Goals and results
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.
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.
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 en-
hanced by large scale computer calculations, and we intend to use a similar double ap-
proach in our studies of graph algebras: Theoretical work aided by massive computations. We expect that innovative programming techniques may be required to this end.
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 sym-
metries 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.