# Experimental Mathematics in Number Theory, Operator Algebras, and Topology

**The project is completed**.

The project attempted to advance three independent, but neighboring, areas of mathematical research (number theory, operator algebras, and topology) by systematic experimentation using the most powerful grids of computers in Denmark and drawing on the highest computer scientific expertise to develop algorithms and methods in active collaboration with leading mathematicians in the three areas.

The recent development of hard- and software in combination with the unique theoretical expertise present in Copenhagen and neighbouring universities at Odense and Lund give us what we see as exceptional opportunities for creating significant breakthroughs both in mathematics and computer science.

Computer science

Number theory

Operator algebras and dynamics

Topology

## Computer science

For experimental mathematics to become a success it is vital to have a tool-chain that allows mathematicians to describe such mathematical experiments in a reasonable abstract high level language, so that time will be spent on mathematics and not computer programming. At the same time there is no doubt that most, if not all, such experiments will belong to the set of algorithms that are known as N P -hard, i.e. they will require very large computing resources, even for small problems, thus the efficiency of the experiment implementation is essential. It is well known that high productivity and high performance are opposed goals in programming languages. The computer science activities will focus on developing a platform that can be used to express mathematical experiments, and support both ease-of-use and flexibility to match a large set of potential mathematical experiments. Once such a platform is developed a high performance backend should also developed so that experiments can be executed in acceptable time. The computer science people in the project are established high performance computing researchers that the last few years has worked with productivity tools in natural science and thus have a strong platform to build the proposed tool chain on.

## Number theory

The theory of mod *p* reductions of modular forms and their attached Galois representations is by now very well understood. Recently, there has been a push by certain people, including IK with collaborators I. Chen and G. Wiese - and others - to extend aspects of this understanding to the question of 'higher congruences', i.e., to reductions mod *p^m* of modular (eigen)forms, *m > 1*. Ultimately, the motivation for this extension is the desire to get more information on the *p*-adic representations attached to eigenforms.

The new work reported on by Chen, Kiming, and Wiese, shows that when *m > 1* the theory appears to become much more complicated: there are for instance apparently not just one but three different notions of modularity for mod *p^m* Galois representations. They also show that Galois representations can be attached to mod *p^m* eigenforms in each of the three corresponding senses. In the weakest sense, one has the reduction mod *p^m* of not just a single modular form, but rather of a `divided congruence form' which is a sum of several forms of different weights. The work also shows that one has a level-reduction type of result (`stripping powers of *p* away from the level'), albeit in general only in the divided congruence sense.

The project will investigate this experimentally with the long-term aim of obtaining an understanding of the differences between the three notions mentioned above. Thus, in connection with the above level-reduction result, one would like to construct examples of genuine divided congruence eigenforms mod *p^m* and thus obtain more information on the weights that occur for the corresponding sums of modular forms. As such examples possibly only occur at very high levels, such an experimental study would involve very high level linear algebra, i.e., diagonalization of Hecke operators in high dimensional spaces. Thus, a considerable amount of computer power as well as - possibly - special purpose algorithms would be needed.

It can be gathered from this that there are in fact many unresolved problems in this new field that can - and should - be investigated computationally. Thus, the proposed specific problem would upon a successful resolution immediately give rise to further computational/experimental questions.

## Operator algebras

Even for the fundamental class of irreducible shifts of finite type, it has not yet been established if invariants exist making the isomorphism problem up to conjucagy decidable, but if one passes to the coarser flow equivalence, a very satisfactory classification result has been provided by Franks using the so-called Bowen-Franks group. A surprisingly efficient strategy for generalizing Franks’ result to more general shift spaces involves an ingenious construction by Matsumoto, allowing the association of a C^∗-algebra to any shift space in a way generalizing the Cuntz and Cuntz-Krieger algebras and having the property of taking flow equivalent shift spaces to stably isomorphic algebras, associating in this way the Bowen-Franks group to K-theory. Inspired by this work Boyle, Carlsen and Eilers were succesful in achieving a flow classification of certain sofic shifts. To apply this result in areas such as automata theory and formal languages, and to determine the range of the ideas in greater generality, an experimental approach leading to a complete classification of flow classes of sofic shifts which are representable on small graphs is required.

