Modeling Course-Based Undergraduate Research Experiences: An Agenda for Future Research and Evaluation

Course-based undergraduate research experiences (CUREs) are being championed as scalable ways of involving undergraduates in science research. Studies of CUREs have shown that participating students achieve many of the same outcomes as students who complete research internships. However, CUREs vary widely in their design and implementation, and aspects of CUREs that are necessary and sufficient to achieve desired student outcomes have not been elucidated. To guide future research aimed at understanding the causal mechanisms underlying CURE efficacy, we used a systems approach to generate pathway models representing hypotheses of how CURE outcomes are achieved. We started by reviewing studies of CUREs and research internships to generate a comprehensive set of outcomes of research experiences, determining the level of evidence supporting each outcome. We then used this body of research and drew from learning theory to hypothesize connections between what students do during CUREs and the outcomes that have the best empirical support. We offer these models as hypotheses for the CURE community to test, revise, elaborate, or refute. We also cite instruments that are ready to use in CURE assessment and note gaps for which instruments need to be developed.Howard Hughes Medical InstituteScience and Mathematics Educatio

On the "Universal" Quantum Area Spectrum

There has been much debate over the form of the quantum area spectrum for a black hole horizon, with the evenly spaced conception of Bekenstein having featured prominently in the discourse. In this letter, we refine a very recently proposed method for calibrating the Bekenstein form of the spectrum. Our refined treatment predicts, as did its predecessor, a uniform spacing between adjacent spectral levels of $8\pi$ in Planck units; notably, an outcome that already has a pedigree as a proposed ``universal'' value for this intrinsically quantum-gravitational measure. Although the two approaches are somewhat similar in logic and quite agreeable in outcome, we argue that our version is conceptually more elegant and formally simpler than its precursor. Moreover, our rendition is able to circumvent a couple of previously unnoticed technical issues and, as an added bonus, translates to generic theories of gravity in a very direct manner.Comment: 7 Pages; (v2) now 9 full pages, significant changes to the text and material added but the general theme and conclusions are unchange

Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives

We generalise the construction of fuzzy CP^N in a manner that allows us to access all noncommutative equivariant complex vector bundles over this space. We give a simplified construction of polarization tensors on S^2 that generalizes to complex projective space, identify Laplacians and natural noncommutative covariant derivative operators that map between the modules that describe noncommuative sections. In the process we find a natural generalization of the Schwinger-Jordan construction to su(n) and identify composite oscillators that obey a Heisenberg algebra on an appropriate Fock space.Comment: 34 pages, v2 contains minor corrections to the published versio

A projective Dirac operator on CP^2 within fuzzy geometry

We propose an ansatz for the commutative canonical spin_c Dirac operator on CP^2 in a global geometric approach using the right invariant (left action-) induced vector fields from SU(3). This ansatz is suitable for noncommutative generalisation within the framework of fuzzy geometry. Along the way we identify the physical spinors and construct the canonical spin_c bundle in this formulation. The chirality operator is also given in two equivalent forms. Finally, using representation theory we obtain the eigenspinors and calculate the full spectrum. We use an argument from the fuzzy complex projective space CP^2_F based on the fuzzy analogue of the unprojected spin_c bundle to show that our commutative projected spin_c bundle has the correct SU(3)-representation content.Comment: reduced to 27 pages, minor corrections, minor improvements, typos correcte

The Information Geometry of the One-Dimensional Potts Model

In various statistical-mechanical models the introduction of a metric onto the space of parameters (e.g. the temperature variable, $\beta$, and the external field variable, $h$, in the case of spin models) gives an alternative perspective on the phase structure. For the one-dimensional Ising model the scalar curvature, ${\cal R}$, of this metric can be calculated explicitly in the thermodynamic limit and is found to be ${\cal R} = 1 + \cosh (h) / \sqrt{\sinh^2 (h) + \exp (- 4 \beta)}$. This is positive definite and, for physical fields and temperatures, diverges only at the zero-temperature, zero-field ``critical point'' of the model. In this note we calculate ${\cal R}$ for the one-dimensional $q$-state Potts model, finding an expression of the form ${\cal R} = A(q,\beta,h) + B (q,\beta,h)/\sqrt{\eta(q,\beta,h)}$, where $\eta(q,\beta,h)$ is the Potts analogue of $\sinh^2 (h) + \exp (- 4 \beta)$. This is no longer positive definite, but once again it diverges only at the critical point in the space of real parameters. We remark, however, that a naive analytic continuation to complex field reveals a further divergence in the Ising and Potts curvatures at the Lee-Yang edge.Comment: 9 pages + 4 eps figure

