Tensor models and embedded Riemann surfaces

Tensor models and, more generally, group field theories are candidates for higher-dimensional quantum gravity, just as matrix models are in the 2d setting. With the recent advent of a 1/N-expansion for coloured tensor models, more focus has been given to the study of the topological aspects of their Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs known as bubbles and jackets. We demonstrate in the 3d case that these graphs are generated by matrix models embedded inside the tensor theory. Moreover, we show that the jacket graphs represent (Heegaard) splitting surfaces for the triangulation dual to the Feynman graph. With this in hand, we are able to re-express the Boulatov model as a quantum field theory on these Riemann surfaces.Comment: 9 pages, 7 fi

Assembling homology classes in automorphism groups of free groups

The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for groups $\Gamma_{k,s}$, where the $\Gamma_{k,s}$ generalize $Out(F_k)=\Gamma_{k,0}$ and $Aut(F_k)=\Gamma_{k,1}$. The symmetric group $\Sigma_s$ acts on $H_*(\Gamma_{k,s})$ by permuting leaves, and for trivial rational coefficients we compute the $\Sigma_s$-module structure on $H_*(\Gamma_{k,s})$ completely for $k \leq 2$. Assembling these classes then produces all the known nontrivial rational homology classes for $Aut(F_n)$ and $Out(F_n)$ with the possible exception of classes for $n=7$ recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of $Aut(F_n)$ and $Out(F_n)$ which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.Comment: Final version for Commentarii Math. Hel

Morita classes in the homology of automorphism groups of free groups

Using Kontsevich's identification of the homology of the Lie algebra l_infty with the cohomology of Out(F_r), Morita defined a sequence of 4k-dimensional classes mu_k in the unstable rational homology of Out(F_{2k+2}). He showed by a computer calculation that the first of these is non-trivial, so coincides with the unique non-trivial rational homology class for Out(F_4). Using the "forested graph complex" introduced in [Algebr. Geom. Topol. 3 (2003) 1167--1224], we reinterpret and generalize Morita's cycles, obtaining an unstable cycle for every connected odd-valent graph. (Morita has independently found similar generalizations of these cycles.) The description of Morita's original cycles becomes quite simple in this interpretation, and we are able to show that the second Morita cycle also gives a nontrivial homology class. Finally, we view things from the point of view of a different chain complex, one which is associated to Bestvina and Feighn's bordification of outer space. We construct cycles which appear to be the same as the Morita cycles constructed in the first part of the paper. In this setting, a further generalization becomes apparent, giving cycles for objects more general than odd-valent graphs. Some of these cycles lie in the stable range. We also observe that these cycles lift to cycles for Aut(F_r).Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper40.abs.htm

Shared Decision-making in the First Educational District of Tennessee: Teachers\u27 and Principals\u27 Perceptions of Actual and Desired Levels of Participation

The purpose of this study was to determine the current involvement of principals and teachers in shared decision making as well as desired levels, and to identify the perceived areas of acceptance and non-acceptance by educators. Eight domains of the Teacher Decision Making Instrument: planning, policy, curriculum/instruction, pupil personnel, staff personnel, staff development, school/community relations, and budget management were used to assess the actual and desired levels of participation in shared decision making by the respondents. A random sample was selected from the public schools of Northeast Tennessee. Seventy-five schools were surveyed which included 75 principals and 1632 teachers. Responses were obtained from 59 principals and 1084 teachers at 59 schools. Data were analyzed using t-tests for independent means, t-tests for dependent (correlated) means and analysis of variance. The analysis and interpretation indicated statistically significant differences between teachers\u27 and principals\u27 perceptions of actual participation in shared decision making with principals perceiving a higher level of involvement than teachers. Significant difference was also found between actual and desired levels of participation with higher desired levels especially in the areas of planning, staff personnel, school/community relations and budget management. No significant difference was found between principals\u27 and teachers\u27 perceptions of desired participation in shared decision making. Significant differences were found between groups\u27 desired level of participation in shared decision making based on age, participants\u27 years in the school, and career ladder status level. No significant differences were found between desired levels of participation in shared decision making based on number of years in education, highest education level, and various school compositions

A planning program and the design for a single enterprise community in the Subarctic

Thesis (M.C.P.) Massachusetts Institute of Technology. Dept. of Architecture, 1956.ACCOMPANYING drawings held by MIT Museum.Includes bibliographies.by James Arthur Hatcher and David Dunsmore Wallace.M.C.P

Integration of remanufacturing issues into the design process

Remanufacturing is the process of returning a used product to like-new condition with a warranty to match. The efficiency and effectiveness of this process greatly depends upon product design; there are certain product properties that may have a positive or negative effect on steps of the remanufacturing process. The concept of 'design for remanufacture' or 'DfRem' is a design task dedicated to improving the remanufacturability of a product. However, it would appear that very few products are currently designed for remanufacture and the reasons behind this have yet to be fully explored. This paper provides an overview of the problem and a discussion of some of the preliminary findings of a study aimed at improving designers' ability to carry out DfRem. The findings provide an early indication of some of the factors affecting the integration of DfRem into the design process

On a theorem of Kontsevich

In two seminal papers M. Kontsevich introduced graph homology as a tool to compute the homology of three infinite dimensional Lie algebras, associated to the three operads `commutative,' `associative' and `Lie.' We generalize his theorem to all cyclic operads, in the process giving a more careful treatment of the construction than in Kontsevich's original papers. We also give a more explicit treatment of the isomorphisms of graph homologies with the homology of moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations on chain complexes of graphs', Mathematische Annalen, 327 (2003) 545-573] we defined a Lie bracket and cobracket on the commutative graph complex, which was extended in [James Conant, `Fusion and fission in graph complexes', Pac. J. 209 (2003), 219-230] to the case of all cyclic operads. These operations form a Lie bi-algebra on a natural subcomplex. We show that in the associative and Lie cases the subcomplex on which the bi-algebra structure exists carries all of the homology, and we explain why the subcomplex in the commutative case does not.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-42.abs.htm