Higher-Dimensional Algebra VII: Groupoidification

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify - or more precisely, groupoidify - the positive part of the quantum group associated to the quiver.Comment: 67 pages, 14 eps figures; uses undertilde.sty. This is an expanded version of arXiv:0812.486

Phaseless three-dimensional optical nano-imaging

We propose a method for optical nano-imaging in which the structure of a three-dimensional inhomogeneous medium may be recovered from far-field power measurements. Neither phase control of the illuminating field nor phase measurements of the scattered field are necessary. The method is based on the solution to the inverse scattering problem for a system consisting of a weakly-scattering dielectric sample and a strongly-scattering nano-particle tip. Numerical simulations are used to illustrate the results.Comment: 10 pages, 2 figure

Nuclear matrix protein 2 antibody-positive adult dermatomyositis: a case report and review of the literature

Dermatomyositis is a clinically heterogenous inflammatory myopathy with unique cutaneous features. Myositis-specific antibodies can aid in diagnosis and anticipation of patient prognosis. Herein, we report a 22-year-old man who presented with multifocal erythematous plaques with violaceous papules on his bilateral elbows, neck, and face. He was diagnosed with biopsy-proven dermatomyositis and determined to be seropositive for nuclear matrix protein 2 antibody (NXP-2). He was treated with systemic corticosteroids, then intravenous methylprednisolone and azathioprine, and ultimately achieved greatest treatment response with intravenous immune globulin therapy

LGBTQ+ Health-a Novel Course for Undergraduate Students.

The concept of providing focused, competency-based LGBTQ+ health education outside the setting of health professional programs, specifically for undergraduates, is quite uncharted. However, the issue at the core of our rationale is one shared by those with and without clinical exposure: how to best support the development of cultural competence in providers who are or will be caring for LGBTQ+ patients. Traditional health professional education programs have enacted a number of curricular initiatives in this regard, designed for advanced learners. By focusing specifically on the undifferentiated learner, we offer a new perspective on the timing of LGBTQ+ health-related education. Our course is not intended to supplant the critical learning and application that must occur in the clinic or hospital room. Rather, we present a framework for cultivating understanding of the healthcare issues faced by the LGBTQ+ community that may help a learner to acquire and apply skills subsequently with greater cultural competence

The Inter-Mammary Sticky Roll: A Novel Technique for Securing a Doppler Ultrasonic Probe to the Precordium for Venous Air Embolism Detection.

Venous air embolism is a devastating and potentially life-threatening complication that can occur during neurosurgical procedures. We report the development and use of the "inter-mammary sticky roll," a technique to reliably secure a precordial Doppler ultrasonic probe to the chest wall during neurosurgical cases that require lateral decubitus positioning. We have found that this noninvasive technique is safe, and effectively facilitates a constant Doppler signal with no additional risk to the patient

Entangled Wavefunctions from Classical Oscillator Amplitudes

In the first days of quantum mechanics Dirac pointed out an analogy between the time-dependent coefficients of an expansion of the Schr\"odinger equation and the classical position and momentum variables solving Hamilton's equations. Here it is shown that the analogy can be made an equivalence in that, in principle, systems of classical oscillators can be constructed whose position and momenta variables form time-dependent amplitudes which are identical to the complex quantum amplitudes of the coupled wavefunction of an N-level quantum system with real coupling matrix elements. Hence classical motion can reproduce quantum coherence.Comment: extended versio

Convenient Categories of Smooth Spaces

A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological spaces" share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of "concrete sheaves on a concrete site". As a result, the categories of such spaces are locally cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use.Comment: 43 pages, version to be published; includes corrected definition of "concrete site