No description available.

This thesis is focused on the problem of growth and integrability in multi-valued dynamics generated by $SL_2 (\mathbb{Z})$ actions. An important example is given by Markov dynamics on the cubic surface $$x^2+ y^2 +z^2 = 3xyz,$$ generating all the integer solutions of this celebrated Diophantine equation, known as Markov triples. To study the growth problem of Markov numbers we use the binary tree representation. This allows us to define the Lyapunov exponents $\Lambda (x)$ as the function of the paths on this tree, labelled by $x \in \mathbb{R}P^1$. We prove that $\Lambda (x)$ is a $PGL_2 (\mathbb{Z})$-invariant function, which is zero almost everywhere but takes all values in $\left[ 0, \ln \varphi \right]$ (where $\varphi$ denotes the golden ratio). We also show that this function is monotonic, and that its restriction to the Markov-Hurwitz set of most irrational numbers is convex in the Farey parametrisation. We also study the growth problem for integer binary quadratic forms using Conway's topograph representation. It is proven that the corresponding Lyapunov exponent $\Lambda_Q(x) = 2 \Lambda(x)$ except for the paths along the Conway river. Finally, we study the tropical version of the Markov dynamics on the tropical version of the Cayley cubic proposed by Adler and Veselov, and show that it is semi-conjugated to the standard action of $SL_2(\mathbb{Z})$ on a torus. This implies the dynamics is ergodic, with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.

No description available.

A physical system is said to be in topological phase if at low energies and long wavelengths the observable quantities are invariant under diffeomorphisms. Such physical systems are of great interest in condensed matter physics and computer science where they can be applied to form topological insulators and fault–tolerant quantum computers. Physical systems in topological phase may be rigorously studied through their algebraic manifestations, (pre)modular categories. A complete classification of these categories would lead to a taxonomy of the topological phases of matter. Beyond their ties to physical systems, premodular categories are of general mathematical interest as they govern the representation theories of quasi–Hopf algebras, lead to manifold and link invariants, and provide insights into the braid group. In the course of this work, we study the classification problem for (pre)modular categories with particular attention paid to their arithmetic properties. Central to our analysis is the question of rank finiteness for modular categories, also known as Wang’s Conjecture. In this work, we lay this problem to rest by exploiting certain arithmetic properties of modular categories. While the rank finiteness problem for premodular categories is still open, we provide new methods for approaching this problem. The arithmetic techniques suggested by the rank finiteness analysis are particularly pronounced in the (weakly) integral setting. There, we use Diophantine techniques to classify all weakly integral modular categories through rank 6 up to Grothendieck equivalence. In the case that the category is not only weakly integral, but actually integral, the analysis is further extended to produce a classification of integral modular categories up to Grothendieck equivalence through rank 7. It is observed that such classification can be extended provided some mild assumptions are made. For instance, if we further assume that the category is also odd–dimensional, then the classification up to Grothendieck equivalence is completed through rank 11. Moving beyond modular categories has historically been difficult. We suggest new methods for doing this inspired by our work on (weakly) integral modular categories and related problems in algebraic number theory. The allows us to produce a Grothendieck classification of rank 4 premodular categories thereby extending the previously known rank 3 classification.

No description available.

<p>This thesis in analytic number theory consists of 3 parts and 13 individual papers.</p><p>In the first part we prove some results in Turán power sum theory. We solve a problem of Paul Erdös and disprove conjectures of Paul Turán and K. Ramachandra that would have implied important results on the Riemann zeta function.</p><p>In the second part we prove some new results on moments of the Hurwitz and Lerch zeta functions (generalized versions of the Riemann zeta function) on the critical line.</p><p>In the third and final part we consider the following question: What is the natural generalization of the classical Poisson summation formula from the Fourier analysis of the real line to the matrix group SL(2,R)? There are candidates in the literature such as the pre-trace formula and the Selberg trace formula.</p><p>We develop a new summation formula for sums over the matrix group SL(2,Z) which we propose as a candidate for the title "The Poisson summation formula for SL(2,Z)". The summation formula allows us to express a sum over SL(2,Z) of smooth functions f on SL(2,R) with compact support, in terms of spectral theory coming from the full modular group, such as Maass wave forms, holomorphic cusp forms and the Eisenstein series. In contrast, the pre-trace formula allows us to get such a result only if we assume that f is also SO(2) bi-invariant.</p><p>We indicate the summation formula's relationship with additive divisor problems and the fourth power moment of the Riemann zeta function as given by Motohashi. We prove some identities on Kloosterman sums, and generalize our main summation formula to a summation formula over integer matrices of fixed determinant D. We then deduce some consequences, such as the Kuznetsov summation formula, the Eichler-Selberg trace formula and the classical Selberg trace formula.</p>

