On the quantization of polygon spaces

Moduli spaces of polygons have been studied since the nineties for their topological and symplectic properties. Under generic assumptions, these are symplectic manifolds with natural global action-angle coordinates. This paper is concerned with the quantization of these manifolds and of their action coordinates. Applying the geometric quantization procedure, one is lead to consider invariant subspaces of a tensor product of irreducible representations of SU(2). These quantum spaces admit natural sets of commuting observables. We prove that these operators form a semi-classical integrable system, in the sense that they are Toeplitz operators with principal symbol the square of the action coordinates. As a consequence, the quantum spaces admit bases whose vectors concentrate on the Lagrangian submanifolds of constant action. The coefficients of the change of basis matrices can be estimated in terms of geometric quantities. We recover this way the already known asymptotics of the classical 6j-symbols

Optical maps in guided genome assembly

With the introduction of DNA sequencing over 40 years ago, we have been able to take a peek at our genetic material. Even though we have had a long time to develop sequencing strategies further, we are still unable to read the whole genome in one go. Instead, we are able to gather smaller pieces of the genetic material, which we can then use to reconstruct the original genome with a process called genome assembly. As a result of the genome assembly we often obtain multiple long sequences representing different regions of the genome, which are called contigs. Even though a genome often consists of a few separate DNA molecules (chromosomes), the number of obtained contigs outnumbers them substantially, meaning our reconstruction of the genome is not perfect. The resulting contigs can afterwards be refined by ordering, orienting and scaffolding them using additional information about the genome, which is often done manually by hand. The assembly process can also be guided automatically with the additional information, and in this thesis we are introducing a method that utilizes optical maps to aid us assemble the genome more accurately. A noticeable improvement of this method is the unification of the contigs, i.e. we are left with fewer but longer contigs. We are using an existing genome assembler called Kermit, which is designed to accept genetic maps as auxiliary long range information. Our contribution is the development of an assembly pipeline that provides Kermit with similar kind of information via optical maps. The initial results of our experiments show that the proposed genome assembly scheme can take advantage of optical maps effectively already during the assembly process to guide the reconstruction of a genome

09171 Abstracts Collection -- Adaptive, Output Sensitive, Online and Parameterized Algorithms

From 19.01. to 24.04.2009, the Dagstuhl Seminar 09171 ``Adaptive, Output Sensitive, Online and Parameterized Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing abstracts of the presentations and a summary of the problem session