__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__

__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__

__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__

__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__

__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__

__Research group website__

__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__

__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__