This is intended to be an elementary introduction to modern combinatorics with special attention paid to its applications in low-dimensional topology and field theory. No prerequisites.

1) R. Stanley, Enumerative combinatorics.
2) I. P. Goulden and D. M. Jackson, Enumerative combinatorics.
3) R. Graham, D. Knuth, O. Patashnik, Concrete mathematics.

1. Combinatorial structures and enumeration of basic objects.
2. Integer sequences and their properties.
3. Power series and generating functions.
4. Lagrange theorem and its applications.
5. Graphs and graph invariants.
6. Enumeration problems in graph theory. Cayley`s theorem.
7. Surfaces and graphs embedded into surfaces.
8. Enumeration problems for embedded graphs.

Traditionally programming is considered as something practically important but very boring, something like looking for errors in long unreadable non-working programs. However, it is quite possible to learn how to write short and elegant programs when they exist; the key idea is that you try to construct a program together with the proof of its correctness. We will try to show you this technique using many examples (from Dijkstra, Gries, and other books). At the same time we will learn some techniques for constructing effective (fast) algorithms.

1) Shen A. Algorithms and programming: theorems and problems, Birkhauser, 1997

1. Basic constructions. Variables, assignments, loops, invariant relations.
2. Arrays. General scheme of one-pass algorithms.
3. Generation of combinatorial objects. Tree traversal, backtracking.
4. Finite automata.
5. Sorting and related problems.
6. Data structures: stacks, queues, etc.
7. Sets and their representations. Hashing, trees, balanced trees.
8. Algorithms on graphs.
9. Recursion: how to use it and how to replace it by iteration.
10. Context-free grammars, recursive parsing.
11. LL(1), LR(1) parsers.

2. The objects of topology: topological and metric spaces, simplectic and cell spaces, manifolds. Topological constructions (product, disjoint union, wedge, cone, suspension, quotient spaces, cell spaces, examples of fiber bundles).

3. Examples of surfaces (2-manifolds), orientability, Euler characteristic. Classification of surfaces (geometric proof for triangulated surfaces).

4. Homotopy and homotopy equivalence, the homotopy groups π_n (.) for n > 1 and their main properties.

5. Vector fields on the plane. Generic singular points. The index of a plane vector field. Vector fields on surfaces. The Poincaré index theorem.

6. Infinite constructions (counterexamples to ``obvious" statements): the Cantor set, the Peano curve, the Brouwer continuum (as a strange attractor related to the equation z³ - 1 = 0 ), Antoine's necklace, and Alexander's horned sphere.

7. Curves in the plane, degree of a point with respect to a curve, Whitney index (winding number) of a curve, classification of immersions, the ``fundamental theorem of algebra". Degree of a map of a circle into itself. Brouwer fixed point theorem.

8. Fundamental group (main properties, simplest computations), covering spaces. Algebraic classification of covering spaces (via subgroups of the fundamental group of the base). Branched coverings, Riemann-Hurwitz theorem.

9. Knots and links in 3-space. Reidemeister moves. The Alexander-Conway polynomial.

Elementary Introduction to Geometric Group Theory

Geometric group theory is the study of groups from the geometric viewpoint. As J.Milnor teaches us, geometry exists both in the group itself and in the spaces it acts on. And this is part of what makes the subject so attractive.

1) J. P. Serre, Trees, Springer, 1980
2) P. Scott, T. Wall, Topological methods in group theory, London Math. Society Lecture Notes 36, Cambridge Univ. Press, 1979

1. Background from group and graph theories.
2. Word problem.
3. Cayley graphs.
4. The word problem and Cayley graphs.
5. Dehn's solution to the word problem for the hyperbolic surface groups.
6. Coxeter groups.
7. Tits' solution to the word problem for Coxeter groups.
8. Cayley graphs and group actions.
9. Milnor theorem.
10. Introduction to Gromov hyperbolicity.;

We present a course giving an overview of basic algebraic structures. It is an essential part of mathematical education in general; besides, our experience shows that it also provides motivation to specialize in algebra.

1) P. M. Cohn, Algebra, Vol. 1, Wiley
2) I. N. Herstein, Topics in Algebra, Wiley

1. Symmetric group.
2. Transformation groups. Burnside's formula.
3. Groups, subgroups, and quotient groups.
4. Free groups. Groups presented by generators and relators.
5. Rings and ideals.
6. Divisibility theory in Euclidean rings.
7. Fields. Complex numbers. Finite fields.
8. Algebras. Quaternions.
9. Matrix algebra. Vector spaces. Basis. Dimension.
10. Grassmann algebra. Determinant.

In this course we try to present the basic concepts of calculus, like real numbers, continuity, mean value theorems, etc., from the modern geometrical viewpoint.

Prerequisites: basic calculus, including integration.

1) R. Courant, F. John. Introduction to Calculus and Analysis, Springer-Verlag, 1988

1. Basics of general topology: topological spaces, continuity, connectivity.
2. Frequently used topological spaces: real numbers (a rigorous definition will be given), Euclidean spaces, spaces of functions, and more.
3. Connected spaces: mean value theorem and applications.
4. Compact spaces: main properties, characterisation of compact sets in various topological spaces, continuous maps between compact spaces, applications (among them various extremal properties, inequalities, etc.).
5. Metric and normed spaces: equivalence of norms in finite dimension, completeness and Hausdorff completion, contraction mapping principle, and applications (e.g., existence and uniqueness theorem for differential equations).
6. Spaces of functions: completeness of various spaces of functions. Conjugate spaces and distributions. Fourier expansion and applications.

Despite the fact that non-Euclidean geometry has found its use in numerous applications (the most striking example being 3-dimensional topology), it has retained a kind of exotic and romantic element. This course intends to be a businesslike introduction to non-Euclidean geometry for nonexperts.

1) I. S. Iversen, Hyperbolic geometry, Cambridge Univ. Press 1993
2) H. S. Coxeter, Non-Euclidean Geometry, Toronto Univ. Press, 1957

1. Prerequisites from linear algebra, point set topology, group theory, metric spaces.
2. Axioms for plane geometry.
3. The inversive models.
4. The hyperboloid and the Klein model.
5. The geometry of the sphere.
6. Some computations in the hyperbolic plane and on the sphere.
7. Hyperbolic isometries.
8. Convex polygons.
9. Isoperimetric inequality in non-Euclidean geometry.
10. Hyperbolic surfaces.

Ordinary differential equations gave rise to different branches of mathematics: they stimulated the development of calculus by Newton, the invention of topology by Poincare and motivated the study of Lie groups. On the other hand, the theory of differential equations is the field of application of all the disciplines mentioned above, with complex analysis and algebraic geometry added to this list. The present course follows the modern exposition given by Arnold in his textbook, 1985. This exposition is enriched by elements of the analytic theory of ODE and very first steps in the general theory of dynamical systems.

1) V. I. Arnold, Ordinary differential equations.

1. Processes described by ODE.
2. Examples from ecology, mechanics, electricity.
3. Elementary methods of integration and symmetries.
4. General theory: existence and uniqueness of solutions in real and complex domains.
5. Small parameter method.
6. Linear theory.
7. Phase flows of linear vector fields and exponentials of linear operators.
8. Linear equations of higher order.
9. Resonance.
10. Small amplitude oscillations.
11. Stability of singular and fixed points.
12. Limit cycles and their stability.
13. Poincare-Bendixson theorem.
14. The Smale horseshoe.
15. Linear equations in the complex domain.
16. Monodromy group.
17. Elements of Frobenius theory.