Victor Kleptsyn

Victor Kleptsyn is a researcher at CNRS, in the Institute of Mathematical Research of Rennes. His working themes are mainly Dynamical Systems and Geometry. His belief is that most arguments, theorems, and proofs in the mathematics should be visual, and easily explicable, at least on the "why should it be true" level of explanation.

 

"Asymptotic problems in combinatorics": There are many interesting problems in the combinatorics that are stated as "how does a random big object look like" or as "how many are there of these objects". For instance, in a randomly chosen sequence of zeros and ones of big length n, the proportions of zeros and ones are most probably close to one half; in a randomly chosen permutation of n elements most probably there is a "large" cycle (of length comparable to n). It turns out that the problems "count the objects" and "find the limit properties of a random object" are often related. We will consider (only with handwaving arguments) some problems where such a link appears:

  • Partitions of a large number n into a sum of non-increasing numbers (or, what is the same, Young diagrams of size n. How does a typical partition look like? What is the number of such partitions (Hardy-Ramanujan formula)?
  • What is the affine length of a curve, and what is the typical shape of a convex broken line, going from (0,1) to (1,0) inside the unit square, if its vertices are restricted to stay on a lattice with step 1/ n? (Bárány-Vershik, Sinai)

If the time permits, we will also discuss the questions related to the domino tilings, mentioning the well-known "arctic circle" theorem.