Machine learning combinatorial statistics and maps

• Applicant: Professor Dr. Christian Stump
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539866596

Algorithmic approaches and computer-assistance have a long history in combinatorics research. Finding well-suited examples and counterexamples is a driving force in formulating conjectures and in proving theorems. In recent years, significant progress has been made in the area of machine learning algorithms, and they are currently becoming an exciting tool for mathematicians in general and for combinatorialists in particular. This project aims at systematically analyzing and studying the combinatorial statistics database FindStat with machine learning techniques by reinforcement learning combinatorial maps and geometric deep learning combinatorial statistics. This database is developed and maintained by the PI together with Martin Rubey, with support of many contributors. The two guiding open problems are longstanding questions in enumerative, bijective and algebraic combinatorics. They concern the famous q,t-Catalan numbers. The first is to find a combinatorial proof of their symmetry and the second is to find a combinatorial definition for type B and for general reflection groups. These two longstanding open problems are perfect candidates to be approached using machine learning techniques. A lot of research has been devoted to solutions and we expect machine learning combinatorial statistics and maps to provide genuinely new combinatorial insight. All research branches of this project belong to the core of the priority program SPP2458. The presented open problems interact with the nine themes and the expected advancements are driven by the availability of the combinatorial data in the FindStat database. The concrete research questions are motivated by structures in the core themes Commutative Algebra, Convexity, and Dynkin Classification, while the heart of the project belongs to classical questions in the Enumeration of combinatorial structures.

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