Doctoral INPhINIT Fellowships – Incoming Call 2024

Goals:

The first goal for this project is to quantify in a precise way the amount of level lowering for weight 2 modular forms that can occur under certain hypothesis. Another main goal for the project is to apply this new characterization to the resolution of new Diophantine equations and the simplification of safe chain of congruences and base change results.

Methodology:

We will approach the first goal of this project via the relation of modular forms of weight 2 with abelian varieties and a classical result of Serre–Tate.

We will carefully study the possible compatible scenarios and, in particular, we aim to characterize when the theorem of Serre-Tate forces at least one of the 2-dimensional blocks in the residual Galois representation attached to a modular abelian variety to ramify.

The second part of this project focus on applications of the previous idea to the modular method (for solving Diophantine Equations of Fermat type) and to safe chains of congruences (a method of Dieulefait to solve cases of Langlands functoriality)

Job position description

The fellow will work as a PhD research student within the research group “Seminari de Teoria de Nombres de Barcelona”, he/she will participate in the different seminars and advanced courses organized regularly by the group while advancing in the research project described in the previous section.

Some activities that we can anticipate the candidate will participate in are the annual workshop of the number theory groups in Barcelona (STNB), conferences, and research schools.

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In 2021 the supervisor Miranda together with Cardona, Peralta-Salas and Presas designed an abstract fluid computer (PNAS, 2021). Our approach integrated techniques from contact geometry, the path method in Symplectic geometry, symbolic dynamics, and fundamental concepts related to the Turing completeness of a dynamical system (a system capable of simulating a Turing machine). This groundbreaking achievement holds significant implications for various reasons, primarily for implying the existence of undecidable paths. The tour de force of these proofs was to use a mirror between geometrical objects (contact manifolds and Reeb
fields) and Fluid Dynamics (Euler equations and Beltrami fields) proved by Etnyre and Ghrist back in 1999. Miranda together with Dyhr (inphinit La Caixa PhD student), Gonzalez-Prieto and Peralta Salas have recently generalized these mirrors to include other geometrical structures on the one side and other solutions of PDE’s on the other side. The understanding of the geometric aspects within the realm of Fluid Dynamics and more general PDE’s remains incomplete. The goal of this thesis is to pursue this investigation between Geometry and Fluid Dynamics going in the direction of improving the abstract design of a hybrid computer between “Fluid computer” and Quantum computer following Feynam’s rules as being currently investigated by Miranda, Gonzalez-Prieto and Peralta- Salas. This naturally takes us to the realm of Topological Quantum Field Theory. The purpose of this thesis is to relate invariants and complexity invariants from the initial geometry reflects into the PDE’s and the TQFT template associated to the generalization of the fluid computer.

The aspiring scholar is poised to become an member of a dynamic group situated at the confluence of Geometry, Dynamics, and Partial Differential Equations (PDEs). The student will be affiliated both the Laboratory of Geometry and Dynamical Systems at UPC and the distinguished team dedicated to Geometry and Dynamical Systems at CRM. The focal point of the forthcoming doctoral thesis will rest squarely at the intersection of these aforementioned disciplines: Geometry, Dynamical Systems, and PDEs but will also venture into new aspects such as Quantum computing and Topological Quantum Field Theory.

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The research of the group is devoted to certain aspects of harmonic analysis and approximation theory. From its foundation in 2009, the CRM group “Harmonic Analysis and Approximation Theory”’ has been growing intensively. The group has incorporated 18 members (PhDs and postdoctoral researchers).

Since the foundation of the group, its members were awarded several prizes. The main ones are ISAAC prize for achievements in analysis, Popov Prize in Approximation Theory, Ramon y Cajal program, Simons and Pembroke fellow, the Humboldt Research Fellowship for Experienced Researchers, and Friedrich Wilhelm Bessel Research Award.

During the last five years, six group members have successfully defended their PhD theses under the supervision of the group leader Dr. Sergey Tikhonov and five postdoctoral researchers were supported by various international funds and organizations.

The group members have organized and actively participated in many programs and conferences. In particular, the principal investigator, Sergey Tikhonov, organized two research programs in INI (Cambridge) and two programs in CRM (Barcelona) as well as many conferences all over the world.

The research output of the group members has been published in the leading scientific journals. In particular, several long-standing problems were solved by group members with the papers published in Annals of Math, Constr. Appr, IMRN, JEMS.

The main goal of the project is to obtain new uncertainty principle relations for the Fourier transform. The uncertainty principle is a fundamental concept in physics (quantum mechanics) and mathematics (Fourier analysis). From the mathematical point of view, we examine the fact that both function and its Fourier transform cannot be simultaneously localized. Our goal is to show quantitative relations for uncertainty principle. As a result, we arrive at several important applications in harmonic analysis and PDEs.

Job position description

The main research topic of this project is at the intersection of three areas of analysis: harmonic analysis, functional analysis, and approximation theory.

The main tasks to be undertaken by the candidate are:

– Acquiring and strengthening the necessary background on constructive theory, harmonic analysis, and complex analysis.

