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

All finite transitive graphs admit self-adjoint free semigroupoid algebras

In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is $B(\mathcal{H})$. This is accomplished through a new construction that reduces this problem to in-degree $2$-regular graphs, which is then treated by applying the periodic Road Coloring Theorem of B\'eal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.Comment: Added missing reference. 16 pages 2 figure

A Guide to Algorithm Design: Paradigms, Methods, and Complexity Analysis

International audiencePresenting a complementary perspective to standard books on algorithms, A Guide to Algorithm Design: Paradigms, Methods, and Complexity Analysis provides a roadmap for readers to determine the difficulty of an algorithmic problem by finding an optimal solution or proving complexity results. It gives a practical treatment of algorithmic complexity and guides readers in solving algorithmic problems. Divided into three parts, the book offers a comprehensive set of problems with solutions as well as in-depth case studies that demonstrate how to assess the complexity of a new problem. Part I helps readers understand the main design principles and design efficient algorithms. Part II covers polynomial reductions from NP-complete problems and approaches that go beyond NP-completeness. Part III supplies readers with tools and techniques to evaluate problem complexity, including how to determine which instances are polynomial and which are NP-hard. Drawing on the authors' classroom-tested material, this text takes readers step by step through the concepts and methods for analyzing algorithmic complexity. Through many problems and detailed examples, readers can investigate polynomial-time algorithms and NP-completeness and beyond

Toeplitz operators in TQFT via skein theory

International audienceTopological quantum field theory associates to a punctured surface $\Sigma$, a level $r$ and colors $c$ in $\{1,\ldots,r-1\}$ at the marked points a finite dimensional hermitian space $V_r(\Sigma,c)$. Curves $\gamma$ on $\Sigma$ act as Hermitian operator $T_r^\gamma$ on these spaces. In the case of the punctured torus and the 4 times punctured sphere, we prove that the matrix elements of $T_r^\gamma$ have an asymptotic expansion in powers of $\frac{1}{r}$ and we identify the two first terms using trace functions on representation spaces of the surface in \su. We conjecture a formula for the general case. Then we show that the curve operators are Toeplitz operators on the sphere in the sense that $T_r^{\gamma}=\Pi_r f^\gamma_r\Pi_r$ where $\Pi_r$ is the Toeplitz projector and $f^\gamma_r$ is an explicit function on the sphere which is smooth away from the poles. Using this formula, we show that under some assumptions on the colors associated to the marked points, the sequence $T^\gamma_r$ is a Toeplitz operator in the usual sense with principal symbol equal to the trace function and with subleading term explicitly computed. We use this result and semi-classical analysis in order to compute the asymptotics of matrix elements of the representation of the mapping class group of $\Sigma$ on $V_r(\Sigma,c)$. We recover in this way the result of \cite{tw} on the asymptotics of the quantum 6j-symbols and treat the case of the punctured S-matrix. We conclude with some partial results when $\Sigma$ is a genus 2 surface without marked points. \end{abstract