PartAct · Partial actions of monoids and partial reflections
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2019-01-07 → 2021-01-06
- EU contribution
- €183,455
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Partial actions of monoids and partial reflections
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.
Data: CORDIS, © European Union
Project objective
Mathematics is at the heart of many areas of research in the Sciences and Social Sciences, with new mathematical achievements feeding the growth of other areas. Most scientists are familiar with the notion of the algebraic construct of a group, and how groups may be used to encode and explore the notion of symmetry, by studying the way on which they act on various structures. However, much of the universe is not symmetric, and neither do we always have actions totally known or defined. In such situations it is monoids, and partial actions, that provide the correct mathematical paradigm. The overall aim of this project is to develop and apply two (related) sets of semigroup theoretical techniques for actions and partial actions.We have four sets of Objectives:Obj. 1. To determine when the partial action of a monoid M on a set X via partial bijections can be lifted to actions by bijections, and answer the corresponding question for inverse monoids.Obj. 2. To determine the conditions under which strong partial actions of monoids on sets with additional structure lift to actions on sets with the same structure.Obj. 3. To develop the theory of associative algebras and C*-algebras constructed from (partial) monoid actions.Obj.4. To determine the algebraic constituents and constructs associated to reflection monoids, in particular, to determine their congruences and ideals, and associated lattices.Each Objective is supported by a Work Package, each of which will take approximately 6 months, and result in a journal output. Our methodology, as usual in pure mathematics, is that of testing examples (including by computation), spotting patterns of behaviour, making and proving conjectures. A careful plan has been designed for the ER, weaving the academic objectives of the proposal with a 2 year programme of training and personal development in both skills and knowledge.
Original text from CORDIS.
Participants
- UNIVERSITY OF YORK · YORK NORTH YORKSHIRECoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union