Nowadays mathematics becomes more and more specialized. However, there exist certain constructions and methods appearing in almost all of its subfields. This is for instance the case of invariants. An important part of the mathematical research relies on the study, the computation and the applications of invariants, sometimes generating spectacular results. The topics of the present project gathers three main fields of mathematics, namely algebra, algebraic geometry and algebraic topology. Our choice is based on the fact that these three fields have deep connections, their research areas and problems being often mutually determined. The unifying element of the project consists in the presence and study of some important invariants of algebraic, geometrical and topological structures, such as: (co)homology and K-theory groups, Betti numbers, moduli spaces, syzygies, Chern classes, Castelnuovo-Mumford regularity, polynomial invariants for knots and links, Brauer groups etc. Among the objectives of this project we mention: the study of moduli spaces of fibred bundles on 3-dimensional elliptic non-Kahler Calabi-Yau manifolds and of Calabi-Yau smoothings for singular manifolds; the study of locally monomial structures and of syzygies for algebraic curves; effective computations for the cohomology with local coefficients and for the monodromy of the Milnor fibre; the study of the Mahler measure for special classes of polynomials; the study of Betti numbers and Castelnuovo-Mumford regularity for standard graded algebras; the study of the invariants for polymatroidal ideals and of the smoothness of morphisms of commutative rings; the study of the co-Galois theory, Artin-Coxeter type groups and of Galois invariant analytical Krasner maps; homological properties of algebras in monoidal categories, actions and coactions of Hopf algebras.
The methods based on the study and computation of invariants play a central role in the approach to the problems of modern mathematics. The goal of the present project is to use and develop these methods in the following five research topics:
1. Local and global invariants of manifolds
2. Invariants for hyperplane arrangements, knots and links
3. Homological invariants for commutative rings
4. (co) – Galois theory and invariants in local fields
5. Hopf algebras and monoidal categories.
Vasile Brinzanescu, Senior Researcher I in the Institute of Mathematics "Simion Stoilow" of the Romanian Academy
Nicolae Ciprian Bonciocat
Dorin Mihail Popescu
The Green Conjecture for Exceptional Curves on a K3 surface, authors Marian Aprodu, Gianluca Pacienza;
Nonvanishing for Koszul Cohomology of Curves, authors Marian Aprodu, Jan Nagel, accepted for publication in Commentarii Mathematici Helvetici;
Smootheness and differentials in positive characteristic, author C. Ionescu;
On the vanishing of the first André-Quillen homology, author C. Ionescu;