Jonathan Rosenberg - Profile on Academia.edu (original) (raw)
Papers by Jonathan Rosenberg
Geometry of Surfaces
Multivariable Calculus with MATLAB®, 2017
Physical Applications of Vector Calculus
Multivariable Calculus and Mattiematica®, 1998
T his chapter represents the culmination of multivariable calculus. We investigate the remarkable... more T his chapter represents the culmination of multivariable calculus. We investigate the remarkable physical applications of vector calculus that provided the original motivation for the development of this subject in the seventeenth, eighteenth, and nineteenth centuries. The vector fields that we examine arise naturally in celestial mechanics, electromagnetism, and fluid flow. We will use the basic concepts of vector calculus to derive fundamental laws of physics in these subjects. In the attached Problem Set I, you will have a chance to use Mathematica to solve some interesting physical problems that would be difficult or impossible to tackle with pencil and paper alone.
Vectors and Graphics
Multivariable Calculus and Mattiematica®, 1998
We start this chapter by explaining how to use vectors in Mathematics, with an emphasis on practi... more We start this chapter by explaining how to use vectors in Mathematics, with an emphasis on practical operations on vectors in the plane and in space. We discuss the standard vector operations, and give several applications to the computations of geometric quantities such as distances, angles, areas, and volumes. The bulk of the chapter is devoted to instructions for graphing curves and surfaces.
Multivariable Calculus and Mathematica
The American Mathematical Monthly, 2000
Multivariable Calculus with MATLAB®, 2017
The study of curves in space is of interest not only as a topic in geometry but also for its appl... more The study of curves in space is of interest not only as a topic in geometry but also for its application to the motion of physical objects. In this chapter, we develop a few topics in mechanics from the point of view of the theory of curves. Additional applications to physics will be considered in Chapter 10.
Multivariable Calculus with MATLAB®, 2017
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Pacific Journal of Mathematics, 1978
Thus WwdjW* -tt(ai)\\ will be small for ε small enough, and a is approximately inner in the stron... more Thus WwdjW* -tt(ai)\\ will be small for ε small enough, and a is approximately inner in the stronger sense. COROLLARY 2.2. Suppose A is a C*-algebra with an approximate identity consisting of projections. Then the flip a for A is approximately inner if and only if it is so in the stronger sense. Proof. If {e r } is an approximate identity for A consisting of projections, then {e r (x) e r } is an approximate identity for A (x) A consisting of <7-fixed projections. Keeping these results in mind, we shall work hereafter with our original definition, since M(A) has better functorial properties than A~. (For instance, the proof of the next lemma breaks down if we use A~ in place of M(A).) Any inner automorphism a must fix ideals, i.e., if J is an ideal in A, then a(J) = J, since J will again be an ideal in M{A). It immediately follows that the same is true for approximately inner automorphisms. Another simple observation that we will need is: LEMMA 2.3. Given C*-algebras A and B with approximately inner automorphisms a and β, respectively, the automorphism α (g) β: A(g) B-> A (x) B determined by a ® b h-> a(a) (x) β{b) is again approximately inner.
Flux quantization for a superconducting ring in the shape of a M\"obius band
We give two derivations of magnetic flux quantization in a superconducting ring in the shape of a... more We give two derivations of magnetic flux quantization in a superconducting ring in the shape of a M\"obius band, one using direct study of the Schr\"odinger equation and the other using the holonomy of flat U(1)-gauge bundles. Both methods show that the magnetic flux must be quantized in integral or half-integral multiples of Phi_0=hc/(2e)\Phi_0=hc/(2e)Phi_0=hc/(2e). Half-integral quantization shows up in "nodal states" whose wavefunction vanishes along the center of the ring, for which there is now some experimental evidence.
