Zeros da função zeta de Riemann e o teorema dos números primos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A family of asymptotically good lattices having a lattice in each dimension

A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n >= 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field Q(zeta(q)) where q is the smallest prime congruent to 1 modulo n.

Digit systems over commutative rings

Let epsilon be a commutative ring with identity and P is an element of epsilon[x] be a polynomial. In the present paper we consider digit representations in the residue class ring epsilon[x]/(P). In particular, we are interested in the question whether each A is an element of epsilon[x]/(P) can be represented modulo P in the form e(0)+ e(1)x + ... + e(h)x(h), where the e(i) is an element of epsilon[x]/(P) are taken from a fixed finite set of digits. This general concept generalizes both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that 0 is an element of the digit set and that P need not be monic, several new phenomena occur in this context.

Teorema de Riemann-Roch, morfismos de Frobenius e a hipótese de Riemann

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Codificação de canais em sistemas de comunicação sem fio baseado em reticulados

Pós-graduação em Engenharia Elétrica - FEIS

Anéis de inteiros de corpos de números e aplicações

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Algebraic constructions of densest lattices

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Explorando o universo dos números primos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Tópicos de teoria dos números algébricos e aplicações em reticulados e equações diofantinas

Pós-graduação em Matemática Universitária - IGCE

Formalização do conjunto dos números racionais e alguns jogos com frações

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

O uso de elementos da criptografia como estímulo matemático na sala de aula

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

O uso de elementos da criptografia como estímulo matemático na sala de aula

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.