# Self-similarity: recursively definable objects in topology, analysis, category theory and algebra

Lead Research Organisation: University of Glasgow
Department Name: School of Mathematics & Statistics

### Abstract

Cut a square in half once horizontally, then once vertically, and you get four small squares. Cut a branch off a tree, and that branch looks something like a small tree itself. Take the whole numbers ending in zero (10, 20, 30, ...), and that looks like a spread-out version of all the whole numbers (1, 2, 3, ...).These are all examples of self-similarity , where an object can be cut up in such a way that the pieces look like smaller copies of itself. Put another way, we have an object that looks like several copies of itself glued together. In more complicated situations there may be two or more objects: for instance, one object X may look like three copies of X stuck to one copy of a second object Y, and Y may look like two copies of X stuck to four copies of Y. This is like simultaneous equations from school mathematics (here, X = 3X + Y and Y = 2X + 4Y).Self-similarity occurs in remarkably diverse parts of mathematics, although it is not always easy to put one's finger on the exact connection between different forms of it. Examples include not only the well-known fractals , but also more mundane objects such as circles, cylinders and balls. I propose a broad and far-reaching research programme to set up a general theory of self-similarity and to apply it in several areas, including algebra, geometry, analysis, and theoretical computer science. What is the point of such a programme? The greatest advances in mathematics are made when apparently unrelated phenomena, often observed in areas that seem to be poles apart, are understood to be instances of a single, general phenomenon. (For example, Newton realized that the motion of a cricket ball and the orbits of the planets around the sun are governed by the same force - gravity - and therefore by the same equations.) This unification of disparate ideas leads to great simplification, suggests new results by analogy, and clarifies thinking. I aim to unify the different types of self-similarity.More specifically, I believe I can find new invariants . An invariant is what enables you to tell two things apart. For instance, you can always tell a jumper from a pair of trousers, even if you are dressing in the dark and your clothes are made of identical, baggy material: a jumper has one more hole. Here the invariant is the number of holes; since the two items have different numbers of holes, they can be distinguished. Now, some of the most striking examples of self-similar objects are fractals, which are infinitely intricate webs of filaments and gaps. Since most fractals have infinitely many holes, this invariant is almost useless for telling fractals apart. To distinguish between fractals we need a much more subtle invariant. I believe I can define one. It comes about by transforming self-similarity of the usual geometric kind into self-similarity of an algebraic kind (like the simultaneous equations above), and is an extension of the invariant known as Euler characteristic.I come to this project with experience in finding ways of describing unusual, complicated structures in a simple, practical way. This is exactly what is needed here. Objects such as fractals may appear forbiddingly complex, but I have begun to show that they can be described in such a way that difficult problems become approachable. To carry out this programme I will need the input of specialists in other fields. This will be achieved through targeted visits to experts and through continuing to give a large number of seminars to varied audiences at different locations, resulting in cross-fertilization of ideas. (This month, for instance, I am giving one seminar to algebraists in Edinburgh and another to complex dynamicists in Liverpool.) Through a combination of developing existing collaborations, initiating new ones, and using my own expertise, I plan to transform our understanding of self-similarity and turn it into a tool of great practical use.

### ORCID iD

Thomas Leinster (Principal Investigator)

### Publications

10 25 50

Berger C (2008) The Euler characteristic of a category as the sum of a divergent series in Homology, Homotopy and Applications

Brient J-C. (2002) The calorimetry at the future e+ e- linear collider in ArXiv e-prints

Fiore M (2010) An abstract characterization of Thompson's group F in Semigroup Forum

Leinster (2014) Basic Category Theory

Leinster T (2016) Maximizing Diversity in Biology and Beyond in Entropy

Leinster T (2012) Integral geometry for the 1-norm in Advances in Applied Mathematics

Leinster T (2012) Notions of Möbius inversion in Bulletin of the Belgian Mathematical Society

Description As a project based in pure mathematics (although with significant applications outside that field), any explanation for a non-specialist audience is bound to be highly approximate. With that caveat, the major discoveries are:

1. Surprising connections between different parts of mathematics. My core research is in category theory, a branch of mathematics designed to look at the big picture and explain general phenomena. Using categorical methods has produced new results and conjectures in other mathematical fields. For instance, it has automatically produced a new geometric quantity, magnitude, that is closely related to ancient quantities such as volume, surface area and perimeter; but fully proving this seems to need methods from the field of partial differential equations, on the other side of the mathematical universe from category theory.