The Cuntz algebras *O_n* correspond to full shifts, and are among the most important and most intensively studied objects in theory of operator algebras. For instance, their automorphisms and, more generally, endomorphisms of *O_n* have been thoroughly studied, including their applications to index theory for subfactors and for C∗-algebras, entropy calculations, wavelets, and quantum field theory. Passing to the much larger class of C∗-algebras of the types alluded to above, it is our objective to extend the analysis of localized endomorphisms (and automorphisms) from the Cuntz algebras to the much more general and challenging class of graph C^∗-algebras. In particular, we seek to understand the relationship between automorphisms of graph algebras and automorphisms of subshifts. Furthermore, we plan to investigate the noncommutative dynamical systems arising from such automorphisms and, in particular, the structure of the corresponding crossed products.

The investigations of automorphisms of the Cuntz algebras have been greatly enhanced by large scale computer calculations, and we intend to use a similar double approach in our studies of graph algebras: Theoretical work aided by massive computations. We expect that innovative programming techniques may be required to this end.

## Topology

One of the absolutely greatest achievements in 20th century mathematics was the classification of all finite simple groups. This classification gives a list, or catalogue, of all finite sets of symmetries that cannot be decomposed into smaller parts. The proof spans thousands of pages, spread over many journal articles. The original theorem already uses a significant computer input for the construction of the largest of the 26 “sporadic” sets of symmetries, e.g., the construction and study of the so-called Monster group of order approximately 8 · 10^53.

One of challenges of 21st century mathematics is to find a conceptual and “p-local” version of this theorem, where a corresponding classification is found for every prime number p, and from which the global classification mentioned above can be recovered. KA, Oliver, and Ventura have embarked on an ambitious program of finding new sporadic, or exotic, finite sets of symmetries at a prime number p, via the theory of so-called p-fusion systems.

Their approach is computer aided, and they have already mapped out all such symmetries at the prime 2 and of order less than 29 . However, to carry this classification program further requires significant computing power, since one both needs to store massive amounts of data, and be able to carry out linear algebra and other operations on very large matrices. However, with sufficient computing power at ones disposal, there is hope to make siginificant progess on this fundamental question in mathematics.

In a project which is unrelated to the former, but with close ties to the K-theoretical and graph theoretical approaches undertaken by SE, JMM has introduced the concept of (r, s)-coloring of simplicial complexes with Dobrinskaya and Notbohm. An (r,s)-coloring of a simplicial complex is a coloring of the vertex set in r colors with no monochrome s-simplices.

Any triangulated compact surface seems to admit a vertex coloring in 4 colors so that none of the 2-simplices are monochrome, ie any triangulated surface seems to be (4,2)-colorable. We would like to test this conjecture on a large number of concrete triangulated surfaces and maybe also on randomly generated surface triangulations. A related open question that we would like to investigate concerns (r,2)-colorings of triangulated 3-spheres. It would be interesting to find an example of a triangulated 3-sphere with no (1000,2)-coloring!

The s-chromatic number of a simplicial complex is the minimum r so that the complex admits an (r,s)-coloring. These s-chromatic numbers can be computed from linear programs that, unfortunately, are so large that until now we have not been able to carry them out in a single interesting case. The hope is that with access to greater computing power it will be possible to compute s-chromatic numbers in some interesting cases.

