Proofs from the BOOK

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Talks:

The aim of the course is to enjoy the beauty of mathematical proofs in different areas: number theory, algebra, analysis etc. The main source of inspiration is THE BOOK by M. Aigner, G. Ziegler ``Proofs from the BOOK'', but as course go on we can find more interesting mathematical stories to tell. The idea is that students choose their favorite topic and give a presentation on it. Here is the list of the talks that we succeded to do this semester:

Date	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 pi
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	Godel’s incompleteness theorems. Part 1
7 May	Iago de Carvalho	Godel’s incompleteness theorems. Part 2
9 May	Iago de Carvalho	Godel’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 pi-lambda dinkin 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

Classes:

• March - June 2024. Tuesdays and Thursdays: 9am - 11am
• Room 863, Department of Mathematics, Edifício Leme, PUC-Rio

Materials:

• Books:
  • M. Aigner, G. Ziegler ‘‘Proofs from the BOOK’’
  • S. Tabachnikov, D. Fuchs ‘‘Mathematical Omnibus: Thirty Lectures on Classic Mathematics’’

Exercises:

• Homework:
  • List 1. Deadline: 1 of April 2024