Mathematical analysis 3, NMAI056, summer term 2025/26
I will closely follow the lectures I gave a year ago but sometimes will update them.
Exam questions are the 12 questions given at the end of the last 13th lecture. Zápočet (credits for the tutorials)
Five HWs are assigned at the end of each lecture, except the last one. Please, send me by e-mail their solutions by the end
of the coming Sunday. Solutions will be discussed on the tutorial on Tuesday. You get credit for solving (in the sense
I explained in the tutorial) at least 3/5 of the HWs.
Lecture 1, February 16, 2026 Metric spaces. Hemisphere is not flat. p-adic ultrametrics
Lecture 2, February 23, 2026 Ostrowski's theorem. Compact metric spaces
Lecture 3, March 2, 2026 Continuity and compactness. The Heine - Borel theorem.
Connectedness
Lecture 4, March 9, 2026 The proof of FTA. Complete spaces. Baire's theorem
Lecture 5, March 16, 2026 Thomassen's proof of the Weak Jordan Theorem. Update:
this is a detailed write-up for Thomassen's proof and Filippov's proof of
the WJT
Lecture 6, March 23, 2026 Applications of Baire's theorem: non-differentiable functions
and growth rates of permutations
Lecture 7, March 30, 2026 The Basel problem and Fourier series
April 6, 2026 - no lecture, Easter Monday
Lecture 8, April 13, 2026 Polya's theorem on random walks in Z^d via generating functions
Lecture 9, April 20, 2026 Introduction to complex analysis 1
Lecture 10, April 27, 2026 Introduction to complex analysis 2
Lecture 11, May 4, 2026 Introduction to complex analysis 3. Update:
this will be a detailed write-up for the proofs of the Cauchy - Goursat
theorem (for rectangles), of Liouville's theorem and of the Analyticity Theorem (not just for entire functions
but for arbitrary ball domain)
Lecture 12, May 11, 2026 Differential equations 1: Picard's and Peano's theorems
Lecture 13, May 18, 2026 Differential equations 2: examples of ODE and PDE.
ODE with separated variables and the 1st order linear ODE
May 2026