This thesis in analytic number theory consists of 3 parts and 13 individual papers. In the first part we prove some results in Turán power sum theory. We solve a problem of Paul Erdös and disprove conjectures of Paul Turán and K. Ramachandra that would have implied important results on the Riemann zeta function. In the second part we prove some new results on moments of the Hurwitz and Lerch zeta functions (generalized versions of the Riemann zeta function) on the critical line. In the third and final part we consider the following question: What is the natural generalization of the classical Poisson summation formula from the Fourier analysis of the real line to the matrix group SL(2,R)? There are candidates in the literature such as the pre-trace formula and the Selberg trace formula. We develop a new summation formula for sums over the matrix group SL(2,Z) which we propose as a candidate for the title "The Poisson summation formula for SL(2,Z)". The summation formula allows us to express a sum over SL(2,Z) of smooth functions f on SL(2,R) with compact support, in terms of spectral theory coming from the full modular group, such as Maass wave forms, holomorphic cusp forms and the Eisenstein series. In contrast, the pre-trace formula allows us to get such a result only if we assume that f is also SO(2) bi-invariant. We indicate the summation formula's relationship with additive divisor problems and the fourth power moment of the Riemann zeta function as given by Motohashi. We prove some identities on Kloosterman sums, and generalize our main summation formula to a summation formula over integer matrices of fixed determinant D. We then deduce some consequences, such as the Kuznetsov summation formula, the Eichler-Selberg trace formula and the classical Selberg trace formula.

The aim of this work is to study the interaction between two classical objects of mathematics: the modular group, and the absolute Galois group. The latter acts on the category of finite index subgroups of the modular group. However, it is a task out of reach do understand this action in this generality. We propose a lattice which parametrizes a certain system of &rdquo / geometric&rdquo / elements in this category. This system is setwise invariant under the Galois action, and there is a hope that one can explicitly understand the pointwise action on the elements of this system. These elements admit moreover a combinatorial description as quadrangulations of the sphere, satisfying a natural nonnegative curvature condition. Furthermore, their connections with hypergeometric functions allow us to realize these quadrangulations as points in the moduli space of rational curves with 8 punctures. These points are conjecturally defined over a number field and our ultimate wish is to compare the Galois action on the lattice elements in the category and the corresponding points in the moduli space.

Es wird die Theorie der generischen pro-$p$ Hecke-Algebren und ihrer Bernstein-Abbildungen entwickelt. Für eine Unterklasse diese Algebren, der \textit{affinen} pro-$p$ Hecke-Algebren wird ein Struktursatz bewiesen, nachdem diese Algebren unter anderem stets noethersch sind, wenn es der Koeffizientenring ist. Hilfsmittel ist dabei der Nachweis der Bernsteinrelationen, der in abstrakter Weise geführt wird und so die bestehende Theorie verallgemeinert. Ferner wird der top. Raum der Orientierungen einer Coxetergruppe eingeführt und im Falle der erweiterten modularen Gruppe $\operatorname{PGL}_2(\mathds{Z})$ untersucht, und ausgenutzt um Kenntnisse über die Struktur der zugehörigen Hecke-Algebra als Modul über einer gewissen Unteralgebra, welche zur Spitze im Unendlichen zugeordnet ist, zu erlangen. Schließlich wird die Frage des Zerfallens des Normalisators eines maximalen zerfallenden Torus innerhalb einer zerfallenden reduktiven Gruppe als Erweiterung der Weylgruppe durch die Gruppe der rationalen Punkte des Torus untersucht, und mittels zuvor erreichter Ergebnisse auf eine kohomologische Frage zurückgeführt. Zur Teilbeantwortung dieser werden dann die Kohomologiegruppen bis zur Dimension drei der Kocharaktergitter der fasteinfachen halbeinfachen Wurzeldaten einschließlich des Rangs 8 berechnet. Mittels der Theorie der $\mathbf{FI}$-Moduln wird daraus die Berechnung der Kohomologie der mod-2-Reduktion der Kowurzelgitter für den Typ $A$ in allen Rängen bewiesen. / The theory of generic pro-$p$ Hecke algebras and their Bernstein maps is developed. For a certain subclass, the \textit{affine} pro-$p$ Hecke algebras, we are able to prove a structure theorem that in particular shows that the latter algebras are always noetherian if the ring of coefficients is. The crucial technical tool are the Bernstein relations, which are proven in an abstract way that generalizes the known cases. Moreover, the topological space of orientations is introduced and studied in the case of the extended modular group $\operatorname{PGL}_2(\mathds{Z})$, and used to determine the structure of its Hecke algebra as a module over a certain subalgebra, attached to the cusp at infinity. Finally, the question of the splitness of the normalizer of a maximal split torus inside a split reductive groups as an extension of the Weyl group by the group of rational points is studied. Using results obtained previously, this questioned is then reduced to a cohomological one. A partial answer to this question is obtained via computer calculations of the cohomology groups of the cocharacter lattices of all almost-simple semisimple root data of rank up to $8$. Using the theory of $\mathbf{FI}$-modules, these computations are used to determine the cohomology of the mod 2 reduction of the coroot lattices for type $A$ and all ranks.

