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Doctoral Program DISCRETE MATHEMATICS Supplement 1 Doctoral Program DISCRETE MATHEMATICS Supplement 1 Speaker: Univ.-Prof. Dipl.-Ing. Dr. Wolfgang Woess Institut für Mathematische Strukturthorie, TU Graz Steyrergasse 30 / III, A-8010 GRAZ, AUSTRIA Tel.: ++43 316 873 7130, Fax.: ++43 316 873 4507 email: [email protected] Deputy Speaker: Ao.Univ.-Prof. Dipl.-Ing. Mag. Dr. Alfred Geroldinger Institut für Mathematik und Wissenschafliches Rechnen, Universität Graz Heinrichstraße 36, A-8010 GRAZ, AUSTRIA Tel.: ++43 316 380 5154 , Fax.: ++43 316 380 9815 email: [email protected] Deputy Speaker: Ao.Univ.-Prof. Dipl.-Ing. Dr. Jörg Thuswaldner Institut für Mathematik und Informationstechnologie, Montanuniversität Leoben Franz-Josef-Strasse 18, A-8700 Leoben, AUSTRIA Tel.: ++43 3842 402 3805, Fax.: ++43 3842 402 3802 email: [email protected] Secretary of the Speaker: Christine Stelzer Institut für Mathematische Strukturthorie, Technische Universität Graz Steyrergasse 30 / III, A-8010 GRAZ, AUSTRIA Tel.: ++43 316 873 7131, Fax.: ++43 316 873 4507 email: [email protected] 57 Supplement 1 (Beilage 1) Description of the Achievements and Goals of the Researchers Contents 1. Project 01: Random walk models on graphs and groups 59 2. Project 02: Probabilistic methods in combinatorial number theory 70 3. Project 03: Additive group theory, zero-sum theory and non-unique factorizations 77 4. Project 04: Fractal analysis and combinatorics of digital expansions 83 5. Project 05: Digital expansions with applications in cryptography 90 6. Project 06: Polynomial diophantine equations - combinatorial and number theoretic aspects 97 7. Project 07: Structural investigations on combinatorial optimization problems 104 8. Project 08: Number systems and fractal structures 115 9. Project 09: Diophantine approximation and combinatorial Problems 123 10. Project 10: Subdivision in nonlinear geometries 131 References of Supplement 1 138 58 59 1. Project 01: Random walk models on graphs and groups 1.1. Principal Investigator and Speaker of the DK-plus: Wolfgang Woess Institut für Mathematische Strukturtheorie Technische Universität Graz Steyrergasse 30 8010 Graz AUSTRIA 1.2. Keywords: random walk, transition operator, spectrum, horocyclic product, group ac- tion, internal aggregation 1.3. Research interests of the Faculty Member. The central topic of the research of W. Woess is “Random Walks on Infinite Graphs and Groups”, which is also the title of the quite successful monograph [150] (= item 1 in 1.11). Here, Random Walks are understood as Markov chains whose transition probabilities are adapted to an algebraic, geometric, resp. combinatorial structure of the underlying state space. The main theme is the interplay between probabilistic, analytic and potential theoretic properties of those random processes and the structural properties of that state space. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk, such as transience/recurrence, decay and asymptotic behaviour of transition probabilities, rate of escape, convergence to a boundary at infinity and harmonic functions. Vice versa, random walks may also be seen as a nice tool for classifying, or at least describing the structure of graphs, groups and related objects. The work of W. Woess is not limited to those aspects that concentrate on the link between random walks and structure theory. One one side, there is also a body of more “pure” work on infinite graphs, group actions, and also formal languages (which entered the scene via the free group). This comprises past and current collaboration with T. Ceccherini-Silberstein. On another side, some recent and less recent work concerns locally compact groups and their actions in relation with the computation of norms of transition operators, and harmonic functions on certain manifolds that arise as so–called horocyclic products: past and current collaboration with S. Brofferio, M. Salvatori, L. Saloff-Coste and A. Bendikov. Woess’ research is interdisciplinary between several Mathematical areas: Probability – Graph Theory – Geometric Group Theory – Discrete Geometry – Discrete Potential Theory – Har- monic Analysis and Spectral Theory. 1.4. Short description of two showcases of PhD Research Projects. 