On the Elliptic Genera of Manifolds of Spin(7) Holonomy

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The \({\mathcal{N} = 1}\) superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group \({\mathcal{S}W(3/2,2)}\). In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper.     

The Global Dimension of the full Transformation Monoid (with an Appendix by V. Mazorchuk and B. Steinberg)

The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid \(\mathfrak {T}_{n}\) (the monoid of all self-maps of an n-element set) is the monoid analogue of the symmetric group. The investigation of its representation theory was begun by Hewitt and Zuckerman in 1957. Its character table was computed by Putcha in 1996 and its representation type was determined in a series of papers by Ponizovski?, Putcha and Ringel between 1987 and 2000. From their work, one can deduce that the global dimension of \(\mathbb {C}\mathfrak {T}_{n}\) is n?1 for n = 1, 2, 3, 4. We prove in this paper that the global dimension is n?1 for all n ≥ 1 and, moreover, we provide an explicit minimal projective resolution of the trivial module of length n?1. In an appendix with V. Mazorchuk we compute the indecomposable tilting modules of \(\mathbb {C}\mathfrak T_{n}\) with respect to Putcha’s quasi-hereditary structure and the Ringel dual (up to Morita equivalence).     

Existence of Néel Order in the S=1 Bilinear-Biquadratic Heisenberg Model via Random Loops

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e., vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connection, which is presented in a general form, and a precise analysis of the singularities of solutions of transport equations when there are trapped geodesics. In the case of closed manifolds, we obtain similar results modulo the obstruction given by twisted conformal Killing tensors, and we also study this obstruction.     

Inductive and Deductive Justification of Knowledge: Flexible Judgments Underneath Stable Beliefs in Teacher Education

Personal epistemological beliefs are considered to play an important role for processes of learning and teaching. However, research on personal epistemology is confronted with theoretical issues as there is conflicting evidence regarding the structure, stability, and context-dependence of epistemological beliefs. We give evidence how theoretical and methodological issues can partly be resolved by distinguishing between relatively stable “epistemological beliefs” and situation-specific “epistemological judgments.” A qualitative content analysis of a series of semistructured interviews (study 1) with pre-service teachers, teachers, and teacher educators as well as a statistical analysis of pre-service teachers’ extensive answers in questionnaires (study 2), both on the topic of “mathematical discovery,” reveal not only beliefs of the participants but also different qualities of judgments. Therefore, in further research both aspects of beliefs should be considered in a more differentiated manner when categorizing belief structures.     

Brownian and advective dynamics in microflow studied by coherent X‐ray scattering experiments

Combining microfluidics with coherent X‐ray illumination offers the possibility to not only measure the structure but also the dynamics of flowing samples in a single‐scattering experiment. Here, the power of this combination is demonstrated by studying the advective and Brownian dynamics of colloidal suspensions in microflow of different geometries. Using an experimental setup with a fast two‐dimensional detector and performing X‐ray correlation spectroscopy by calculating two‐dimensional maps of the intensity auto‐correlation functions, it was possible to evaluate the sample structure and furthermore to characterize the detailed flow behavior, including flow geometry, main flow directions, advective flow velocities and diffusive dynamics. By scanning a microfocused X‐ray beam over a microfluidic device, the anisotropic auto‐correlation functions of driven colloidal suspensions in straight, curved and constricted microchannels were mapped with the spatial resolution of the X‐ray beam. This method has not only a huge potential for studying flow patterns in complex fluids but also to generally characterize anisotropic dynamics in materials.     

Modeling a halogen dance reaction mechanism: A density functional theory study

Since the discovery of the halogen dance (HD) reaction more than 60 years ago, numerous insights into the mechanism have been unveiled. To date however, the reaction has not been investigated from a theoretical perspective. Density functional theory (DFT) was used to model the potential energy surface linking the starting reagents to the lithiated products for each step in the mechanism using a thiophene substrate. It was found that the lithium‐halogen exchange mechanism is critical to understand the HD mechanism in detail and yielded the knowledge that S_N2 transition states (TS) are favored over the four‐center type for the lithium‐bromine exchange steps. The overall driving force for the HD is thermodynamics, while the kinetic factors tightly control the reaction path through temperature. The S_N2 lithium‐bromide TS are barrierless, except the second, which is the limiting step. Finally, the model for the HD is discovered to be a pseudo‐clock type, due to a highly favorable bromide catalysis step and the reformation of 2‐bromothiophene. © 2016 Wiley Periodicals, Inc.