Group Leader: Luis Narváez Macarro
Description:
The goal of RL11 is to deepen the interactions between Algebra, Geometry and Topology, and to address some important current issues in this cluster.
Besides its theoretical nature, this line is strongly connected with the quantum cryptography, where deep theoretical results and techniques are needed.
Our research in the last years has dealt with D-module Theory as a tool in Algebraic Geometry and Singularity Theory; majorations of exponential sums, Galois representations and the Inverse Galois Problem, in the context of Arithmetic Algebraic Geometry and Number Theory; combinatorics; Geometric Group Theory and low dimensional Topology, Homotopy Theory and K-Theory and compact Lie groups.
We have worked on the definition of a new type of L-function that allows to get improved bounds for exponential sums and for the number of points on some varieties over finite fields results, proved a famous open problem in Artin groups and developed a polynomial algorithm for braid combing, among
other subjects.
Members:
M.C. Fernández
Ramón Flores
Juan González-Meneses
Clara Grima
Fernando Muro
Research portfolio:
Post-quantum cryptography
A. Group theory (Artin groups and related families)
B. Arithmetic geometry (supersingular elliptic curves)
C. Almost perfect non-linear (APN) functions over boolean fields
D.Quantum and classical complexity in problems related to algebraic structures
E. Computational algebra (Gröbner bases)
F. Coding theory
G. Cryptography.
Security. Error-correcting codes. Resource management.
D-modules and Hodge modules
A. Algebraic geometry
B. Hodge theory
C. Derived categories
D. Computational algebra and effective calculations
E. Hypergeometric systems
F. Mirror symmetry of families of divisors
Effective symbolic computation, mirror symmetry in physics, partial differential equations, computation of maximum likelihood estimates
Braid groups and generalizations
A. Group theory
B. Combinatorics in group theory
C. Geometric group theory
D. Low-dimensional topology
E. Algorithmics
F. Cryptography
Cryptography, Computation
Computational Methods in Algebra, D-modules, Representation Theory and Optimization
A. D-modules and singularities
B. Combinatorial representation and invariant theory
C. Computational Methods in Lie Algebras, Leibniz Algebras and Malcev Algebras
D. Computational Methods in linear and non-linear Integer Programming
Computer Science, Complexity of computation, Optimization and control.
Topological data analysis
A. Simplicial complexes for point clouds
B. Persistent homology
C. Barcodes
Machine learning, shape retrieval, time series, medicine
Knot Theory
A. Low-dimensional topology
B. Graph Theory
C. Homotopy Theory
D. Group Theory
E. Algorithmics
Study of DNA, Quantum Physics (via Jones polynomial), Chemistry (structure of molecules).
Algebra, singularities, number theory and applications
A. Algebraic Geometry and D-module theory
B. Singularities
C. Representation theory
D. Number theory
Computer Science, Complexity of computation
Related projects:
- Arithmetic Geometry, D-Modules and Singularities
- Algebraic Geometry and Arithmetic Geometry: Differential methods, singularities, cohomology and elliptic curves
- Higher structures in Differential Geometry and Homotopy Theory
- Arithmetic Geometry and Applications
- Computational Methods in Algebra, D-modules, Representation Theory and Optimization
- Cross-cutting challenges in homotopy theory, knots, and groups
- Spanish Topology Network
- Braids: Knots, Garside Groups and Mapping Class Groups
- Triangulated structures in algebra, geometry, and topology
- Groups and topology
- Algebra, singularities, number theory and applications
- Combinatorics of Networks and Computation
- Cross-cutting challenges in homotopy theory, knots, and groups
Related transfer:
- Irregular Hodge filtration of hypergeometric systems
- Hodge ideals of free divisors
- Gauss-Manin system of the Dwork familiy
- Realization of sporadic finite simple groups via families of exponential sums
- Quantum vs Classical Complexity in Numerical Semigroups
- Cryptanalysis of Public Key Protocols in cryptography using braid groups
- Mathematical algorithms applied to Mathematical Education
- Solving Kauffman Conjecture (stated in 1983)
- A new approach to extreme Khovanov homology
- Khovanov spectrum of periodic links admits a homology group action
- Homotopy theory of classifying spaces
- Graphs, right-angled Artin groups and cryptography
- Solving a problem open for 20 years: Intersection of parabolic subgroups in spherical Artin groups.
