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"Geometric Foundations of Analysis", Spring 2016.
The course description. The purpose of the between basic courses such as Real Analysis, advanced courses such as Topology, Differential are going to introduce concepts of topological geometrical properties in that broad context. course is to construct a Real Multivariable Anal Geometry, Calculus on and metric spaces and natural bridge ysis and more Manifolds. We to study main

List of recommended textbooks. [R] W.Rudin. Principles of mathematical analysis. [T] T. Tao, Analysis. [M] J.Munkres. Topology. Program of the course. 1. A rigorous definition of real numbers using Dedecind cuts. Complex numbers, Euclidean spaces, spaces of functions. 2. Introductory notions of topology: topological spaces, open and closed sets, continuous functions. Frequently used topological spaces. 3. Metric spaces. Converging sequences, Cauchy sequences in metric spaces. Complete metric spaces. Contraction mapping principle. 4. Connected and path-connected topological and metric spaces. Intermediate value theorem. 5. Compact sets in metric spaces, various equivalent definitions. Compactness, continuity and extremal properties. 6. Mean value theorem. L'Hospital's rule. Taylor's rule. 7. Sequences of functions. Uniform convergence, differentiation and integration. The Stone-Weierstrass theorem. 8. Power series, Fourier series. 9. Functions of several variables. The inverse function theorem, the implicit function theorem.

Grading policy. During the semester you will get 11 homework assignements, each consists of 5 problems. For a correct solution of every problem you get 2 points. Thus, for all homeworks you may get up to 110 points. Occasionally bonus problems (for 3 points each) might be added to homeworks. The due date for each homework is the next


class. For the semester you can use two homeworks extensions, each for one week. Both the midterm and the final exam are in-class, open books, open notes. The midterm contains 5 problems for 4 points each, the final ­ 5 problems for 6 points each. Also you get 1 bonus point every time you are the first to correct the professor's mistake/mistype in class.The grading table has the following form:

A+ A AB+ B BC+ C CD

140 130 120 110 100 90 80 70 60 50

Contact information. mathinmoscow (at) gmail.com, Skype: mathinmoscow.