Michal Jan Adamaszek , postdoc

Michal Adamaszek joined the group as a postdoc in December 2014. His research is concentrated around combinatorial algebraic topology with connections to graph theory, applied topology and face enumeration. He is also interested in algorithmic and probabilistic aspects of these topics and often does computer experiments as part of his work. |

Kasper K. Andersen , associate professor

Kasper Andersen (KA) was very recently recruited from Copenhagen to the University of Lund to help build a research group in algebraic topology there. Originally a PhD student of Møller, he has been a very active experimental mathematician since the mid-nineties and has obtained or facilitated several important results this way. Apart from the work mentioned above, KA has, with Oliver and Ventura, recently advanced the program of classifying simple 2-fusion systems using a computer-aided approach, verifying the conjectured classification for low orders and ranks. Since KA is employed outside of Denmark he will act as an international expert associated to the project. |

### Søren Eilers , professor

Søren Eilers organizes the project and leads the efforts targeting the focus area “operator algebras and dynamics” together with Wojciech Szymanski. SE was recently appointed to the first professorial chair at the University of Copenhagen expressly directed at the exploration of the crossfield between operator algebras and dynamics and has used experimental approaches with great success in the study of dynamical systems to allow for a further developent within operator algebras. With Gunnar Restorff and Efren Ruiz, he has reinvigorated the classification theory for certain non-simple C-algebras associated to dynamics and with Mike Boyle and Toke Meier Carlsen given the first flow classification results for sofic shifts. He has also pioneered an experimental approach to certain combinatorial problems, and an article by him and Mikkel Abrahamsen has recently appeared in the leading journal “Experimental Mathematics”. |

### Rune Johansen , postdoc

Rune Johansen works with Søren Eilers and Wojciech Szymanski in the research area operator algebras and dynamics. His PhD thesis contained a significant experimental component in the investigation of Adler's question concerning renewal systems in symbolic dynamics. He has also used computer experiments to investigate a colouring problem for LEGO-bricks and the classification of beta-shifts. Currently, he is working on experimental investigations of automorphisms of graph algebras. |

### Ian Kiming , professor

Ian Kiming (IK) leads the algebra and number theory group in Copenhagen and works in various areas of algebraic number theory/arithmetic geometry. |

### Frank H. Lutz , postdoc

Frank Lutz's main field of research is experimental topology. He has been involved in various projects on constructing extremal or otherwise interesting triangulations of manifolds. With Jesper Møller he works on colorings of simplicial complexes and manifolds. Together with Bruno Benedetti and Karim Adiprasito he has initiated and is developing random discrete Morse theory. Mimi Tsuruga and FL recently combinatorialized the Akbulut-Kirby handle body description of the Cappell-Shaneson spheres. The resulting triangulations are used, in collaboration with Konstantin Mischaikow and Vidit Nanda, as non-trivial examples for testing the homology software CHomP. With Menachem Lazar, Robert MacPherson, Jeremy Mason and David Srolovitz he works on the topological microstructure analysis of metals and steel. |

### Jesper Michael Møller , professor

Jesper Michael Møller (JMM) is a member of the topology group and the Centre for Symmetry and Deformation in Copenhagen and has, among many other things, in joint work Andersen, Grodal and Viruel solved the important problem of classifying p-compact groups (for p odd). He has been interested in applying methods of experimental mathematics to topological problems for more than a decade and have pioneered this approach with several students at PhD and master level. For instance, he has verified the strong Quillen conjecture in the context of Euler characteristic of p-subgroup categories. |

### Nadim Rustom , PhD student

Nadim Rustom is working in algebraic number theory. His main research interest is the arithmetic properties of modular forms and their connections with Galois representations, specifically problems connected to reduction modulo prime powers. He has relied on computer experimentation to generate data in order to verify and formulate conjectures concerning the structure of graded algebras over modular forms of various levels and over various rings, and to investigate questions on finiteness of Galois representations modulo higher powers of primes. |

### Maria Ramirez Solano , postdoc

Maria Ramirez-Solano works in the research area of operator algebras and tilings. She is currently working with Uffe Haagerup on the experimental side of the investigation of the Thompson group F. Part of her PhD thesis involved computer experiments while investigating a non-standard hierarchical tiling. |