– Pursuing original results within the research lines previously depicted under the guidance of his supervisor and elaborating research articles reporting the achievements to be published in peer-reviewed international journals,

– Taking active part in the scientific life of the Barcelona Harmonic analysis group.

Requirements for eligibility are capacity of autonomous work and a solid and demonstrated education in mathematical analysis.

The overall training activities can be divided into several categories. First, collaborating with the supervisor, his team, and analysis community in CRM, the applicant will acquire new knowledge, valuable for this research, in the field of classical approximation theory and harmonic analysis. Moreover, the holder of the position will prepare several scientific papers to publish in high rank journals. Participation in seminars, conferences, graduate courses organized by Barcelona Graduate School of Mathematics is an important part of the process. The applicant will be supervised by Prof. Sergey Tikhonov (ORCID0000-0001-5061-4308), an ICREA Research Professor at CRM.

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In recent years the theory of p-adic L-functions and Euler systems has exploded into a rich and extremely active subject. It is our plan to develop frontier research in this field by introducing new ideas which we hope will allow us to obtain new insights on the conjectures of Stark, Birch and Swinnerton-Dyer (BSD) and Bloch and Kato (BK).

A p-adic L-function is a rigid-analytic function whose values, at special points, are related to classical L-functions. An Euler system is a compatible family of motivico-geometric elements. They are interrelated if there is a reciprocity law allowing to recover one from another by means of a regulator map in the sense of Perrin-Riou. Until recently there were very few Euler systems and p-adic L-functions for which one could establish a reciprocity law, mainly due to the fundamental works of Kato and Perrin-Riou. At present the picture is much better understood thanks to the works of Zerbes, Loeffler, Pilloni, Urban, Skinner, Bertolini, Darmon and Rotger, among others.

Furthermore, celebrated applications to the conjectures of BSD and BK have been obtained, establishing plenty of new theoretical and experimental evidence in rank 0 and 1, and gaining the first non-trivial insights in the mysterious scenario of rank 2 thanks to works of Darmon, Lauder and Rotger.

Most of the literature, however, is devoted to applications of Euler systems and p-adic L-functions at smooth points in the rigid-analytic space parametrizing the Euler system. The main goal and unifying principle of our Ph.d project is delving into the study of Euler systems, p-adic L-functions and associated reciprocity laws at points that fail to be smooth in the parameter space. These points exhibit intriguing phenomena, require novel techniques, and are called to obtain ground-breaking results on BSD and BK that were not previously available, pushing them into unforeseen new directions.

We plan to develop our research activities with the CRM, UAB, UB and UPC.

Job position description

The main tasks to be undertaken by the candidate are:

–Acquiring and strengthening the necessary background on algebraic number theory, local and global class field theory, classical and p-adic L-functions, and rational points on algebraic varieties.

-Pursuing original results within the research lines previously depicted under the guidance of her supervisor and elaborating research articles reporting the achievements to be published in peer-reviewed International journals

-Taking active part in the scientific life of the Barcelona Number Theory Group.

Requirements for eligibility are capacity of autonomous work and a solid and demonstrated education in algebraic geometry and algebraic Number Theory.

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Many problems that humankind faces arise from large human aggregations in the form of cities and megalopolis. An increasing number of people is living in urban areas, which poses formidable challenges regarding infrastructure, mobility, transport of goods, sustainability and inequality. And although we collectively created those challenges, we do not know how they came to be, nor how to fix them.

Urban systems present statistical patterns caused by the interaction between smaller units. A fundamental problem is how human settlements scatter through a territory and cluster into large agglomerations. The current structure of cities has emerged from the competition between top-down interventions and bottom-up self-organization, but frustration between different interactions leads to non-unique solutions for the morphology and non-unique distribution of functions. In addition, different feedback mechanisms may lead to the reinforcement of segregation and exclusion. Fractal structures have been known to describe the outcome of such processes, whereas Zipf’s law has been claimed to account for the overall population of such aggregations. Self-similar patterns seem also to be present for many urban variables, but the relationship between processes that gave rise to such configurations is unknown. In addition, correlations between the different subsystems are most of the time ignored.

This project aims to address these issues from a three-fold approach: empirical, computational and theoretical. The use of large databases of population, infrastructure, mobility, etc. will lead to statistical facts that will be confronted with computational modeling as well as with theoretical insights. In their turn, theoretical and computational insight will trigger new empirical questions. It will be fundamental to consider that the different urban descriptors are not independent but strongly entangled. It is expected that the project will unveil fundamental principles of urban science.

We are looking for predoctoral researchers with a broad interest in understanding the complexity of urban processes and with a cross-disciplinary attitude, with a degree in mathematics, physics, computer science, or a related discipline. Strong background in applied mathematics, stochastic processes, statistical physics, complex-systems science, or computational modeling will be highly appreciated. The successful candidate will join the CRM Complex-Systems Group and will work under the co-supervision of Prof. Elsa Arcaute (Centre for Advanced Spatial Analysis, University College London).

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