Dans ce mémoire de thèse, on étudie le morphisme de monoïde $mu$du monoïde libre sur l'alphabet des entiers $nb$,`a valeurs dans le groupe modulaire $PSL_2(zb)$,considéré comme monoïde, défini pour tout entier $a$ par $mu(a)=begin{pmatrix} 0 & -1 1 & a+1 end{pmatrix}.$Les nombres de Catalan apparaissent naturellement dans l'étudede sous-ensembles du noyau de $mu$.Dans un premier temps, on met en évidence deux systèmes de réécriture, l'un sur l'alphabet fini ${0,1}$, l'autresur l'alphabet infini des entiers $nb$ et on montreque ces deux systèmes de réécriture définissent des présentations de monoïde de $PSL_2(zb)$ par générateurs et relations.Par ailleurs, on introduit le morphisme d'indice associé `a l'abélianisé du rev^etement universel de $PSL_2(zb)$,le groupe $B_3$ des tresses `a trois brins. Interprété dans deux contextes différents,le morphisme d'indice est associé au nombre de "demi-tours".Ensuite, dans les quatrième et cinquième parties, on dénombre des sous-ensembles du noyau de $mu_{|{0,1}}$ etdu noyau de $mu$, bigradués par la longueur et l'indice. La suite des nombres de Catalan et d'autres diagonales du triangle de Catalan interviennentsimplement dans les résultats.Enfin, on présente l'origine géométrique de cette étude : on explicite le lien entre l'objectif premier de la thèse qui était l'étudedes polygones convexes entiers d'aire minimale et notre intéret pour le monoïde engendré par ces matrices particulières de $PSL_2(zb)$. / In this thesis, we study a morphism of mono"id $mu$ between the free mono"id on the alphabet of integers $nb$and the modular group $PSL_2(zb)$ considered as a mono"id, defined for all integer $a$by $mu(a)=begin{pmatrix} 0 & -1 1 & a+1 end{pmatrix}.$ The Catalan Numbers arised naturally in the study ofsubsets of the kernel of the morphism $mu$.Firstly, we introduce two rewriting systems, one on the finite alphabet ${0,1}$, and the other on the infinite alphabet of integers $nb$. We proove that bothof these rewriting systems defines a mono"id presentation of $PSL_2(zb)$ by generators and relations.On another note, we introduce the morphism of loop associated to the abelianised of the universal covering group of $PSL_2(zb)$, the group $B_3$ ofbraid group on $3$ strands. In two different contexts, the morphism of loop is associated to the number of "half-turns".Then, in the fourth and the fifth parts, we numerate subsets of the kernel of $mu_{|{0,1}}$ and of the kernel of $mu$,bi-graduated by the morphism of lengthand the morphism of loop. The sequences of Catalan numbers and other diagonals of the Catalan triangle come into the results.Lastly, we present the geometrical origin of this research : we detail the connection between our first aim,which was the study of convex integer polygones ofminimal area, and our interest for the mono"id generated by these particular matrices of $PSL_2(zb)$.