Transactions of the American Mathematical Society, 1980
An analogue of the "Langlands conjecture" is proved for a large class of connected unimodular Lie... more An analogue of the "Langlands conjecture" is proved for a large class of connected unimodular Lie groups having square-integrable representations (modulo their centers). For nilpotent groups, it is shown (without restrictions on the group or the polarization) that the L2-cohomology spaces of a homogeneous holomorphic line bundle, associated with a totally complex polarization for a flat orbit, vanish except in one degree given by the "deviation from positivity" of the polarization. In this degree the group acts irreducibly by a square-integrable representation, confirming a conjecture of Moscovici and Verona. Analogous results which improve on theorems of Satake are proved for extensions of a nilpotent group having square-integrable representations by a reductive group, by combining the theorem for the nilpotent case with Schmid's proof of the Langlands conjecture. Some related results on Lie algebra cohomology and the "Harish-Chandra homomorphism" for Lie algebras with a triangular decomposition are also given. 0. Introduction. Since the appearance of Kirillov's thesis [21] on nilpotent Lie groups, the key unifying idea in the study of unitary representations of more or less arbitrary connected Lie groups has been the association of irreducible or at least primary representations with coadjoint orbits or "generalized orbits." This one principle is the basis for what one may call the Kirillov-Kostant "orbit method," which encompasses many of the deepest results of the representation theory of both solvable and semisimple Lie groups. (See, for example, [3], [5], [14], [21], [22], [25]-[28], [35], [36], [40], [43], [46], . This is not by any means a complete list-it is only a small sample of the literature to suggest the scope of the subject.) At the very least, it seems that for any connected Lie group G, all the representations needed to decompose the regular representation should be obtainable from some sort of "quantization process" involving polarizations for elements of the (real) dual g* of the Lie algebra g of G that satisfy some sort of integrality condition. The most familiar instance of this construction is the one used by Kirillov to construct all the irreducible unitary representations of a nilpotent Lie group, and which one can also use to construct the (unitary) principal series of a complex (or more generally, quasi-split) semisimple group. In this situation, one starts with an element A G g* and a real polarization b, for X, that is a Lie subalgebra b of g that is
Pacific Journal of Mathematics, 1995
Lecture Notes in Mathematics, 1984
Bulletin of the American Mathematical Society, 1999
Transactions of the American Mathematical Society, 1996
We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-v... more We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group G G . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on G G extends to be “holomorphic” on an appropriate complexification of (a part of) G ^ \hat G . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of C ∗ ( G ) C^{*}(G) when G G is two-step nilpotent.
Mathematical Sciences Research Institute Publications, 1994
The scalar curvature κ is the weakest curvature invariant one can attach (pointwise) to a Riemann... more The scalar curvature κ is the weakest curvature invariant one can attach (pointwise) to a Riemannian n-manifold M n. Its value at any point can be described in several different ways: (1) as the trace of the Ricci tensor, evaluated at that point. (2) as twice the sum of the sectional curvatures over all 2-planes e i ∧ e j , i < j, in the tangent space to the point, where e 1 ,. .. , e n is an orthonormal basis. (3) up to a positive constant depending only on n, as the leading coefficient in an expansion [22, Theorem 3.1]
Wiadomości Matematyczne, 2012
Princeton University Press eBook Package 2014, 2001
Advances in Theoretical and Mathematical Physics, 2014
Recently Baraglia showed how topological T-duality can be extended to apply not only to principal... more Recently Baraglia showed how topological T-duality can be extended to apply not only to principal circle bundles, but also to non-principal circle bundles. We show that his results can also be recovered via two other methods: the homotopy-theoretic approach of Bunke and Schick, and the noncommutative geometry approach which we previously used for principal torus bundles. This work has several interesting byproducts, including a study of the Ktheory of crossed products by O(2) = Isom(R), the universal cover of O(2), and some interesting facts about equivariant K-theory for Z/2. In the final section of this paper, some of these results are extended to the case of bundles with singular fibers, or in other words, non-free O(2)-actions.
Communications in Mathematical Physics, 2014
We analyze the brane content and charges in all of the orientifold string theories on space-times... more We analyze the brane content and charges in all of the orientifold string theories on space-times of the form E × R 8 , where E is an elliptic curve with holomorphic or anti-holomorphic involution. Many of these theories involve "twistings" coming from the B-field and/or sign choices on the orientifold planes. A description of these theories from the point of view of algebraic geometry, using the Legendre normal form, naturally divides them into three groupings. The physical theories within each grouping are related to one another via sequences of T-dualities. Our approach agrees with both previous topological calculations of twisted K R-theory and known physics arguments, and explains how the twistings originate from both a mathematical and a physical perspective.
