We propose in this White Paper a concept for a space experiment using cold atoms to search for ultra-light dark matter, and to detect gravitational waves in the frequency range between the most sensitive ranges of LISA and the terrestrial LIGO/Virgo/KAGRA/INDIGO experiments. This interdisciplinary experiment, called Atomic Experiment for Dark Matter and Gravity Exploration (AEDGE), will also complement other planned searches for dark matter, and exploit synergies with other gravitational wave detectors. We give examples of the extended range of sensitivity to ultra-light dark matter offered by AEDGE, and how its gravitational-wave measurements could explore the assembly of super-massive black holes, first-order phase transitions in the early universe and cosmic strings. AEDGE will be based upon technologies now being developed for terrestrial experiments using cold atoms, and will benefit from the space experience obtained with, e.g., LISA and cold atom experiments in microgravity.KCL-PH-TH/2019-65, CERN-TH-2019-126

Let Λ be a finite-dimensional associative algebra. The torsion classes of mod Λ form a lattice under containment, denoted by tors Λ. In this paper, we characterize the cover relations in tors Λ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of tors Λ in terms of representation theory. Finally, we show that, in general, the algebra Λ is not characterized by its lattice tors Λ. In particular, we study the torsion theory of a quotient of the preprojective algebra of type An. We show that its torsion class lattice is isomorphic to the weak order on An.

In this article we give an algorithm for determining the generators and relations for the rings of semi-invariant functions on irreducible components of Rep A (β) when A is a (acyclic) gentle string algebra and β is a dimension vector. These rings of semi-invariants turn out to be semigroup rings to which we can associate a so-called matching graph. Under this association, generators for the semigroup can be seen by certain walks on this graph, and relations are given by certain configurations in the graph. This allows us to determine degree bounds for the generators and relations of these rings. We show further that these bounds also hold for acyclic string algebras in general.

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In this article, we construct the generic modules in each irreducible component of variety of β-dimensional modules of a triangular gentle algebra. The construction is completely combinatorial and allows for determination of canonical decomposition of irreducible components as well as calculation of the dimension of the higher self-extension spaces for generic modules.