Character Formulae and Partition Functions in Higher Dimensional Conformal Field Theory

A discussion of character formulae for positive energy unitary irreducible representations of the the conformal group is given, employing Verma modules and Weyl group reflections. Product formulae for various conformal group representations are found. These include generalisations of those found by Flato and Fronsdal for SO(3,2). In even dimensions the products for free representations split into two types depending on whether the dimension is divisible by four or not.Comment: 43 pages, uses harvmac,version 2 2 references added, minor typos correcte

Attentional load and sensory competition in human vision: Modulation of fMRI responses by load fixation during task-irrelevant stimulation in the peripheral visual field.

Perceptual suppression of distractors may depend on both endogenous and exogenous factors, such as attentional load of the current task and sensory competition among simultaneous stimuli, respectively. We used functional magnetic resonance imaging (fMRI) to compare these two types of attentional effects and examine how they may interact in the human brain. We varied the attentional load of a visual monitoring task performed on a rapid stream at central fixation without altering the central stimuli themselves, while measuring the impact on fMRI responses to task-irrelevant peripheral checkerboards presented either unilaterally or bilaterally. Activations in visual cortex for irrelevant peripheral stimulation decreased with increasing attentional load at fixation. This relative decrease was present even in V1, but became larger for successive visual areas through to V4. Decreases in activation for contralateral peripheral checkerboards due to higher central load were more pronounced within retinotopic cortex corresponding to 'inner' peripheral locations relatively near the central targets than for more eccentric 'outer' locations, demonstrating a predominant suppression of nearby surround rather than strict 'tunnel vision' during higher task load at central fixation. Contralateral activations for peripheral stimulation in one hemifield were reduced by competition with concurrent stimulation in the other hemifield only in inferior parietal cortex, not in retinotopic areas of occipital visual cortex. In addition, central attentional load interacted with competition due to bilateral versus unilateral peripheral stimuli specifically in posterior parietal and fusiform regions. These results reveal that task-dependent attentional load, and interhemifield stimulus-competition, can produce distinct influences on the neural responses to peripheral visual stimuli within the human visual system. These distinct mechanisms in selective visual processing may be integrated within posterior parietal areas, rather than earlier occipital cortex

Anomalous dimension and local charges

AdS space is the universal covering of a hyperboloid. We consider the action of the deck transformations on a classical string worldsheet in $AdS_5\times S^5$. We argue that these transformations are generated by an infinite linear combination of the local conserved charges. We conjecture that a similar relation holds for the corresponding operators on the field theory side. This would be a generalization of the recent field theory results showing that the one loop anomalous dimension is proportional to the Casimir operator in the representation of the Yangian algebra.Comment: 10 pages, LaTeX; v2: added explanations, reference

On the Role of Chaos in the AdS/CFT Connection

The question of how infalling matter in a pure state forms a Schwarzschild black hole that appears to be at non-zero temperature is discussed in the context of the AdS/CFT connection. It is argued that the phenomenon of self-thermalization in non-linear (chaotic) systems can be invoked to explain how the boundary theory, initially at zero temperature self thermalizes and acquires a finite temperature. Yang-Mills theory is known to be chaotic (classically) and the imaginary part of the gluon self-energy (damping rate of the gluon plasma) is expected to give the Lyapunov exponent. We explain how the imaginary part would arise in the corresponding supergravity calculation due to absorption at the horizon of the black hole.Comment: 18 pages. Latex file. Minor changes. Final version to appear in Modern Physics Letters

The Information Geometry of the Ising Model on Planar Random Graphs

It has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterisation of the phase structure, particularly in the case where there are two such parameters -- such as the Ising model with inverse temperature $\beta$ and external field $h$. In various two parameter calculable models the scalar curvature ${\cal R}$ of the information metric has been found to diverge at the phase transition point $\beta_c$ and a plausible scaling relation postulated: ${\cal R} \sim |\beta- \beta_c|^{\alpha - 2}$. For spin models the necessity of calculating in non-zero field has limited analytic consideration to 1D, mean-field and Bethe lattice Ising models. In this letter we use the solution in field of the Ising model on an ensemble of planar random graphs (where $\alpha=-1, \beta=1/2, \gamma=2$) to evaluate the scaling behaviour of the scalar curvature, and find ${\cal R} \sim | \beta- \beta_c |^{-2}$. The apparent discrepancy is traced back to the effect of a negative $\alpha$.Comment: Version accepted for publication in PRE, revtex