Geometry of Surfaces
Multivariable Calculus with MATLAB®, 2017
Physical Applications of Vector Calculus
Multivariable Calculus and Mattiematica®, 1998
T his chapter represents the culmination of multivariable calculus. We investigate the remarkable... more T his chapter represents the culmination of multivariable calculus. We investigate the remarkable physical applications of vector calculus that provided the original motivation for the development of this subject in the seventeenth, eighteenth, and nineteenth centuries. The vector fields that we examine arise naturally in celestial mechanics, electromagnetism, and fluid flow. We will use the basic concepts of vector calculus to derive fundamental laws of physics in these subjects. In the attached Problem Set I, you will have a chance to use Mathematica to solve some interesting physical problems that would be difficult or impossible to tackle with pencil and paper alone.
Vectors and Graphics
Multivariable Calculus and Mattiematica®, 1998
We start this chapter by explaining how to use vectors in Mathematics, with an emphasis on practi... more We start this chapter by explaining how to use vectors in Mathematics, with an emphasis on practical operations on vectors in the plane and in space. We discuss the standard vector operations, and give several applications to the computations of geometric quantities such as distances, angles, areas, and volumes. The bulk of the chapter is devoted to instructions for graphing curves and surfaces.
Multivariable Calculus and Mathematica
The American Mathematical Monthly, 2000
Multivariable Calculus with MATLAB®, 2017
The study of curves in space is of interest not only as a topic in geometry but also for its appl... more The study of curves in space is of interest not only as a topic in geometry but also for its application to the motion of physical objects. In this chapter, we develop a few topics in mechanics from the point of view of the theory of curves. Additional applications to physics will be considered in Chapter 10.
Multivariable Calculus with MATLAB®, 2017
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Pacific Journal of Mathematics, 1978
Thus WwdjW* -tt(ai)\\ will be small for ε small enough, and a is approximately inner in the stron... more Thus WwdjW* -tt(ai)\\ will be small for ε small enough, and a is approximately inner in the stronger sense. COROLLARY 2.2. Suppose A is a C*-algebra with an approximate identity consisting of projections. Then the flip a for A is approximately inner if and only if it is so in the stronger sense. Proof. If {e r } is an approximate identity for A consisting of projections, then {e r (x) e r } is an approximate identity for A (x) A consisting of <7-fixed projections. Keeping these results in mind, we shall work hereafter with our original definition, since M(A) has better functorial properties than A~. (For instance, the proof of the next lemma breaks down if we use A~ in place of M(A).) Any inner automorphism a must fix ideals, i.e., if J is an ideal in A, then a(J) = J, since J will again be an ideal in M{A). It immediately follows that the same is true for approximately inner automorphisms. Another simple observation that we will need is: LEMMA 2.3. Given C*-algebras A and B with approximately inner automorphisms a and β, respectively, the automorphism α (g) β: A(g) B-> A (x) B determined by a ® b h-> a(a) (x) β{b) is again approximately inner.
Flux quantization for a superconducting ring in the shape of a M\"obius band
We give two derivations of magnetic flux quantization in a superconducting ring in the shape of a... more We give two derivations of magnetic flux quantization in a superconducting ring in the shape of a M\"obius band, one using direct study of the Schr\"odinger equation and the other using the holonomy of flat U(1)-gauge bundles. Both methods show that the magnetic flux must be quantized in integral or half-integral multiples of Phi_0=hc/(2e)\Phi_0=hc/(2e)Phi_0=hc/(2e). Half-integral quantization shows up in "nodal states" whose wavefunction vanishes along the center of the ring, for which there is now some experimental evidence.