1. Horocyclic and wreath products: group actions and random walks. This work goes back to two roots: one is a long paper of Cartwright, Kaimanovich and Woess [30] which studies random walks on the stabilizer of an end in a homogeneous tree in the spirit of products of random affine transformations. The other (itself twofold) is in the paper of Soardi and Woess [136] (one of the most frequently cited among Woess’ papers), where amenability and unimodularity of totally disconnected groups are linked with random walks on graphs, and where Woess posed the following problem: does there exist a vertex-transitive graph that is not quasi-isometric with a Cayley graph ? Diestel and Leader Diestel and Leader proposed an example to answer the above question positively. It can be described as the horocyclic product of two homogeneous trees with degrees q + 1 and r + 1, respectively where q = r (that is, a horosphere in the product of two trees). It was only very recently that Eskin,6 Fisher and 60 Whyte [49] succeeded to prove that these Diestel-Leader graphs are indeed not quasi-isometric with any Cayley graph. On the other hand, as pointed out by R. I. Möller, when q = r, it is a Cayley-graph of the so-called lamplighter group, the wreath product of a finite group with the infinite cyclic group. All these facts opened the doors to work with multiple flavour. First of all, random walks on DL-graphs could be studied by Bertacchi [17] by adapting the methods of [30]. The simple desciption of a Cayley graph of the lamplighter group (whose geometry had not been understood so clearly before) lead to a quite complete body of work regarding the Martin boundary and the spectrum of transition operators [items 7 and 8 in 1.11 below] and the long case study [item 3 in 1.11], where horocyclic products of an arbitrary number of trees are studied under several different viewpoints, including “pure” combinatorial group theory besides spectral and boundary theory. This is a starting point for further promising research to be undertaken. Orient the regular tree T = T (d1, d2) such that every vertex has indegree d1 and outdegree d2, and consider the associated “horocycles” plus horocyclic products of two or more such trees, possibly with different di. From the above, the case d1 > d2 = 1 is well understood, where the involved groups are all amenable, which is not true when d1, d2 2. When are those extended horocyclic products amenable,≥ resp., when do they satisfy a strong isoperimetric ineqality ? This appears to be the simplest of the questions to be posed, and a good starting point for a young PhD student to embark on this type of research by learning tools such as the action of vertex-stabilizers, the topology of isometry groups of graphs, related harmonic analysis, probability, etc. What can be said about group actions on and quasi-isometries of these products ? How can one extend the methods of [item 3 in 1.11] to show that they are Cayley graphs when they are amenable ? Did such groups appear in Geometric Group Theory before in different contexts ? How are these graphs related with wreath products, Baumslag-Solitar and related groups ? What is the behaviour at infinity of such random walks ? This should be preceded by a study of random walks on Aut(T (d1, d2)) in the spirit of [30]. Another interesting feature of the DL-graphs is that the “simple random walk” transition op- erator has pure point spectrum. This goes back to work of Girgorchuk and Żuk [70] and Dicks and Schick [43] regarding lampligher groups, which simplifies and extends via the elementary understanding of the geometry of DL-graphs. One should mention here that the phenomenon of a pure point spectrum had been quite new for random walks on groups (contrary to frac- tals). Is the spectrum of random walks on the above more general horocyclic products of trees again pure point ? Via the link between lamplighter groups and DL-graphs, this is also related with recent work of Lehner, Neuhauser and Woess [item 2 in 1.11], who revealed a direct connection between the spectrum of random walk on wreath products and on percolation clusters. To summarize, we all know that it is a subtle task to find and propose good topics for PhD research in Pure Mathematics. On one hand, answers should not be obvious, while on the other, a responsible advisor should only propose questions which appear to have the potential for successful work within reasonable time. The above topic seems to have that potential, and to offer promising work to even more than one PhD student. 2. Random walks, the rotor-router model and other internal aggregation processes The rotor- router model is an automaton that generates a deterministic motion of one or more particles on a graph. See e.g. Levine and Peres [110] (& the references given there), as well as the nice-to-read article by Patrick [120]. Given a locally finite graph, an arrow – the rotor – is 61 placed at each vertex, pointing to one of its neighbors. A walker starts at the root and follows the directions of the arrows; after each step, the last arrow is rotated so that it then points to the next neighbor of its initial vertex. (A cyclic ordering for the neighbors of each vertex has to be pre-chosen and followed by the rotor.) This model has striking similarities with random walk. In particular, there is the random-walk-driven process called internal diffusion limited aggregation (IDLA) introduced by Diaconis and Fulton in 1991. In IDLA, successive particles are emitted from a root vertex in a graph.
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