In this paper, we examine the moment membership problem, investigating both its decidability and undecidability.

We examine notions of universality and recursion in grammars, writing systems and the occurence of words, and find that they come in two main flavours: mechanistic and emergent. Preprints:202402.1330

If a vector (whole) is symmetric and/or positive, can these properties be pushed to the local tensors (parts)? We investigated this question in “Tensor decompositions on simplicial complexes with invariance” just published in Journal of Symbolic Computation 124, 102299 (2024) (The image is DALL·E’s rendering of this short explanation)

We had a wonderful day on the snow with views on the Kalkkoegel and Innsbruck

Operator systems are based on the cone of positive semidefinite matrices. What happens if we generalize this to other convex cones? We introduce general conic systems and show that many important theorems still hold. (arXiv:2312.13983)

What do universal Turing machines, universal spin models, NP completeness, top of a preorder and denseness of a subset have in common? They are all universal in a certain (precise) sense.

arXiv:2307.06851

Being undecidable and NP-hard are two ways of being hard - the former for machines with unbounded time, the latter for machines with poly-bounded time.

Does bounding an undecidable problem result in an NP-hard problem? Quite often. SciPost Phys. 14, 173 (2023)

We see classical spin models as languages and classify them in the Chomsky hierarchy. These allows us to say that the grammar of such physical interactions is at most context-sensitive.

We enjoyed a beautiful summer evening doing a BBQ at the Inn beach close to the university.

It often is much easier to represent a tensor when allowing an approximation error, even if this error is arbitrarily small. This leads to the notion of a border rank. In this paper, we show that this is also true when considering positive and invariant tensor decompositions. (arXiv:2304.13478)

We went on a group retreat to the Adolf Pichler Hütte, at the “lap“ of the Kalkkögel. Despite it only being mid September, the first snow was already there.

Purifying establishes a relation to the ideal world. The opposite of purifying —landing in our imperfect, non-ideal world— is described by the matrix convex hull. Here we show that semiclassical magic squares can be purified to quantum Latin squares. (arXiv:2209.10230)

That depends on how you measure it. We’ve thought of a new way of measuring it, which captures the complexity of the local structure of its function graph (arxiv:2208.08301)

New paper by Tobias Fritz and Andreas Klingler uploaded on arXiv. Andreas also presented this work at ACT 2022 (click here for the recording).

On May 20 we presented some challenging problems at the stand “Haben all schweren Probleme eine effiziente Lösung? A million Dollar question!“ on the Lange Nacht der Forschung.

Andreas Klingler has been awarded a DOC Fellowship of the ÖAW for his PhD, with the title “Decompositions of tensors with invariance, positivity and approximations“. Congratulations, Andreas! PS: Andreas has quite a lot of experience with these topics already, see e.g. tensor decompositions with symmetry in the approximate case, the application to polynomials or approximate cpsd rank.

We needed a new group photo together with Tomáš :) It’s great to now have everyone on board :)