arXiv subjects

Jeffrey M. Rabin

Publications and source records attributed to Jeffrey M. Rabin.

13 recordsLinked to original sources

arXiv2024

Students' Experience of Cultural Differences Between Mathematics and Physics

How students use mathematics in their physics classes has been studied extensively in the physics education literature. In addition to specific mathematical methods in specific physics contexts, possible effects of more general "cultural" differences between the two disciplines have also been explored. However, there has been little examination of students' own awareness and interpretation of these differences. We explore the undergraduate student experience of these "cultural" contrasts, focusing on how they impact learning and problem-solving. Through a qualitative study, including surveys and interviews with students double-majoring in mathematics and physics (or majoring in one and minoring in the other), we investigate students' awareness of distinct pedagogical approaches, mathematical justifications, and organization of concepts in mathematics versus physics classes. We find that students do recognize and navigate these "cultural" differences, often employing specific coping strategies. We identify specific themes from our data and comment on how students feel that these themes impact their learning. We suggest that increased faculty and student awareness of the identified differences in educational practice could facilitate knowledge transfer between mathematics and physics.

physics.ed-ph↗

arXiv2023

The combinatorics of supertorus sheaf cohomology

Affine superspace $\mathbb{C}^{1 \mid n}$ has a single bosonic coordinate $z$ and $n$ fermionic coordinates $θ_1, \dots, θ_n$. Let $M$ be the supertorus obtained by quotienting $\mathbb{C}^{1 \mid n}$ by the abelian group generated by the maps $S: (z,θ_1, \dots, θ_n) \mapsto (z + 1, θ_1, \dots, θ_n)$ and $T: (z, θ_1, \dots, θ_n) \mapsto (z + t, θ_1 + α_1, \dots, θ_n + α_n)$ where $t \in \mathbb{C}$ has positive imaginary part and $α_1, \dots, α_n$ are independent fermionic parameters. We compute the zeroth and first cohomology groups of the structure sheaf $\mathcal{O}$ of $M$ as doubly graded $\mathfrak{S}_n$-modules, exhibiting an instance of Serre duality between these groups. We use skein relations and noncrossing matchings to give a combinatorial presentation of $H^0(M,\mathcal{O})$ in terms of generators and relations.

math.CO↗

arXiv2020

Asymmetric expansion preserves hyperbolic convexity

In an earlier paper we showed that the radial expansion of a hyperbolic convex set in the Poincaré disk about any point inside it results in a hyperbolic convex set. In this work, we generalize this result by showing that the asymmetric expansion of a hyperbolic convex set about any point inside it also results in a hyperbolic convex set.

math.DG↗

arXiv2020

Radial expansion preserves hyperbolic convexity and radial contraction preserves spherical convexity

On a flat plane, convexity of a set is preserved by both radial expansion and contraction of the set about any point inside it. Using the Poincaré disk model of hyperbolic geometry, we prove that radial expansion of a hyperbolic convex set about a point inside it always preserves hyperbolic convexity. Using stereographic projection of a sphere, we prove that radial contraction of a spherical convex set about a point inside it, such that the initial set is contained in the closed hemisphere centred at that point, always preserves spherical convexity.

math.DG↗

arXiv2014

Serre Duality, Abel's Theorem, and Jacobi Inversion for Supercurves Over a Thick Superpoint

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincare duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic Groups and Class Fields, combined with an invariance result for the topology on the dual of the space of repartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators, as a right module over itself. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.

math.AG↗

arXiv2011

Orbifolds, the A, D, E Family of Caustic Singularities, and Gravitational Lensing

We provide a geometric explanation for the existence of magnification relations for the A, D, E family of caustic singularities, which were established in recent work. In particular, it was shown that for families of general mappings between planes exhibiting any of these caustic singularities, and for any non-caustic target point, the total signed magnification of the corresponding pre-images vanishes. As an application to gravitational lensing, it was also shown that, independent of the choice of a lens model, the total signed magnification vanishes for a light source anywhere in the four-image region close to elliptic and hyperbolic umbilic caustics. This is a more global and higher-order analog of the well-known fold and cusp magnification relations. We now extend each of these mappings to weighted projective space, which is a compact orbifold, and show that magnification relations translate into a statement about the behavior of these extended mappings at infinity. This generalizes multi-dimensional residue techniques developed in previous work, and introduces weighted projective space as a new tool in the theory of caustic singularities and gravitational lensing.