2. Surprising applications to the measurement of biodiversity. By following up leads outside my own immediate subject area, I discovered that the work I was doing could be applied to answer a question that has been debated in the ecological literature for more than half a century: how does one quantify the diversity of a biological community? This is, of course, not the last word in the debate, but the answer that I and my collaborators were able to provide has generated a great deal of excitement and interest. We are currently developing it further, developing methods (and software) to identify diversity hotspots in real-world communities.
Exploitation Route As described in item 2 above, there is currently significant interest in developing and applying our new methods of biodiversity measurement. Also, of course, the mathematical results that I and my collaborators have published in the course of this grant can be built on and further developed. Here, I would single out the new theory of integral geometry in the 1-norm (Advances in Applied Mathematics 49 (2012), 81-96) and the conjecture on the magnitude of a convex subset of Euclidean space (Geometriae Dedicata 164 (2013), 287-310).
Sectors Environment

URL http://www.maths.ed.ac.uk/~tl/

Description This project appeared to be very much basic, foundational mathematics, but turns out to have had surprising applications. For instance, for 50 years, ecologists have been debating the best way to quantify biodiversity. (Do you simply count the number of species, or do you take account of population balance? Should a community of ten dramatically different species count as more diverse than ten species of slug?) It turns out that the quantities that arose naturally in my pure-mathematical research - specifically, cousins of topological Euler characteristic - provided a very good answer. This was published in Ecology, one of the top journals in that subject, and has led to a very active collaboration with a range of biological scientists. This is still basic science, but is already being actively taken up by conversation ecologists. For instance, a 12-million-euro EU grant proposal currently under consideration (led by Prof Douglas Yu, East Anglia) plans to use our methods in conducting large-scale assessments of global biodiversity. This demonstrates how quickly mathematics that is apparently very pure can be used in pressing real-world problems. I have been active in disseminating my work outside academia, thus providing a cultural contribution. For instance, I have given a public lecture on the work described above (Edinburgh, 2013), and I write for a popular research mathematics blogs that regularly draws in readers from outside academia.
First Year Of Impact 2008

Description BBSRC International Workshop
Amount £10,200 (GBP)
Organisation Biotechnology and Biological Sciences Research Council (BBSRC)
Sector Public
Country United Kingdom of Great Britain & Northern Ireland (UK)
Start 06/2012
End 07/2012

Description Carnegie Research Grant
Amount £2,200 (GBP)
Organisation Carnegie Trust
Sector Charity/Non Profit
Country United Kingdom of Great Britain & Northern Ireland (UK)
Start 06/2012
End 07/2012

Description Centre de Recerca Matemàtica
Amount £15,000 (GBP)
Organisation Autonomous University of Barcelona (UAB)
Department Mathematics Research Centre
Country Spain, Kingdom of
Start 06/2012
End 07/2012

Description Edinburgh Mathematical Society
Amount £1,000 (GBP)
Organisation Edinburgh Mathematical Society
Sector Learned Society
Country United Kingdom of Great Britain & Northern Ireland (UK)
Start 01/2010
End 12/2010

Description Glasgow Mathematical Journal Trust
Amount £1,200 (GBP)
Organisation University of Glasgow
Department Glasgow Mathematical Journal Learning and Research Support Fund
Country United Kingdom of Great Britain & Northern Ireland (UK)
Start 01/2011
End 12/2011

Description Boyd Orr
Organisation University of Glasgow
Department Boyd Orr Centre for Population and Ecosystem Health
Country United Kingdom of Great Britain & Northern Ireland (UK)
PI Contribution Since 2010, I have been a member of this highly interdisciplinary research team, comprising ecologists, epidemiologists, animal breeding experts, veterinary scientists, mathematical modellers, and mathematicians. I have played a central part in the development of projects on biological diversity. I won funding for an away weekend on the theme of diversity, involving Centre members and some external scientists.
Collaborator Contribution The Centre provides an invaluable context for interdisciplinary research, as well as miscellaneous funding for activities.
Impact Yes, multi-disciplinary: see above. BBSRC International Workshop grant for 5-week research programme at Centre de Recerca Matemàtica, Barcelona, 2012.
Start Year 2010

Description n-Category Cafe
Organisation University of Texas at Austin
Department n-Category Cafe
Country United States of America