## Mathematical analysis 3, NMAI056, summer term 2022/23

I will closely follow
the lectures
I gave a year ago but I will update them and will translate them in English.

Exam: The exam questions are given at the end of the last 13th lecture below.

lecture 1, February 15, 2023 Metric spaces. Hemisphere is not flat. p-adic ultrametrics

lecture 2, February 22, 2023 Ostrowski's theorem. Compact metric spaces

lecture 3, March 1, 2023 Continuity and compactness. The Heine--Borel theorem.
connectedness. FTAlg

lecture 4, March 8, 2023 The proof of FTAlg. Complete spaces. Baire's theorem

lecture 5, March 15, 2023 A proof of weak Jordan's (Circuit)
Theorem a la Thomassen

lecture 6, March 22, 2023 (updated March 22 - the proof of completeness of the MS C(I)
corrected) Applications of Baire's theorem:
non-differentiable continuous function, transcendental growth rates
of permutation classes

lecture 7, March 29, 2023 Solving the Basel problem by
Fourier series

lecture 8, April 5, 2023 G. Pólya's 1921 theorem on random walks
in **Z**^d

lecture 9, April 12, 2023 Introduction to complex analysis 1

lecture 10, April 19, 2023 Introduction to complex analysis 2

lecture 11, April 26, 2023 Introduction to complex analysis 3

lecture 12, May 3, 2023 Existence theorems for solutions of
differential equations: Picard's and Peano's

lecture 13, May 17, 2023 Some particular differential equations

May 2023