math-ph↗

arXiv2009

D-modules on 1|1 Supercurves

It is known that to every 1|1 dimensional supercurve X there is associated a dual supercurve \hat{X}, and a superdiagonal Δin their product. We establish that the categories of D-modules on X, \hat{X}, and Δare equivalent. This follows from a more general result about D-modules and purely odd submersions. The equivalences preserve tensor products, and take vector bundles to vector bundles. Line bundles with connection are studied, and examples are given where X is a superelliptic curve.

math.AG↗

arXiv2005

Periodicity and Growth in a Lattice Gas with Dynamical Geometry

We study a one-dimensional lattice gas "dynamical geometry model" in which local reversible interactions of counter-rotating groups of particles on a ring can create or destroy lattice sites. We exhibit many periodic orbits and and show that all other solutions have asymptotically growing lattice length in both directions of time. We explain why the length grows as $\sqrt{t}$ in all cases examined. We completely solve the dynamics for small numbers of particles with arbitrary initial conditions.

nlin.CG↗

arXiv2002

On Decomposing N=2 Line Bundles as Tensor Products of N=1 Line Bundles

We obtain the existence of a cohomological obstruction to expressing N=2 line bundles as tensor products of N=1 bundles. The motivation behind this paper is an attempt at understanding the N=2 super KP equation via Baker functions, which are special sections of line bundles on supercurves.

hep-th↗

arXiv2000

Magnification relations in gravitational lensing via multidimensional residue integrals

We investigate the so-called magnification relations of gravitational lensing models. We show that multidimensional residue integrals provide a simple explanation for the existence of these relations, and an effective method of computation. We illustrate the method with several examples, thereby deriving new magnification relations for galaxy lens models and microlensing (point mass lensing).

astro-ph↗

arXiv1997

Geodesics and BPS States in N=2 Supersymmetric QCD

Conformal mapping techniques are used to determine analytically the geodesics on the Seiberg-Witten Riemann surface which correspond to the BPS dyon states in N=2 SUSY QCD with gauge group SU(2) and up to three flavors of massless fundamental matter. The results are exact for zero and two flavors, and approximate in the weak-coupling limit for one and three flavors. The presence of states of magnetic charge 2, in the three-flavor case only, is confirmed.

hep-th↗

arXiv1996

Old and New Fields on Super Riemann Surfaces

The ``new fields" or ``superconformal functions" on $N=1$ super Riemann surfaces introduced recently by Rogers and Langer are shown to coincide with the Abelian differentials (plus constants), viewed as a subset of the functions on the associated $N=2$ super Riemann surface. We confirm that, as originally defined, they do not form a super vector space.

hep-th↗

arXiv1993

Super Elliptic Curves

A detailed study is made of super elliptic curves, namely super Riemann surfaces of genus one considered as algebraic varieties, particularly their relation with their Picard groups. This is the simplest setting in which to study the geometric consequences of the fact that certain cohomology groups of super Riemann surfaces are not freely generated modules. The divisor theory of Rosly, Schwarz, and Voronov gives a map from a supertorus to its Picard group, but this map is a projection, not an isomorphism as it is for ordinary tori. The geometric realization of the addition law on Pic via intersections of the supertorus with superlines in projective space is described. The isomorphisms of Pic with the Jacobian and the divisor class group are verified. All possible isogenies, or surjective holomorphic maps between supertori, are determined and shown to induce homomorphisms of the Picard groups. Finally, the solutions to the new super Kadomtsev-Petviashvili (super KP) hierarchy of Mulase-Rabin which arise from super elliptic curves via the Krichever construction are exhibited.

hep-th↗