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MOSCOW CENTER FOR CONTINUOUS MATHEMATICAL EDUCATION FOR UNDERGRADUATES AND GRADUATE STUDENTS |
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Combinatorics
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.
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) C. Kosniowski, First course in algebraic topology, Cambridge Univ. Press, 1980 2)E. H. Spanier, Algebraic topology, McGraw-Hill, 1966 1. The language of topology. Continuity, homeomorphism, compactness for subsets of Rn (from the epsilon-delta language to the language of neighborhoods and coverings). 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 πn (.) 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.
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.
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.
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.
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.
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