Invariant chains in algebra and discrete geometry

• Applicant: Professor Dr. Tim Römer
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539867266

In algebra, geometry, combinatorics and areas of applications of these disciplines there exist in many interesting situations the phenomena that objects of interest occur in ascending chains A(n) and there are suitable and compatible group actions on these objects. Example are generic determinantal ideals in polynomial rings with an increasing number of variables or symmetric polytopes as well as cones in real vector spaces of increasing dimensions. Often there exist a limit object B of the chain, which can be useful to study properties of the elements of the chain and vice versa. In the two examples, the union of the elements of the chain is such a limit object (defined in a suitable way). Three questions are then of importance: (1) Determine the asymptotic behavior of properties of interest of the A(n); (2) Study corresponding properties of B; (3) Show a connection between the properties of A(n) and the ones of B. An answer to (1) can be seen as interesting local information and one to (2) as some kind of global information. (3) can then be considered as a local-global principle which one would like to understand. In the recent years, this approach was extremely successfully applied to various situation in the areas mentioned above with applications, for example, also in machine learning, optimization, representation theory. Results include, for example, equivariant versions related to several of Hilbert’s classical theorems in commutative algebra and classical algebraic geometry like Hilbert’s basis theorem. In polyhedral geometry equivariant versions of Caratheodory’s theorem and Gordan’s lemma were shown and a first approach to an equivariant statement to the Minkowski-Weyl theorem was established. The overall goal beyond the project is to develop foundations of commutative algebra up to symmetry and polyhedral geometry up to symmetry in a systematic way. In this proposal we consider a framework developed by the applicant with his co-authors in a series of papers to extend the known theory. More precisely, it is aimed to study: (a) the rationality of the equivariant Ehrhart series of symmetric chains of rational polytopes; (b) an equivariant Minkowski-Weyl theorem (in a "complete" way); (c) the independent set theorem in algebraic statistics from a purely combinatorial point of view; (d) primary decompositions of symmetric chains of (monomial) ideals in polynomial rings. The goals of this proposal are at the core of four of the nine central themes of the Priority Program, which are: commutative algebra, convexity, enumeration, and lattice points. Applications are also expected in other topics of the SPP, for example in (algebraic) statistics.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies