Maps decorated by the Ising model are a remarkable instance of a model of non-uniform maps with very nice enumerative properties. In this talk, I will first explain how one can obtain a differential equation for the generating function of Ising-decorated cubic maps in arbitrary genus, related to the Kadomtsev--Petviashvili (KP) hierarchy. In particular, this leads to an efficient algorithm to enumerate Ising cubic maps in high genus. I will also present and compare implementations of this algorithm in Maple and SageMath. This is based on a joint work with Mireille Bousquet-Mélou and Baptiste Louf.[-]

The mini-course is structured into three parts. In the first part, we provide a general overview of the tools available in the summation package Sigma, with a particular focus on parameterized telescoping (which includes Zeilberger's creative telescoping as a special case) and recurrence solving for the class of indefinite nested sums defined over nested products. The second part delves into the core concepts of the underlying difference ring theory, offering detailed insights into the algorithmic framework. Special attention is given to the representation of indefinite nested sums and products within the difference ring setting. As a bonus, we obtain a toolbox that facilitates the construction of summation objects whose sequences are algebraically independent of one another. In the third part, we demonstrate how this summation toolbox can be applied to tackle complex problems arising in enumerative combinatorics, number theory, and elementary particle physics.[-]

The mini-course is structured into three parts. In the first part, we provide a general overview of the tools available in the summation package Sigma, with a particular focus on parameterized telescoping (which includes Zeilberger's creative telescoping as a special case) and recurrence solving for the class of indefinite nested sums defined over nested products. The second part delves into the core concepts of the underlying difference ring theory, offering detailed insights into the algorithmic framework. Special attention is given to the representation of indefinite nested sums and products within the difference ring setting. As a bonus, we obtain a toolbox that facilitates the construction of summation objects whose sequences are algebraically independent of one another. In the third part, we demonstrate how this summation toolbox can be applied to tackle complex problems arising in enumerative combinatorics, number theory, and elementary particle physics.[-]

The mini-course is structured into three parts. In the first part, we provide a general overview of the tools available in the summation package Sigma, with a particular focus on parameterized telescoping (which includes Zeilberger's creative telescoping as a special case) and recurrence solving for the class of indefinite nested sums defined over nested products. The second part delves into the core concepts of the underlying difference ring theory, offering detailed insights into the algorithmic framework. Special attention is given to the representation of indefinite nested sums and products within the difference ring setting. As a bonus, we obtain a toolbox that facilitates the construction of summation objects whose sequences are algebraically independent of one another. In the third part, we demonstrate how this summation toolbox can be applied to tackle complex problems arising in enumerative combinatorics, number theory, and elementary particle physics.[-]

T.C. Brown and A.R. Freedman proved that the set $\mathcal{P}_{2}$ of products of two primes contains no dense cluster; technically, $\mathcal{P}_{2}$ has a zero upper Banach density, defined as $\delta^{*}(\mathcal{P}_{2}) =\lim_{H\mapsto \infty} \limsup_{x\mapsto \infty} \frac{1}{H} Card \{n\in \mathcal{P}_{2}:x< n\leq x+H\}$.
Pramod Eyyunni, Sanoli Gun and I jointly studied the local behaviour of the product of two shifted primes $\mathcal{Q}_{2}=\{(q-1)(r-1):q,r \, primes\}$. Assuming a classical conjecture of Dickson, we proved that $\delta^{*}(\mathcal{Q}_{2}) = 1/6$. Notice that we know no un-conditional proof that $\delta^{*}(\mathcal{Q}_{2})$ is positive. The application, which was indeed our motivation, concerns the study of the local behaviour of the set $\mathcal{V}$ of values of Euler's totient function. Assuming Dickson's conjecture, we prove that $\delta^{*}(\mathcal{V})\geq 1/4$. The converse inequality $\delta^{*}(\mathcal{V})\leq 1/4$ had been proved in the previous millenium by K. Ford, S. Konyagin and C. Pomerance.[-]

A square in a matrix $\mathcal M =(a_{ij})$ is a 2X2 sub-matrix of $\mathcal M$ with entries $a_{ij}, a_{i+s,j}, ai,j+s, a_{i+s,j+s}$s for some $s\geq 1$. An Erickson matrix is a square binary matrix that contains no squares with constant entries. In [Eri96], Erickson asked for the maximum value of $n$ for which there exists an n x n Erickson matrix. In [AM08] Axenovich and Manske gave an upper bound of around $2^{2^{40}}$. This gargantuan bound was later improved by Bacher and Eliahou in [BE10] using computational means to the optimal value of 15.
In this talk we present the study of a zero-sum analogue of the Erickson matrices problem where we consider binary matrices with entries in {-1,1}. For this purpose, of course, we need to take into account the discrepancy or deviation of the matrix, defined as the sum of all its entries, that is
$disc(\mathcal M)= \sum_{1\leq i\leq n \; \; 1\leq j\leq m}a_{i,j}$.
A zero-sum square is a square $\mathcal S$ with $disc(\mathcal S) = 0$. A natural question is, for example, the following: is it true that for sufficiently large $n$ every $n\times n \{-1,1\} - matrix \, \mathcal M$ with $disc(\mathcal M) = 0$ contains a zero-sum square? We answered positive to this question. Since, our proof uses an induction argument, in order for the induction to work we prove the following stronger statement: For $n \geq 5$ and $m \in \{n,n+1\}$, every $ n \times m \{-1, 1\}$ -matrix $M$ with $\left | disc(M) \right |\leq n$ contains a zero-sum square except for the triangular matrix (up to symmetries), where a triangular matrix is a matrix with all entries above the diagonal equal to -1 and all remaining entries equal to 1.This is a joint work with Edgardo Roldn-Pensado and Alma R. Arvalo.[-]

Discussing recent work joint with M. Rudnev [2], I will discuss the modern approach to the sum-product problem in the reals. Our approach builds upon and simplifies the arguments of Shkredov and Konyagin [1], and in doing so yields a new best result towards the problem. We prove that
$max(\left | A+A \right |,\left | A+A \right |)\geq \left | A \right |^{\frac{4}{3}+\frac{2}{1167}-o^{(1)}}$ , for a finite $A\subset \mathbb{R}$. At the heart of our argument are quantitative forms of the two slogans ‘multiplicative structure of a set gives additive information', and ‘every set has a multiplicatively structured subset'.[-]