom
---quantum numbers and the periodic system of elements

3. Representations of the Lie algebra sl(2,C):
---finite-dimensional irreducible modules
---Casimir operator

4. Additional subjectmatter:
---Representations of the simplest Kac-Moody algebra and combinatorial analysis.

A problem could be algorithmically unsolvable. On the other hand, a theoretically solvable problem could require so much time to be solved that from the practical viewpoint it does not differ from an unsolvable one. Only if a fast algorithm exists, can the problem be regarded as practically solvable, or feasible. Sometimes it is hard to make a distinction between these possibilities, but the theory of computation helps us in many important cases. We will try to explain the main results of this theory (computational models, undecidability, complexity classes P and NP, cryptography applications, etc.)

1) M. Sipser, Theory of Computation

1. General recursion theory. Unsolvable problems.
2. Decidable and enumerable sets.
3. Computation models. Turing machines and associative calculi.
4. Computation time and space for different models.
5. Uniform and nonuniform complexity. Circuits: size and depth.
6. Diagonal construction, hierarchy theorems.
7. Polynomial time, the class NP.
8. Universal NP-problems: examples.
9. Polynomial hierarchy. PSPACE. Universal PSPACE problems.
10. Games, PSPACE and EXPTIME.
11. Randomized algorithms. BPP. Primality testing.
12. Interactive proofs. IP=PSPACE.
13. Communication complexity.

This intermediate course is designed as a comprehensive introduction to one of the most active and attractive fields of contemporary mathematics with a remarkable range of applications to Lie groups, geometry, physics and coding theory.

The prerequisites needed for this course are primarily : basic algebra, linear algebra, elementary hyperbolic geometry and an interest in learning new things by solving a variety of problems. All we need can be found in
1) S.Lang, Undergraduate algebra, Springer 1989 (first three chapters)
2) S.Lang, Algebra,Addis.-Wesley ,1970 (chapters 13,14)
3) H.Coxeter, Introduction to geometry, Wiley&Sons, 1970

1. Discrete transformation groups.Fundamental domain. Quotients by discrete groups.
2. Cocompact discrete groups. Some instructive classical examples.
3. Discrete reflection groups in spherical,euclidean and hyperbolic geometries.
4. Polytopes with nonobtuse angles. Coxeter polytopes as fundamental domains.
5. Classification of the finite reflection groups.
6. Abstract Coxeter groups and their braid -friends.
7. Basics on the hyperbolic geometry in the Klein model.
8. Finite Coxeter polytopes in hyperbolic space and associated cocompact discrete reflection groups.
9. Circle packings and Andreev`s theorem (after Thurston)
10. Any cocompact reflection group is a Coxeter group.
11. Tits representation .
12. The root system attached to any Coxeter group.
13. Classification of finite and affine root systems.
14. Combinatorial theory of Coxeter groups.