Dimensions of Deformation Rings and Automorphy Lifting Theorems

这个研究项目研究代数数论中的问题。数论和自同构形式之间的联系是一个非常活跃的研究领域，它是周期函数的推广，并导致了数论中长期存在的问题的解决。这样的工作可以应用于求解丢芬图方程——寻求整数解的整数系数代数方程——它在从加密协议到纠错的广泛应用中发挥着作用。此外，在代数数论领域开发的方法可以在密码学中有其他令人惊讶的应用。这个项目旨在加深这一重要学科的知识。该项目研究的问题与数论中的经典问题有关，包括利奥波德猜想和高维类比。所使用的方法与自同构提升定理有关，这些定理在费马大定理的证明和Serre猜想的证明等重要发展中发挥了至关重要的作用。该项目旨在增强这些方法并使其更加灵活，这将导致在多项式方程解及其与自同构形式的关系领域的更多应用。

This research project studies questions in algebraic number theory. The connection between number theory and automorphic forms, which are generalizations of periodic functions, is a very active area of investigation and has led to solutions of longstanding problems in number theory. Such work has applications to solving Diophantine equations -- algebraic equations with integer coefficients for which integer solutions are sought -- which play a role in a wide range of applications from encryption protocols to error correction. Additionally, methods developed in the area of algebraic number theory can have other surprising applications to cryptography. This project aims to deepen knowledge in this important subject.The questions that the project studies are related to classical problems in number theory, including the Leopoldt conjecture and higher dimensional analogs. The methods to be used relate to automorphy lifting theorems, which have played a crucial role in important developments such as the proof of Fermat's Last Theorem and the proof of Serre's conjecture. The project aims to augment these methods and to make them more flexible, which will lead to more applications in the field of solutions of polynomial equations and its relation to automorphic forms.