Proofs from THE BOOK
Graduate/undergraduate course · PUC-Rio, March–June 2024 · Tuesdays and Thursdays 9–11am, room 863, Edifício Leme
The aim of the course is to enjoy the beauty of mathematical proofs in different areas: number theory, algebra, analysis and beyond. The main source of inspiration is the book by M. Aigner and G. Ziegler, Proofs from THE BOOK, complemented by other interesting mathematical stories. Students choose their favorite topic and give a presentation on it.
Materials
Books: M. Aigner, G. Ziegler, Proofs from the BOOK; S. Tabachnikov, D. Fuchs, Mathematical Omnibus: Thirty Lectures on Classic Mathematics.
Homework: List 1 (PDF).
Talks given in the course
| Date (2024) | Speaker | Title |
|---|---|---|
| 5 March | Yulia Petrova | Six proofs of the infinity of prime numbers |
| 7 March | Yulia Petrova | Bertrand's postulate |
| 12 March | Yulia Petrova | Buffon's needle, problem of the number of collisions of two blocks and π |
| 14 March | Sergio Andrade | Um pi real e um pi complexo: demonstração do Problema de Basileia e da Equação de Euler para Números Complexos |
| 19 March | Yulia Petrova | Every large point set has an obtuse angle |
| 21 March | Yulia Petrova | Steiner's simetrization and applications: isodiametric inequality, isoperimetric inequality and Brunn–Minkowski inequality |
| 26 March | João Arthur | Three applications of Euler's formula |
| 2 April | Sergio Andrade | Cauchy and Fourier proofs of the Basel problem |
| 4 April | Anselmo de Souza Pontes | Period 3 implies all other periods |
| 9 April | Sergio Andrade | Lemma Borel–Cantelli and Gamma function |
| 11 April | Anselmo de Souza Pontes | Period 3 implies chaos |
| 16 April | Bruno Baldissera | On Lemma of Littlewood and Offord |
| 18 April | Yulia Petrova | Discussion of problems from List 1 |
| 25 April | Carlos Henrique | Aproximações por racionais |
| 30 April | João Arthur | Representando números como soma de dois quadrados |
| 2 May | Iago de Carvalho | Gödel's incompleteness theorems. Part 1 |
| 7 May | Iago de Carvalho | Gödel's incompleteness theorems. Part 2 |
| 9 May | Iago de Carvalho | Gödel's incompleteness theorems. Part 3 |
| 14 May | Yulia Petrova | Shuffling cards |
| 16 May | Ivan Ogloblin | Every finite division ring is a field |
| 21 May | Bruno Baldissera | Sobre a conjectura de Borsuk |
| 23 May | Rony Geymar | O teorema π-λ de Dynkin e a unicidade da medida de Lebesgue |
| 28 May | João Arthur | Polinômios ciclotômicos e versões fracas de teoremas fortes |
| 18 June | Carlos Henrique | Números p-ádicos e o teorema de Ostrowski |
| 20 June | Carlos Henrique | Fórmulas de Euler em Teoria dos Números |
| 25 June | Ivan Ogloblin | Of friends and politicians |
| 27 June | Bruno Baldissera | A conjectura da sensitividade |
| 2 July | Rony Geymar | In praise of inequalities |
| 4 July | Rony Geymar | Fundamental theorem of algebra |