The network empraces a wide spectrum of research topics including algebraic, transcendental, topologic, combinatorial and arithmetic methods:
(1) Local algebra: Hilbert functions, filtrations, graduations, Rees rings, methods in positive characteristic, multiplier ideals and its applications to moduli theory, resolution of singularities and minimum model program.
(2) D-modules and differential methods. Differential Galois theory. Applications to Physics.
(3) Hilbert schemes, vector bundles and principal bundles on curves. Geometry and classification of complex manifolds. Abelian varieties. Moduli problems, degeneration, compactification…
(4) Hodge theory and its applications to algebraic geometry and singularities, coverings spaces, and knot theory. Alexander invariants.
(5) Singularities and degenerations from topological, metric, motivic, arithmetic and Hodge theory points of view. In particular, topology of quasi-projective manifolds, hyperplanes arrangements, complements of hypersurfaces. Non-isolated and quotient singularities. Polar and discriminant varieties. Singularities of foliations.
(6) Geometric structures: Kähler, hyperkähler, Sasakian, holomorphic symplectic. Simplectic geometry, Floer theory, and its applications to algebraic geometry and singularities.
(7) Resolution of singularities.
(8) Non-Archimedean methods in theory of singularities and degenerations. Valuations and valuation spaces. Berkovich spaces.
(9) Newton-Okounkov bodies and toric geometry.
(10) Arc spaces in zero and positive characteristics. Motivic and invariant integration of singularities. Interaction with logic methods, in particular model theory.
(11) Zeta functions in algebraic geometry and singularities. Igusa conjectures.
(12) Singularities of real and complex holomorphic and differential applications. Characteristic classes in singular spaces.
(13) Elliptic curves, their torsion subgroups, Galois representations, modular forms, equivariant Tamagawa number conjecture, and related Birch and Swinnerton-Dyer like conjectures.
(14) Applications of non-Archimedean analysis to Arakelov theory. Applications of hybrid topology to algebraic geometry and singularities. Study of singular metrics.
(15) Arithmetic algebraic geometry, including the study of abelian extensions of fields of functions of algebraic curves: applications to arithmetic.
(16) Theories of cohomology in algebraic geometry. Local, étale and l-adic, motivic, crystalline, and
prismatic cohomology. Grothendieck duality. Cohomology of stacks and formal schemes; vanishing theorems.
(17) Derived categories in algebraic geometry, derived algebraic geometry.
(18) Cryptography and code theory (error correcting codes with algebraic geometry techniques, and applications to cryptography). Post-quantum cryptography with algebraic geometry methods.
(19) Combinatorial methods and effective methods.
(20) Algebraic geometry and singularities applied to computer graphics and data science, robotics, biochemistry or phylogenetics.
