Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing

Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM

On the Whitney distortion extension problem for Cm(Rn)C^m(\mathbb R^n) and C∞(Rn)C^{\infty}(\mathbb R^n) and its applications to interpolation and alignment of data in Rn\mathbb R^n

Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $\phi:U\to \mathbb R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\phi$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almost isometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. (b) Let $\phi:U\to \mathbb R^n$ be $C^{\infty}$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\phi$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. Our results complement those of [14,15,20] where there, $E$ is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in $\mathbb R^n$.Comment: This is part three of four papers with C. Fefferman (arXiv:1411.2451, arXiv:1411.2468, involve-v5-n2-p03-s.pdf) dealing with the problem of Whitney type extensions of $\delta>0$ distortions from certain compact sets $E\subset \Bbb R^n$ to $\varepsilon>0$ distorted diffeomorphisms on $\Bbb R^n

Convex Geometry and its Applications

The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of other algorithms in computer science. High-dimensional geometry, both the discrete and convex branches of it, has experienced a striking series of developments in the past 10 years. Several examples were presented at this meeting, for example the work of Rudelson et al. on conjunction matrices and their relation to confidential data analysis, that of Litvak et al. on remote sensing and a series of results by Nazarov and Ryabogin et al. on Mahler’s conjecture for the volume product of domains and their polars

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