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Fundamental aspects of the representation theory of diagram algebras
Chwas Abas Ahmed
Abstract
Broadly speaking, diagram algebras are finite-dimensional algebras with a basis represented by diagrams. The multiplication of these basis elements is described by concatenating the corresponding diagrams. Some classical examples of diagram algebras include the symmetric group algebra and the Brauer algebra. Diagram algebras arise naturally in both representation theory, in the context of Schur-Weyl duality, and statistical mechanics. In this talk, I will provide a gentle introduction to diagram algebras and explain why they are of interest. I will then focus on some fundamental aspects of the representation theory of a particular type of diagram algebra called tonal partition algebras. |
| Escher & Droste in Erbil
Peter Stevenhagen
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Galois number fields with a fixed Polya index
Amir Akbary
Abstract
Let $K$ be a number field. For a prime power $q$, the Ostrowski ideal $\Pi_q(K)$ is the product of prime ideals of K with norm $q$. The Polya group
${\rm Po}(K)$ is the subgroup of the class group ${\rm Cl}(K)$ generated by classes $[\Pi_q(K)]$ of Ostrowski ideals. We discuss some finiteness results for number fields $K$ with a fixed Polya index $[{\rm Cl}(K): {\rm Po}(K)]$ in certain families of Galois number fields. We are motivated by the classical problem of ``one class in each genus," originally formulated by Gauss in the context of binary quadratic forms of negative even discriminant.
This is joint work with Abbas Maarefparvar (University of Lethbridge). |
| Bogomolov property for Galois representations
Lea Terracini
Abstract
A subset of algebraic numbers is said to have the Bogomolov property
(B) if the Weil height of its non-torsion elements is uniformly bounded from below. I will discuss property (B) for algebraic fields cut out by some Galois representations, present some new findings and outline the main proof techniques.
This is a joint work with F. Amoroso and A. Conti.
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An overview on some problems of unlikely intersections
Laura Capuano
Abstract
The Zilber-Pink conjecture on unlikely intersections for (semi)abelian varieties describes the behaviour of the intersection of an algebraic variety of a (semi)abelian variety with the union of all the ?special? subvarieties of the ambient space. This conjecture generalizes many classical results such as Faltings? Theorem (Mordell Conjecture) and Raynaud?s Theorem (Manin-Mumford Conjecture) and have been studied by several authors in the last two decades. In this talk I will give a general introduction to these problems and describe some applications to other problems of Diophantine nature.
| On some Diophantine equations involving some particular sequences
Alain Togbe
Abstract
Let $b\geq 2$ be a positive integer. Let $r$ and $s$ be two integers with $r\geq 1$, $s =\pm 1$
and $\Delta= r^2+4s > 0$, let $\{U_n\}_{n\geq0}$ be the Lucas sequence given by
$$
U_{n+2} = rU_n+1+ sU_n,\quad \mbox{with}\;\; U_0 = 0\;\; \mbox{and}\;\; U_1 = 1.
$$
In this talk, we will discuss the solutions of the Diophantine equations
$$
U_n\pm U_m = (b\pm 1)\cdot b^{\ell}\pm 1.
$$
by giving effective bounds for the variables $n, m$ and $\ell$ in terms of $b, r$ and $s$. Moreover, we will give all the solutions of these equations in some particular cases.
This talk is based on a joint work with
M. Faye and N. Adédji.
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Non-vanishing of Zeta functions and $L$-functions
Chantal David
Abstract |
Non-vanishing of Zeta or $L$-functions at the point $s = \frac{1}{2}$ is a fundamental question in analytic number theory. For Zeta functions of Abelian extensions of $\mathbb{Q} (or equivalently $L$- functions of primitive Dirichlet characters), a folklore conjecture of Chowla predicts that those functions never vanish. We will explain in this talk the significance of the question, concentrating on the basic cases of quadratic and cubic Dirichlet characters.
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| On the total cut complexes of chordal graphs
Rahim Zaare-Nahandi
Abstract |
On Erdős' last equation
Florian Luca
Abstract |
Let $n$ be a positive integer. The Diophantine equation
$$n(x_1+x_2+\dots +x_n)=x_1x_2\dots x_n,\qquad 1 \le x_1\le x_2\le \dots \le x_n$$ in positive integers $x_1,\ldots,x_n$ is called Erdős' last equation. Denoting the number of solution in $n$ by $f(n)$
we show that $\liminf f(n)=n^{o(1)}$ as $n\to\infty$. Let $g(n)$ denote the maximal value of $x_n$ as $x_1\le \cdots \le x_n$ range over all the $f(n)$ solutions of Erdős' equation.
We show that for some positive constants $n_0$ and $\kappa$ the inequality
$$
g(n)\ge \kappa \frac{(\log\log n)^2}{\log\log\log n} \quad \textrm{holds~when} \quad n\ge n_0.
$$
We also find all the solutions with $x_n\le 10$. There are exactly $1382$ such and they all have $n\le 367416$.
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| Playing with monomials: Algebra, Geometry and Combinatorics
Rashid Zaare-Nahandi
Abstract |
Many problems in science can be transformed to problems on polynomials and then problems on monomials. Monomials which are algebraic objects, have interpretations in Geometry as simplicial complexes and in Combinatorics as hyper-graphs. In other hand, some of problems on monomials are similar to mind games. In this talk, some of these problems will be presented. |
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Divisibility of reduction in groups of rational numbers
Andam Ali Mustafa
Abstract |
Given a finitely generated multiplicative group of rational numbers, we will discuss the computation of the density of the set of prime numbers for which the order of the reduction group is divisible by a given integer.
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| Some topics in analytic number theory
Michel Waldschmidt
Abstract |
Analytic number theory is a very active domain of research.
We survey a selection of some of the many recent results.
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