### Wojciech Szymanski , associate professor

Wojciech Szymanski (WS), was recruited to the University of Southern Denmark in 2008 from a position in Australia to continue his ground-breaking work on the study of -automorphisms and graph C-algebras, thus bringing unique expertise to Denmark. With Conti, he has recently achieved significant progress in the study of the endomorphisms of the Cuntz algebra On which globally preserve certain additional objects. In particular, they have answered an old question of Cuntz’s about the structure of the restricted Weyl group of On by finding a direct relation to the group of automorphisms of the full two-sided shift. |

### Brian Vinter , professor

Brian Vinter (BV) leads the University of Copenhagen’s E-Science center and has been active in cluster-computing research since 1994. He has provided significant contributions to research in the field, specifically within Distributed Shared Memory, and has been a pioneer in establishing grids in Denmark and has served as Nordic representative in the steering committee for the OECD Global Science forum in the working group for Grid computing. |

The experimental mathematics colloquium usually took place on Fridays from 1 to 2. The seminar was open to all mathematical subjects, and to talks ranging from descriptions of finished experiments to presentations of open problems that could be interesting to investigate experimentally.

## 2016

- Jan 15 at 13:15, AUD 9, Mike Boyle (Maryland), Decidability of equivalence of poset blocked matrices over a finite group ring, for Cuntz-Krieger algebras and flow equivalence

## 2015

- Dec 4 at 13:15, Patric Östergård (Aalto), Coloring the Square of the 8-Cube

- Nov 27 at 13:15, Steen Markvorsen (DTU), A Finsler geometric attack on wildfires
- Nov 20 at 13:15, Moritz Firsching (FU Berlin), Experimenting with polytopes and getting it exactly right
- Sept. 18 at 13:15, Søren Eilers (KU): Some early gems of experimental mathematics
- Feb. 24 at 13.15, Jan Hladky (Czech Academy of Sciences): Loebl-Komlos-Sos conjecture

## 2014

- Jan. 31, Johan Nilsson (Universität Bielefeld): Counting and Constructing LEGO Buildings
- Jan. 14 at 14.15, Michał Adamaszek (University of Bremen): Hard squares, Euler characteristic and combinatorics
- Jan. 10, Bruno Benedetti (Freie Universität): Random discrete Morse theory and a quantitative approach to triangulations
- Jan. 8 at 14.15, Sylwia Antoniuk (Adam Mickiewicz University): Collapse of the random triangular group

## 2013

- Dec. 13, Jeremy Mason (Boğaziçi University): Rigorous quantification of the grain growth microstructure in 2D and 3D
- Nov. 22, Casper Guldberg (KU): On homotopy self-equivalences of configuration spaces
- Nov. 1, Jim Stankevicz (KU): Torsion on Elliptic Curves
- Oct. 11, Uffe Haagerup (KU): A computational approach to the amenability problem for the Thompson group F
- May 24, Brendan Berg (UMD): Textile Systems
- May 17, James Avery and Rune Johansen (KU): Constructing Automorphisms of Graph C*-algebras
*May 15, (double feature in the OA seminar), Wojciech Szymanski (SDU): Automorphisms of graph algebras.*- May 3, Nadim Rustrom (KU): Generators of graded rings of modular forms.
- Apr 5, Søren Eilers (KU): Identifying constants with the PSLQ algorithm.
- Mar 15, Jesper Michael Møller (KU): Chromatic polynomials of simplicial complexes
- Feb 22, Maria Ramirez Solano (KU): Doing experimental mathematics in tiling theory, circle packing theory, and geometric transversal theory
- Jan 24, Frank Lutz (Technische Universität Berlin): Experimental Topology for the Structural Analysis of Materials

## 2012

- Nov 16, Henrik Kragh Sørensen (AU): Reflections on exploratory experimentation in mathematics
- Nov 16, Mikkel Willum Johansen (KU): Experimental mathematics as synthetic aposteriori knowledge