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Programme INPhINIT "la Caixa". Research lines at the ICMAT.

INE 1: Algebraic Geometry and Mathematical Physics

 

Job title/Job Position title

PhD in Algebraic Geometry and Mathematical Physics

Research project/Research Group Description

This line covers various topics in the field of Algebraic Geometry and its interactions with Differential Geometry, Symplectic Geometry, Topology, Mathematical Physics and Number Theory. The institute has world leading experts working on themes such as the study of moduli spaces parametrizing geometric structures of various kinds, and the study of singularities. These are all topics of much international activity in mathematical research.

Job position description

The job will be to carry out a PhD Thesis in Algebraic Geometry and Mathematical Physics with any Faculty Member with expertise on this topic as PhD Advisor. The list of Faculty Members can be found in https://www.icmat.es/people/faculty

Group Leader

Oscar García Prada

Research group website

https://www.icmat.es/research/lines/line1

 

LINE 2: Differential Geometry, Symplectic Geometry and Geometric Mechanics

 

Research project/Research Group Description

PhD in Differential Geometry, Symplectic Geometry and Geometric Mechanics.

Job position description

The research of the group is divided into two lines, one corresponding to the differential geometric aspects of symplectic geometry, and the other centering on its applied aspects, witha focus on geometric mechanics and control theory. The first line fits broadly into the theme of the study of the global aspects of manifolds, and include topological properties of symplectic and contact manifolds, manifolds with special holonomy, rational homotopy theory of differentiable manifolds, and geometric structures of non-Riemannian type (path geometries, ...). The second line of research focuses on geometric mechanics and control theory. Research themes include geometric field theories, Poisson geometry (groupoids, algebroids, ...), symplectic integration and numerical linear algebra (algorithms for matrix computations), and non-linear dynamics (matrix analysis, matric polynomials, ...). A common interest in gauge theoretic problems and techniques unites the two lines of research.

Job position description

The job will be to carry out a PhD Thesis in Differential Geometry, Symplectic Geometry and Geometric Mechanics with any Faculty Member with expertise on this topic as PhD Advisor. The list of Faculty Members can be found in https://www.icmat.es/people/faculty

Group Leader

David Martín de Diego

Research group website

https://www.icmat.es/research/lines/line2

 

LINE 3: Mathematical Analysis

 

Research project/Research Group Description

PhD in Mathematical Analysis.

Job position description

Harmonic analysis is a corner stone in the solution of many problems coming from the natural and social sciences. The analysis group at the ICMAT works actively on directional maximal operators, convergence of Fourier series and integrals, noncommutative Calderón-Zygmund theory, harmonic analysis on discrete groups, restriction of the Fourier transform to surfaces, inverse problems, geometric measure theory and weighted norm inequalities. These areas have applications in the recovery of potentials from scattered plane waves, elliptic and dispersive partial differential equations, number theory, as well as quantum probability and quantum information theory.

Job position description

The job will be to carry out a PhD Thesis in Mathematical Analysis with any Faculty Member with expertise on this topic as PhD Advisor. The list of Faculty Members can be found in https://www.icmat.es/people/faculty

Group Leader

Antonio Córdoba

Research group website

https://www.icmat.es/research/lines/line31

 

LINE 4: Differential Equations and Applications

 

Research project/Research Group Description

PhD in Differential Equations and Applications.

Job position description

This research line has grown to be one of the richest areas of mathematics. Ordinary and partial differential equations are very powerful tools for modelling many diverse phenomena, in physics, mechanics, chemistry, biology, economics, etc. Moreover, the emergence of computers has made these models even more efficient and useful by predictive numerical simulations. The researchers of the ICMat involved in this area work in the well-posedness and formation of singularities of PDEs, mathematical theory and numerical computations of Euler, Navier-Stokes and related equations of fluid mechanics, as well as the study of the mathematical theory of kinetic models related to statistical physics and convective instabilities in geophysical problems.

Job position description

The job will be to carry out a PhD Thesis in Differential Equations and Applications with any Faculty Member with expertise on this topic as PhD Advisor. The list of Faculty Members can be found in https://www.icmat.es/people/faculty

Group Leader

Diego Córdoba

Research group website

https://www.icmat.es/research/lines/line32

 

LINE 5: Statistics, Probability and Operations Research

 

Research project/Research Group Description

PhD in Statistics, Probability and Operations Research.

Job position description

This line aims to cover a wide spectrum of mathematical problems including statistical modelling, stochastic models in queuing and mathematical biology, and operations research. Statistics, Probability and Operations Research Group (SPOR) covers a broad range of topics including data science and engineering, machine learning, stochastic differential equations, optimal control, small area estimation, stochastic process inference, mathematical biology models, risk analysis, game theory, decision analysis and Bayesian methods, being one of the most productive groups in Europe in its area of specialty. SPOR supports the whole spectrum of evidence based decision making, from mathematical foundations, to data analysis, to inference, to analysis, to decision support, with applications in fraud detection, security, cybersecurity, epidemiology, insurance, poverty estimation and social robotics, to name but a few.

Job position description

The job will be to carry out a PhD Thesis in Statistics, Probability and Operations Research with any Faculty Member with expertise on this topic as PhD Advisor. The list of Faculty Members can be found in https://www.icmat.es/people/faculty

Group Leader

Antonio Gomez Corral

Research group website

https://www.icmat.es/spor/

 

LINE 6: Group theory

 

Research project/Research Group Description

PhD in Group theory.

Job position description

The research in this line deals with the global aspect of finite and infinite groups. Our main theme is the Asymptotic Group Theory. The group maintains close collaboration with researchers belonging to other centers, both within Spain and abroad (in particular the United Kingdom, Germany, US, Israel, and Italy among others).

Job position description

The job will be to carry out a PhD Thesis in Group theory with any Faculty Member with expertise on this topic as PhD Advisor. The list of Faculty Members can be found in https://www.icmat.es/people/faculty

Group Leader

Andrei Jaikin

Research group website

https://www.icmat.es/research/lines/line5

 

LINE 7: Number Theory

 

Research project/Research Group Description

PhD in Number Theory.

Job position description

The Number Theory Group at the ICMAT studies arithmetic problems with an interdisciplinary point of view, using techniques from harmonic analysis, combinatorics, probability theory, algebraic geometry, ergodic theory or even theoretical physics.

Among the problems and questions tackled by the group we mention:

The applications of harmonic analysis in the Euclidean and hyperbolic setting to analytic number theory.

Additive combinatorics, which is a relatively recent term coined to comprehend the developments of the more classical combinatorial number theory, mainly focussed on problems related to the addition of integers and, more generally, on subsets of abelian groups.

Generalizations and extensions of classical conjectures in number theory like the equivariant Birch-Swinerton Dyer conjecture and the equivariant Tamagawa number conjecture.

The arithmetic study of toric varieties, modular curves and modular varieties as well as elliptic curves and abelian varieties.

The development of Arakelov theory.

The computation of rational points in curves and varieties.

The connections among formal groups, generalized Riemann--Hurwitz--type zeta functions and number theory.

The relation between the theory of motives, higher K-theory and arithmetic geometry.

Of course all these problems and topics are related and depend on each other. For instance harmonic analysis is heavily used in studying combinatorial problems and part of the development in Arakelov theory has been taylored to the study of modular varieties.

Job position description

The job will be to carry out a PhD Thesis in Number Theory with any Faculty Member with expertise on this topic as PhD Advisor. The list of Faculty Members can be found in https://www.icmat.es/people/faculty

Group Leader

https://www.icmat.es/miembros/burgos/

Research group website

https://www.icmat.es/research/lines/line4