Transactions of the American Mathematical Society, 1980
An analogue of the "Langlands conjecture" is proved for a large class of connected unimodular Lie... more An analogue of the "Langlands conjecture" is proved for a large class of connected unimodular Lie groups having square-integrable representations (modulo their centers). For nilpotent groups, it is shown (without restrictions on the group or the polarization) that the L2-cohomology spaces of a homogeneous holomorphic line bundle, associated with a totally complex polarization for a flat orbit, vanish except in one degree given by the "deviation from positivity" of the polarization. In this degree the group acts irreducibly by a square-integrable representation, confirming a conjecture of Moscovici and Verona. Analogous results which improve on theorems of Satake are proved for extensions of a nilpotent group having square-integrable representations by a reductive group, by combining the theorem for the nilpotent case with Schmid's proof of the Langlands conjecture. Some related results on Lie algebra cohomology and the "Harish-Chandra homomorphism" for Lie algebras with a triangular decomposition are also given. 0. Introduction. Since the appearance of Kirillov's thesis [21] on nilpotent Lie groups, the key unifying idea in the study of unitary representations of more or less arbitrary connected Lie groups has been the association of irreducible or at least primary representations with coadjoint orbits or "generalized orbits." This one principle is the basis for what one may call the Kirillov-Kostant "orbit method," which encompasses many of the deepest results of the representation theory of both solvable and semisimple Lie groups. (See, for example, [3], [5], [14], [21], [22], [25]-[28], [35], [36], [40], [43], [46], . This is not by any means a complete list-it is only a small sample of the literature to suggest the scope of the subject.) At the very least, it seems that for any connected Lie group G, all the representations needed to decompose the regular representation should be obtainable from some sort of "quantization process" involving polarizations for elements of the (real) dual g* of the Lie algebra g of G that satisfy some sort of integrality condition. The most familiar instance of this construction is the one used by Kirillov to construct all the irreducible unitary representations of a nilpotent Lie group, and which one can also use to construct the (unitary) principal series of a complex (or more generally, quasi-split) semisimple group. In this situation, one starts with an element A G g* and a real polarization b, for X, that is a Lie subalgebra b of g that is
Pacific Journal of Mathematics, 1995
Lecture Notes in Mathematics, 1984
Bulletin of the American Mathematical Society, 1999
Transactions of the American Mathematical Society, 1996
We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-v... more We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group G G . Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on G G extends to be “holomorphic” on an appropriate complexification of (a part of) G ^ \hat G . We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of C ∗ ( G ) C^{*}(G) when G G is two-step nilpotent.
Mathematical Sciences Research Institute Publications, 1994
The scalar curvature κ is the weakest curvature invariant one can attach (pointwise) to a Riemann... more The scalar curvature κ is the weakest curvature invariant one can attach (pointwise) to a Riemannian n-manifold M n. Its value at any point can be described in several different ways: (1) as the trace of the Ricci tensor, evaluated at that point. (2) as twice the sum of the sectional curvatures over all 2-planes e i ∧ e j , i < j, in the tangent space to the point, where e 1 ,. .. , e n is an orthonormal basis. (3) up to a positive constant depending only on n, as the leading coefficient in an expansion [22, Theorem 3.1]
Wiadomości Matematyczne, 2012
Princeton University Press eBook Package 2014, 2001
Advances in Theoretical and Mathematical Physics, 2014
Recently Baraglia showed how topological T-duality can be extended to apply not only to principal... more Recently Baraglia showed how topological T-duality can be extended to apply not only to principal circle bundles, but also to non-principal circle bundles. We show that his results can also be recovered via two other methods: the homotopy-theoretic approach of Bunke and Schick, and the noncommutative geometry approach which we previously used for principal torus bundles. This work has several interesting byproducts, including a study of the Ktheory of crossed products by O(2) = Isom(R), the universal cover of O(2), and some interesting facts about equivariant K-theory for Z/2. In the final section of this paper, some of these results are extended to the case of bundles with singular fibers, or in other words, non-free O(2)-actions.
Communications in Mathematical Physics, 2014
We analyze the brane content and charges in all of the orientifold string theories on space-times... more We analyze the brane content and charges in all of the orientifold string theories on space-times of the form E × R 8 , where E is an elliptic curve with holomorphic or anti-holomorphic involution. Many of these theories involve "twistings" coming from the B-field and/or sign choices on the orientifold planes. A description of these theories from the point of view of algebraic geometry, using the Legendre normal form, naturally divides them into three groupings. The physical theories within each grouping are related to one another via sequences of T-dualities. Our approach agrees with both previous topological calculations of twisted K R-theory and known physics arguments, and explains how the twistings originate from both a mathematical and a physical perspective.