Publications

Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.

Vertically parametrised polynomial systems are a particular nice class of parametrised polynomial systems for which a lot of interesting algebraic information is encoded in its combinatorics. Given a fixed polynomial system, we empirically study what constitutes a good vertically parametrised polynomial system that gives rise to it and how to construct said vertically parametrised polynomial system. For data, we use all polynomial systems in ODEbase, which we have transcribed to an OSCAR readable format, and made available as a Julia package OscarODEbase.

We present a framework for generalizing tropical homotopy continuation for computing stable intersections of tropical hypersurfaces, tropical linear spaces, and tropical inverted linear spaces. We also report on beginning implementations in OSCAR.

We describe a Laurent phenomenon for the Cayley plane, which is the homogeneous variety associated to the cominuscule representation of \(E_6\). The corresponding Laurent phenomenon algebra has finite type and appears in a natural sequence of LPAs indexed by the \(E_n\) Dynkin diagrams for \(n\leq 6\). We conjecture the existence of a further finite type LPA, associated to the Freudenthal variety of type \(E_7\).

We present a numerical solver based on the opensource finite volume library OpenFOAM that solves the filtered two-fluid equations for fluidized gas-particle flows using a wide range of diverse models developed in recent literature. We illustrate the phase coupling algorithm and discuss the implementation of the model of Sarkar et al. (2016). Finally, we compare the predictions from different constitutive models against experimental results using very coarse grids.