Intriguing Problems from Combinatorics, Algebra, Analysis and Probability

This lecture is addressed to a wide audience: from graduate students in Mathematical Sciences or related areas, to PhD students, young researchers and professionals. It will be shown that diverse and intriguing mathematical problems from the above-mentioned areas can be solved quite elegantly by using ideas and techniques from Probability. Sometimes these are the only available solutions.

Some of the following topics will be discussed in detail:

• Combinatorial and algebraic identities.
• A problem involving two dice – fair or unfair?
• Toss 15 dice: if your sum equals your product, you win 1 million! Ready to play?
• Bernoulli’s LLN and Weierstrass’s theorem by Bernstein’s polynomials.
• Old Uspensky’s problem and its far-reaching extensions.
• The Buridan Donkey story. Random walks in a random environment. Markov chains.
• Many ways to interpret and solve the equation X + Y = XY.
• Values of the Riemann zeta function via Cauchy’s distribution.
• A few exercises and one unsolved problem.