Partial actions of monoids and partial reflections

Project Information

PartAct

Grant agreement ID: 799419

DOI 10.3030/799419

Project closed

EC signature date 27 February 2018

Start date 7 January 2019

End date 6 January 2021

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost € 183 454,80

EU contribution € 183 454,80

Coordinated by

UNIVERSITY OF YORK
United Kingdom

Periodic Reporting for period 1 - PartAct (Partial actions of monoids and partial reflections)

Reporting period: 2019-01-07 to 2021-01-06

Summary of the context and overall objectives of the project

Humans have a deep-rooted sensitivity to symmetry in the world, which has been studied by psychology and neuroscience, and applied in art and architecture from the dawn of humanity. On the other hand, symmetry is a fundamental tool in science, and has greatly helped our understanding of physics, chemistry, or even biology. All this was achieved by translating the concept symmetry to the language of mathematics. The prevailing idea was to consider those transformations of some mathematical or physical structure which leave its key features unchanged, which led to the notion of groups. This approach, however, fails to capture partial symmetries, similarities between parts of the whole. There are structures which we perceive to be highly symmetric, though do not have any symmetries in this classical sense, such as that of fractals, mathematical shapes that exhibit the same pattern at increasingly smaller scales. The same property is often found in nature: trees, blood vessels, frost crystals. As a result, modern mathematics brought with it the need for a theory of partial symmetries. The appropriate framework was found to be via inverse semigroups, a mathematical concept first defined in the 50s.

Inverse semigroups generalize groups as partial symmetries generalize symmetries, and while they exhibit many similar properties, are vastly more complicated. The more generic structure is also more difficult to work with, and as a result, the theory is much less developed than that of groups. Still, they seem an indispensable structure in mathematics, appearing naturally in various areas within and outside of mathematics, from computer science to quasi-crystals. The overall objective of the fellowship was to expand are knowledge base on the theory of inverse semigroups. In particular we focused on their actions by partial bijective maps, and how these connect to other parts of mathematics. Our results greatly enriched our understanding of the questions investigated and in particular settled a problem that has been open for over 40 years.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The concept of an inverse semigroup was first defined in the 50s, independently by Preston and Wagner, in an attempt to describe the algebraic structure of bijections between subsets of a fixed set. A set of such partial bijections closed under composition and taking inverses forms an inverse semigroup, and conversely, every inverse semigroup can be represented this way. As such, the relationship between inverse semigroups and partial symmetries is a generalization of the relationship between groups and symmetries. Furthermore, similarly as to how groups act by bijections, inverse semiroups act by partial bijections.

During the fellowship, we investigated various facets of actions of inverse semigroups by partial maps. One of these is such actions can be used to construct associative and C*-algebras. We gave a full characterization of the simplicity of the former (which also has implications for the latter), and applied these results to obtain an algorithmic characterization when the action is by certain partial maps on infinite regular trees, these are called Nekrashevych algebras. These results settle a 40-year-old open problem posed by Douglas Munn, conclude a decade research on the simplicity of étale groupoid algebras, and make very significant progress in understanding the simplicity of Nekrashevych algebras and their C*-algebra counterparts. These results have been summarized in two papers:

Simplicity of inverse semigroup and étale groupoid algebras (with B. Steinberg), Adv. Math. 386 (2021) 107611, freely accessible on arXiv:2006.13787

On the simplicity of Nekrashevych algebras of contracting self-similar groups (with B. Steinberg), submitted to Journal für die reine und angewandte Mathematik, freely accessible on arXiv:2008.04220

We also investigated several question about inverse semigroups defined by geometric properties. Namely, motivated by the success of geometric group theory, we studied the algorithmic properties of inverse semigroups that have a hyperbolic, and respectively, tree-like geometry. We have proved several results, in particular, managed to extend one direction of the celebrated Müller-Schupp theorem for groups. The work is summarized in the following paper:

Algorithmic properties of inverse monoids with hyperbolic and tree-like Schützenberger graphs (with R. Gray and P. V. Silva), freely accessible on arXiv:1912.00950

In addition to the papers, we have disseminated the results at several seminars and conferences:

Simplicity Nekrashevych algebras of contracting self-similar groups: seminar talks at the Mathematical Institute, Göttingen, (online, Jan 14, 2021), Western Sydney University, (online, Oct 22, 2020), Alfréd Rényi Institute of Mathematics, (online, Sept 28, 2020).

Simplicity of contracted inverse semigroup algebras: seminar talks at University of Szeged, (Hungary, Sept 2, 2020), University of York, (online, May 20, 2020), conference talk at International Conference on Semigroups and Applications, (Cochin, India, Dec 9, 2019).

Algorithmic properties of tree-like and hyperbolic inverse monoids: seminar talks at University of York, (UK, Oct 17, 2019), Heriot-Watt University, (Edinburgh, UK, May 22, 2019), City University of New York, (USA, May 10, 2019), University of St Andrews, (UK, Apr 4, 2019), conference talk at Semigroups and Groups, Automata, Logics, (Cremona, Italy, Jun 10, 2019).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Our results were highly acclaimed by the scientific community, resulted in invitations for altogether 10 talks of the ER (and several other for her coathors), and we have published/expecting to publish in the most pretigous journals. Our latest preprint, despite only being 8 months old, already has 2 independent citations, which is rare in pure mathematics. As this shows, the results are expected to have a large impact on the scientific community.

As part of the project, the ER organized a project workshop, which is currently postponed due to COVID-19, but the ER and the University of York are committed to realizing it once restrictions are lifted, hopefully in the second half of 2021. The project workshop ties together 3 topics that appear in the fellowship research: semigroups, geometry and C*-algebras, and it is (to our best of knowledge) going to be the first workshop in its kind. The list of speakers include experts from all 3 areas, and we expect that the workshop will facilitate many intradisciplinary converstations, research connections and possible collaborations in the future. This could shape research in the interface of these areas for year to come.

A famous fractal with lots of partial symmetries: the Sierpinski triangle