# Three mathematicians are awarded project grants by the Independent Research Fund Denmark

Peter Jørgensen, Simon Kristensen and Andreas Basse O'Connor awarded project grants from IRFD.

DNRF Chair, Prof. Peter Jørgensen has been awarded 5.454.019 DKK to pursue the project "Higher Dimensional Cluster Theory", which will complement Peter's existing two projects dedicated to Higher Homological Algebra (funded by AUFF) and Calabi-Yau Categories (funded by DNRF). The project will cover a postdoc and a phd student that are scheduled to join Peter's "Aarhus Homological Algebra" group in 2022.

Classic cluster theory is controlled by two-dimensional combinatorial structures like triangulations of surfaces. This proposal will generalise cluster theory to higher dimensions, with the key aim of defining higher dimensional cluster algebras. The methodology builds on known examples of higher dimensional cluster categories and tropical friezes.

The total project consists of three independent projects: (Dynkin), (RepFin), and (Surface). Project (Dynkin) concerns the higher dimensional generalisation of cluster theory to Dynkin type A, project (RepFin) concerns the generalisation to drepresentation finite type, and project (Surface) concerns the cluster theory of generalised surfaces, for instance cyclic polytopes with an internal “hole”. The proposal has potential applications in combinatorics because of its links to higher Stasheff-Tamari orders and in geometry because of the links between cluster theory and knot invariants.

Assoc. Prof. Simon Kristensen has been awarded 2.876.500 DKK to pursue the the project "Irrationality and transcendence of numbers". The project will cover a phd student, scheduled to start in 2022.

The aim of the research project is to extend existing and develop new methods for the study of the irrationality and transcendence of numbers given in terms of some representation. The representations to be studied include infinite series, infinite products and general continued fractions. We will work on general criteria for irrationality and transcendence for series, products and continued fractions given in terms of algebraic integers. In addition, we will work on new approaches to determining the irrationality of the Riemann zeta-function at odd integers.

Assoc. Prof. Andreas Basse-O'Connor has been awarded 2.372.477 DKK to pursue the project "Phase Transitions for the Boolean Satisfaction Problem". The project is affiliated with the DIGIT interdisciplinary research center and will cover a phd student, scheduled to start in 2022.

The boolean satisfaction (SAT) problem is an important decision problem involving a large number of constraints. This problem has attracted a lot of interest in computer science, physics and mathematics, and it is a long-standing conjecture (the satisfiability conjecture) that there exists a phase transition where the random k-SAT problem goes from being under- to overconstrained. After many years of intense work, Ding, Sly and Sun succeeded in 2015 to prove the satisfiability conjecture when k is large. In this project we aim to complete the Ding-Sly-Sun Theorem by showing that the satisfiability conjecture indeed holds for all k, and in particular, it holds for the important 3-SAT problem. The main difficulties in proving this theorem are due to a high level of long-range dependence when k is small. Our methodology is heavily inspired by the one-step replica symmetry breaking predictions from physics, but will also involve newly developed techniques from long-range dependence.