SINGULAR2025

Objective:

The project revolves around different aspects of singularities and cryptography from a wide range of viewpoints. We work in a settled collaboration between the projects at BCAM, UCM and UNIZAR, and other research groups at the international level. The main questions of this proposal are structured as follows: (1) Topology and arithmetics of the singularities. Topology of singularities and Milnor fibrations are intertwined with arithmetical aspects; e.g., in the monodromy conjecture, which we study in classes of singularities with special structure. We will study the Bernstein polynomials in the meromorphic case; power structures on Grothendieck rings; the topology of quasi-projective varieties, in particular, Zariski pairs; hyperplane and varieties arrangements; and low-dimensional topology. Cohomological methods will play an important role. (2) Symplectic study of singularities and degenerations. Symplectic topology, Floer theory, and mirror symmetry provide new tools that led to the solution of long-standing problems in our field (e.g., the family case of Zariski's conjecture). Symplectic A'Campo spaces, designed to apply the Floer-theoretic methods to singularity theory, have led to new developments in mirror symmetry. We plan to draw from this interplay to attack the SYZ conjecture in Mirror Symmetry, the Zariski Conjecture for non-isolated singularities, and important questions about arc spaces or in the study of symplectic curves and Lagrangian spines. (3) Metric study of Singularities. The metric study of real and complex singularities and degenerations evolved a lot in recent years producing important results such as the inner Lipschitz classification of surface singularities or the Lipschitz study of multiplicity. We will develop the MD homology and apply it to study degenerations in many contexts such as arithmetic, algebraic or Riemannian geometry or model theory. (4) Algebraic study of singularities. One of the motivating reasons is the absence of the resolution of singularities in positive characteristic, which requires one to rely on new tools, such as ones coming from the valuation theory or from the Frobenius endomorphism, a distinctive ingredient that can be exploited to measure singularities in various ways. Even when resolutions are known, e.g., for normal surface singularities, one benefits a lot from algebraic invariants, such as the Milnor and delta invariants, jumping numbers, and Jacobian and multiplier ideals. Our program also includes calculations of matrix factorizations, the study of dicritical divisors, the description of Q-resolutions and toric resolutions, and free divisors. (5) Cryptography. The recent advances in quantum computing led NIST (the National Institute of Standard and Technology, USA) to develop the standards for Post Quantum Cryptography, the new public key cryptography resisting quantum computer attacks. The members of our team participated in NIST competitions with several schemes, including one of the winners (Kyber), and we will keep working on improvement of existing and development of new schemes. In conclusion, thanks to the varied expertise of the members of the project, we expect significant progress in theoretical questions, bringing a deeper understanding of the relations among the topological, arithmetic, symplectic, metric and algebraic aspects of singularities, and on a more practical side of development of cryptographic schemes.