Objectives and Research lines

 Our research is organized into two main blocks: Geometry and Applications (led by Eva Miranda) and Algebra and Applications (led by Josep Àlvarez). These blocks interact dynamically, sharing ideas across themes like singularity theory, cohomology, Floer homology and computational methods, with the goal of advancing both theoretical and applied aspects of Mathematics and Physics. 

1.Singular Symplectic Geometry, its Mirrors, and Applications
This line explores the impact of singularities on geometrical structures, with applications ranging from celestial mechanics to fluid dynamics. Building on Miranda’s pioneering theory of b-symplectic structures, we address long-standing open problems like the Weinstein and Arnold conjectures and develop key tools in Floer homology.  We also plan to consider classification problems concerning group actions and integrable systems, using cutting-edge geometrical and topological techniques in contact and symplectic geometry.  We plan to use geometrical and topological techniques in contact and symplectic geometry to tackle open challenges in the Euler and Navier-Stokes equations, which are foundational to understanding the complexity of fluid motion.
Concrete Goals:

• • GEO1: Generalized singular Weinstein conjecture

  • GEO2: Floer homology and Arnold conjecture

  • GEO3: Poisson manifolds as limits of E-symplectic manifolds

  • GEO4: Group actions, quantization, and the [Q,R]=0 conjecture

  • GEO5: Singular Mirrors and Fluid Dynamics

  • GEO6: Applications to the complexity of Fluid Dynamics

  • GEO7: Applications to Physics, including General Relativity, galaxy morphology, and astrodynamics

2. GEOBOT: Applications of Differential Geometry to Computer Science and Robotics
This innovative line leverages advances in geometry and algebra to drive progress in robotics and computer science. A key focus is the development of a fluid computer that mimics computational processes via fluid dynamics, opening exciting possibilities for creating hybrid computers that combine fluid mechanics with quantum field theory, potentially outperforming current quantum computing models. We also apply geometric techniques to simulate realistic cloth behavior for robotic manipulation, driving advancements in neural networks and robot control.
Concrete Goals:

• • GEOBOT1: The hybrid computer

  • GEOBOT2: Mechanical Modelling of Cloth and Planning for Robotic Manipulation of Cloth

1. Algebraic and Geometric Aspects of Singularities
This line investigates singularities from an algebraic perspective, focusing on key structures like Bernstein-Sato polynomials, Hodge spectral numbers, and multiplier ideals. We aim to solve deep problems in stratification of singularities, a central topic in algebraic geometry with far-reaching implications for deformation theory. Additionally, we study the geography of fibred varieties, focusing on the role of slope inequalities in understanding the geometry of singular varieties, an important question in both mathematics and theoretical physics.
Concrete Goals:

• • ALG1: Stratification by roots of Bernstein-Sato polynomials

  • ALG2: Limit distribution of Hodge spectral numbers

  • ALG3: D-modules over singular varieties

  • ALG4: Geography of fibred varieties

  • ALG5: Prime ideals in three-dimensional regular local rings

2.BIOALG: Applications to Phylogenetics
This line uses algebraic geometry and statistics to solve problems in phylogenetics, the study of evolutionary relationships among species. By employing algebraic methods, we go beyond traditional parameter estimation, tackling complex evolutionary networks with models tailored for amino acid substitutions and phylogenetic reconstruction. Our work has applications in evolutionary biology and artificial intelligence, addressing challenges such as model selection and identifiability in complex data sets.
Concrete Goals:

• • BIOALG1: Algebraic substitution models

  • BIOALG2: Identifiability and phylogenetic reconstruction