11,953 research outputs found
Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity
No full textLet ρ1 and ρ2 be a pair of residual, odd, absolutely irreducible two-dimensional Galois representations of a totally real number field F. In this article we propose a conjecture asserting existence of "safe" chains of compatible systems of Galois representations linking ρ1 to ρ2. Such conjecture implies the generalized Serre's conjecture and is equivalent to Serre's conjecture under a modular version of it. We prove a weak version of the modular variant using the connectedness of certain Hecke algebras, and we comment on possible applications of these results to establish some cases of Langlands functoriality.Fil: Dieulefait, Luis. Universidad de Barcelona; EspañaFil: Pacetti, Ariel Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
Teoría de pseudo-representaciones y representaciones asociadas
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Luis Victor Dieulefait[en] We study in this work the concept of pseudo-representation and its relevance in the Theory of Representations. In particular, we work with pseudo-representations induced by representations of algebras; and we give the necessary conditions so that, given a pseudo-representation, we can build a representation associated with it
Logaritme discret i pairings en corbes el·líptiques amb aplicacions a la criptografia
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Luis Victor Dieulefait[en] Cryptography is based on practically unsolvable problems computationally. We will study elliptic curves and the discrete logarithm problem that is defined over these curves. We will see some ways to take advantage of this problem to obtain encoding systems and finally we will see 2 ways to attack the curve reducing the discrete logarithm problem over elliptic curves to the discrete logarithm problem over finite fields
Quantum key distribution and other related protocols
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2024, Director: Luis Victor Dieulefait[en] The main goal of this work is to present an introduction to quantum computation and Quantum Key Distribution (QKD). The first chapter is dedicated to the basics of quantum mechanics needed to understand further concepts. Then the definition of quantum circuits is presented, along two examples: Quantum telepor-
tation and the Deutsch-Josza problem.In the chapter about QKD, we will explain in detail the BB84 algorithm and some eavesdropping techniques, and we will take a look at the theory behind the EPR Protocol. Finally, we will explain the CHSH Game
Logaritme discret i pairings en corbes el·líptiques amb aplicacions a la criptografia
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Luis Victor Dieulefait[en] Cryptography is based on practically unsolvable problems computationally. We will study elliptic curves and the discrete logarithm problem that is defined over these curves. We will see some ways to take advantage of this problem to obtain encoding systems and finally we will see 2 ways to attack the curve reducing the discrete logarithm problem over elliptic curves to the discrete logarithm problem over finite fields
Algunas variantes del algoritmo cuántico de Shor
No full textTreballs Finals de Grau d'Enginyeria Informàtica, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Luis Victor Dieulefait[en] The aim of this project is to study the Shor’s factorization algorithm, as well as some of its variants, both from a theoretical and practical point of view. First, the mathematical foundations on which it is based are presented, as well as the formalization of the notation used in quantum computing. Next, Shor’s algorithm and some variants are detailed in order to make it more efficient. Finally, a practical Python implementation of Shor’s quantum algorithm is carried out using IBM’s Qiskit library and also another implementation of Ekerå’s algorithm in SageMath
Enters algebraics i treballs de Kummer sobre l’últim teorema de Fermat
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Luis Victor Dieulefait[en] Fermat’s Last Theorem asserts that the equation has not non-zero integral solutions for . In this project we study the Kummer’s proof of the first case of the Theorem along with developing the basic tools of the algebraic number theory. This first case is that the exponent in the equation is a so-called regular prime and n does not divide any of
Sieve theory and applications
No full textTreballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Luis Victor Dieulefait[en] The object of this Master Thesis is the study of 1) sieve theory 2) a theorem due to J. Chen (1966) stating that every even large enough number is the sum of an odd prime and a product of at most two primes.
The proof of said theorem makes great use of sieving techniques. Thus, this Master Thesis’ purpose is to introduce the required sieving methods in order to fully understand their application in said theorem’s proof, as well as providing a proof of the theorem
Sieve theory and applications
No full textTreballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Luis Victor Dieulefait[en] The object of this Master Thesis is the study of 1) sieve theory 2) a theorem due to J. Chen (1966) stating that every even large enough number is the sum of an odd prime and a product of at most two primes.
The proof of said theorem makes great use of sieving techniques. Thus, this Master Thesis’ purpose is to introduce the required sieving methods in order to fully understand their application in said theorem’s proof, as well as providing a proof of the theorem
Teorema de Waring
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Luis Victor Dieulefait[en] In number theory, Waring’s problem (1770) asks whether for each natural number k exists an associated positive integer such that every natural number is the sum of at most s natural numbers to the power of . The statement was proved by Hilbert in 1909.
We present an overview of Hilbert-Waring theorem. First, we introduce the modern notation and find several lower and upper bounds using elementary methods. Next, we offer a proof of the theorem based on Schnirelmann’s density. Finally, we summarize the current state of the problem
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