Discrete mathematics seminar in UZH - talks given in Spring Semester 2015

• May 21st: Distances in percolation and TASEP, Lucas Gerin (École polytechnique). .

The aim of this talk is to describe a connection between the geometry of 2d percolation (each edge of the square lattice lattice is removed with equal probability) and the discrete-time TASEP, a 1d particle system modeling non-equilibrium phenomena.

(joint work with A.L.Basdevant, N.Enriquez, J.B.Gouere).

• May 12th: Hook length formulae and difference operators, Huan Xiong (UZH). .

Motivated by Nekrasov-Okounkov formula on hook lengths, Han conjectured that $\frac{1}{n!}\sum_{|\lambda|= n} f_\lambda^2\sum_{\square\in\lambda}h_{\square}^{2k}$ is always a polynomial of $$n$$ for any $$k\in \mathbb{N},$$ where $$h_{\square}$$ denotes the hook length of the box $$\square$$ in the partition $$\lambda$$ and $$f_\lambda$$ denotes the number of standard Young tableaux of shape $$\lambda$$. This conjecture was generalized and proved by R. Stanley (Ramanujan J., 23, 91--105, 2010).

In this talk, we introduce two kinds of difference operators defined on functions of partitions and study their properties. As an application, we obtain a formula to compute $\frac{1}{(n+|\mu|)!}\sum_{|\lambda/\mu|= n}f_{\lambda} f_{\lambda/\mu}F(h_{\square}^2: {\square}\in\lambda)$ and therefore show that it is indeed a polynomial of $$n$$, where $$\mu$$ is any given partition, $$F$$ is any symmetric function, and $$f_{\lambda/\mu}$$ denotes the number of standard Young tableaux of shape $$\lambda/\mu$$. Our formulae could lead to many classical results on partitions, including marked hook formula, Han-Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula.

This is a joint work with Guo-Niu Han(CNRS, Strasbourg).

• April 28th: Explicit formulas of a generalized Ramanujan sum, Patrick Kühn (UZH). .

Explicit formulas are often used in analytic number theory to describe the behaviour of the non-trivial zeros of the Riemann zeta-function and their connection to some well-known arithmetic functions and, in particular, prime numbers.

In this talk, results of some explicit formulas involving a generalized Ramanujan sum are derived. Moreover, an analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown.

• March 31st: Generalisation of two theorems of Andrews to overpartitions, Jehanne Dousse (LIAFA, Université Paris Diderot). .

A partition of a positive integer n is a nonincreasing sequence of positive integers (called parts) whose sum is n. The Rogers-Ramanujan identities state that for every n, the number of partitions of n into parts congruent to 1 or 4 modulo 5 is equal to the number of partitions of n such that the difference between two consecutive parts is at least 2. More generally, Rogers-Ramanujan type identities establish equalities between certain types of partitions with congruence conditions and partitions with difference conditions. In 1968 and 1969, Andrews proved two very general Rogers-Ramanujan type identities which generalise Schur's theorem. Those identitites went on to become important theorems in the theory of partitions, with applications in combinatorics, representation theory and quantum algebra.

In this talk, we will show that we can generalise Andrews' identities to overpartitions (partitions in which the last occurence of a part may be overlined) using a new technique which consists in going back and forth between q-difference equations on generating functions and recurrence equations on their coefficients.

• March 24th: Reduced words of Coxeter groups, Philippe Nadeau (CNRS, Université Lyon I). .

Coxeter groups form a class of groups generated by involutions which occur naturally in various areas of mathematics. Given an element $$w$$ in such a group $$W$$, of particular importance are its reduced expressions (or words) , i.e. its factorizations $$w=s_1\ldots s_k$$ with $$k$$ minimal, where all $$s_i$$ belong to the finite set of generating involutions.

The set of reduced words for a given element has an intricate combinatorial structure, and I'll first survey old and new results about it. I will talk about work in progress with C. Hohlweg and N. Williams around the Brink-Howlett automaton for $$W$$, which recognizes the set of all reduced words in $$W$$. I will also explain how to construct such an automaton for reduced words of the subset $$W^{fc}\subseteq W$$ of so-called fully commutative elements of $$W$$.

• March 3rd: Random geometric graphs on compact symmetric spaces of rank one, Pierre-Loïc Méliot (Université Paris-Sud). .

Let $$X$$ be a compact metric space. We choose $$N$$ points $$v_1,\dots,v_N$$ at random on $$X$$, independently and according to a probability measure $$m$$ on $$X$$; and we connect two points $$v_i$$ and $$v_j$$ if they are separated by a distance smaller than some level $$D$$. The resulting graph is called the random geometric graph of level $$D$$ and with $$N$$ points on $$X$$.

We shall study the spectrum of the random adjacency matrix of this graph, and prove limit theorems for it (law of large numbers, central limit theorem) when $$X$$ is a sufficiently “nice” manifold, namely, a two-point homogeneous compact Riemannian manifold. The computations will rely on the representation theory of the isometry group of $$X$$, and on the manipulation of so-called spherical functions, which we shall recall and explain in details.

• February 24th: Dual combinatorics of Jack polynomials, Valentin Féray (UZH). .

Jack polynomials are some one-parameter deformation of Schur functions, a well-known basis of the symmetric function ring. They have been introduced in 1970 and have been widely studied since.

Here, we follow a more recent approach, due to Michel Lassalle. This approach extends some work of Kerov and Olshanski on representation theory of symmetric groups. We will present this approach, state some conjecture also due to Michel Lassalle, explained some solved cases and finally present some lead to tackle the general case. This involves the combinatorics of graphs embedded in (oriented and non-oriented) surfaces.

Thank you to all speakers and attendees !

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