Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory

The aim of this Arbeitsgemeinschaft was to introduce young researchers with various backgrounds to the multifaceted and mutually perpetuating connections between ergodic theory, topological dynamics, combinatorics, and number theory

Arbeitsgemeinschaft: Limits of Structures

The goal of the Arbeitsgemeinschaft is to review current progress in the study of very large structures. The main emphasis is on the analytic approach that considers large structures as approximations of infinite analytic objects. This approach enables one to study graphs, hypergraphs, permutations, subsets of groups and many other fundamental structures

On Approximability, Convergence, and Limits of CSP Problems

This thesis studies dense constraint satisfaction problems (CSPs), and other related optimization and decision problems that can be phrased as questions regarding parameters or properties of combinatorial objects such as uniform hypergraphs. We concentrate on the information that can be derived from a very small substructure that is selected uniformly at random. In this thesis, we present a unified framework on the limits of CSPs in the sense of the convergence notion of Lovasz-Szegedy that depends only on the remarkable connection between graph sequences and exchangeable arrays established by Diaconis-Janson. In particular, we formulate and prove a representation theorem for compact colored r-uniform directed hypergraphs and apply this to rCSPs. We investigate the sample complexity of testable r-graph parameters, and discuss a generalized version of ground state energies (GSE) and demonstrate that they are efficiently testable. The GSE is a term borrowed from statistical physics that stands for a generalized version of maximal multiway cut problems from complexity theory, and was studied in the dense graph setting by Borgs et al. A notion related to testing CSPs that are defined on graphs, the nondeterministic property testing, was introduced by Lovasz-Vesztergombi, which extends the graph property testing framework of Goldreich-Goldwasser-Ron in the dense graph model. In this thesis, we study the sample complexity of nondeterministically testable graph parameters and properties and improve existing bounds by several orders of magnitude. Further, we prove the equivalence of the notions of nondeterministic and deterministic parameter and property testing for uniform dense hypergraphs of arbitrary rank, and provide the first effective upper bound on the sample complexity in this general case

Packing conditions in metric spaces with curvature bounded above and applications

General metric spaces satisfying weak and synthetic notions of upper and lower curvature bounds will be studied. The relations between upper and lower bounds will be pointed out, especially the interactions between a packing condition and different forms of convexity of the metric. The main tools will be a new and flexible definition of entropy on metric spaces and a version of the Tits Alternative for groups of isometries of the metric spaces under consideration. The applications can be divided into classical and new results: the former consist in generalizations to a wider context of the theory of negatively curved Riemannian manifolds, while the latter include several compactness and continuity results

Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets

Let P: \F \times \F \to \F be a polynomial of bounded degree over a finite field \F of large characteristic. In this paper we establish the following dichotomy: either $P$ is a \emph{moderate asymmetric expander} in the sense that |P(A,B)| \gg |\F| whenever A, B \subset \F are such that |A| |B| \geq C |\F|^{2-1/8} for a sufficiently large $C$, or else $P$ takes the form $P(x,y) = Q(F(x)+G(y))$ or $P(x,y) = Q(F(x) G(y))$ for some polynomials $Q,F,G$. This is a reasonably satisfactory classification of polynomials of two variables that moderately expand (either symmetrically or asymmetrically). We obtain a similar classification for weak expansion (in which one has |P(A,A)| \gg |A|^{1/2} |\F|^{1/2} whenever |A| \geq C |\F|^{1-1/16}), and a partially satisfactory classification for almost strong asymmetric expansion (in which |P(A,B)| = (1-O(|\F|^{-c})) |\F| when |A|, |B| \geq |\F|^{1-c} for some small absolute constant c>0). The main new tool used to establish these results is an \emph{algebraic regularity lemma} that describes the structure of dense graphs generated by definable subsets over finite fields of large characteristic. This lemma strengthens the Sz\'emeredi regularity lemma in the algebraic case, in that while the latter lemma decomposes a graph into a bounded number of components, most of which are \eps-regular for some small but fixed $\epsilon$, the former lemma ensures that all of the components are O(|\F|^{-1/4})-regular. This lemma, which may be of independent interest, relies on some basic facts about the \'etale fundamental group of an algebraic variety

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

The goal of this present manuscript is to introduce the reader to the nonstandard method and to provide an overview of its most prominent applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom