Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update

Topics in hyperbolic groups

Hyperbolic groups are a class of groups introduced by Gromov in 1987, which form an important part of geometric group theory. In Chapter 1, we give an introduction to this subject. In Chapter 2, we use the theory of complexes of groups to show that the integral homology and cohomology groups of a hyperbolic group are computable by a Turing machine. In Chapter 3, we present the boundary of a hyperbolic group as an inverse limit of topological spaces and use this to give computable estimates for properties of the boundary. In Chapter 4, we investigate symbolic dynamic properties concerning hyperbolic groups. In paricular, we give symbolic codings for the actions on the boundary of a hyperbolic and actions on the geodesic flow on a hyperbolic group. In Chapter 5 we investigate the problem of determining when graphs are Cayley graphs. The graphs which we are concerned with are regular and semi-regular planar graphs

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Improving Model-Based Software Synthesis: A Focus on Mathematical Structures

Computer hardware keeps increasing in complexity. Software design needs to keep up with this. The right models and abstractions empower developers to leverage the novelties of modern hardware. This thesis deals primarily with Models of Computation, as a basis for software design, in a family of methods called software synthesis. We focus on Kahn Process Networks and dataflow applications as abstractions, both for programming and for deriving an efficient execution on heterogeneous multicores. The latter we accomplish by exploring the design space of possible mappings of computation and data to hardware resources. Mapping algorithms are not at the center of this thesis, however. Instead, we examine the mathematical structure of the mapping space, leveraging its inherent symmetries or geometric properties to improve mapping methods in general. This thesis thoroughly explores the process of model-based design, aiming to go beyond the more established software synthesis on dataflow applications. We starting with the problem of assessing these methods through benchmarking, and go on to formally examine the general goals of benchmarks. In this context, we also consider the role modern machine learning methods play in benchmarking. We explore different established semantics, stretching the limits of Kahn Process Networks. We also discuss novel models, like Reactors, which are designed to be a deterministic, adaptive model with time as a first-class citizen. By investigating abstractions and transformations in the Ohua language for implicit dataflow programming, we also focus on programmability. The focus of the thesis is in the models and methods, but we evaluate them in diverse use-cases, generally centered around Cyber-Physical Systems. These include the 5G telecommunication standard, automotive and signal processing domains. We even go beyond embedded systems and discuss use-cases in GPU programming and microservice-based architectures