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Michael Harrison

Publications and source records attributed to Michael Harrison.

26 records · Page 2Linked to original sources
arXiv2019

Fibrations of $\mathbb{R}^3$ by oriented lines

A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $\mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.

math.GT↗
arXiv2019

Contact structures induced by skew fibrations of R^3

A smooth fibration of $\mathbb{R}^3$ by oriented lines is given by a smooth unit vector field $V$ on $\mathbb{R}^3$, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if $\nabla V$ vanishes only in the direction of $V$. Nondegeneracy is a form of local skewness, though in fact any nondegenerate fibration is globally skew. Nondegenerate and skew fibrations have each been recently studied, from both geometric and topological perspectives, in part due to their close relationship with great circle fibrations of $S^3$. Any fibration of $\mathbb{R}^3$ by oriented lines induces a plane field on $\mathbb{R}^3$, obtained by taking the orthogonal plane to the unique line through each point. We show that the plane field induced by any nondegenerate fibration is a tight contact structure. For contactness we require a new characterization of nondegenerate fibrations, whereas the proof of tightness employs a recent result of Etnyre, Komendarczyk, and Massot on tightness in contact metric 3-manifolds. We conclude with some examples which highlight relationships among great circle fibrations, nondegenerate fibrations, skew fibrations, and the contact structures associated to fibrations.

math.GT↗
arXiv2015

Skew flat fibrations

A fibration of ${\mathbb R}^n$ by oriented copies of ${\mathbb R}^p$ is called skew if no two fibers intersect nor contain parallel directions. Conditions on $p$ and $n$ for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of ${\mathbb R}^3$ by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of $S^3$ by Gluck and Warner. We show that Salvai's classification has a topological variation which generalizes to characterize all continuous fibrations of ${\mathbb R}^n$ by skew oriented copies of ${\mathbb R}^p$. We show that the space of fibrations of ${\mathbb R}^3$ by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of $S^2$. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of ${\mathbb C}^n$ (${\mathbb H}^n$) by skew oriented copies of ${\mathbb C}^p$ (${\mathbb H}^p$).

math.AT↗
arXiv2015

The Archimedean Projection Property

Let $H$ be a hypersurface in $\mathbb R^n$ and let $π$ be an orthogonal projection in $\mathbb R^n$ restricted to $H$. We say that $H$ satisfies the $Archimedean$ $projection$ $property$ corresponding to $π$ if there exists a constant $C$ such that $Vol(π^{-1}(U)) = C \cdot Vol(U)$ for every measurable $U$ in the range of $π$. It is well-known that the $(n-1)$-dimensional sphere, as a hypersurface in $\mathbb R^n$, satisfies the Archimedean projection property corresponding to any codimension 2 orthogonal projection in $\mathbb R^n$, the range of any such projection being an $(n-2)$-dimensional ball. Here we construct new hypersurfaces that satisfy Archimedean projection properties. Our construction works for any projection codimension $k$, $2 \leq k \leq n - 1$, and it allows us to specify a wide variety of desired projection ranges $Ω^{n-k} \subset \mathbb R^{n-k}$. Letting $Ω^{n-k}$ be an $(n-k)$-dimensional ball for each $k$, it produces a new family of smooth, compact hypersurfaces in $\mathbb R^n$ satisfying codimension $k$ Archimedean projection properties that includes, in the special case $k = 2$, the $(n-1)$-dimensional spheres.

math.DG↗
arXiv2013

Benefits to the U.S. from Physicists Working at Accelerators Overseas

We illustrate benefits to the U.S. economy and technological infrastructure of U.S. participation in accelerators overseas. We discuss contributions to experimental hardware and analysis and to accelerator technology and components, and benefits stemming from the involvement of U.S. students and postdoctoral fellows in global scientific collaborations. Contributed to the proceedings of the Snowmass 2013 Community Summer Study.

physics.soc-ph↗
arXiv2013

A note on the five valued conjectures of Johansen and Helleseth and zeta functions

For the complete five-valued cross-correlation distribution between two $m$-sequences ${s_t}$ and ${s_{dt}}$ of period $2^m-1$ that differ by the decimation $d={{2^{2k}+1}\over {2^k+1}}$ where $m$ is odd and $\mbox{gcd}(k,m)=1$, Johansen and Hellseth expressed it in terms of some exponential sums. And two conjectures are presented that are of interest in their own right. In this correspondence we study these conjectures for the particular case where $k=3$, and the cases $k=1,2$ can also be analyzed in a similar process. When $k>3$, the degrees of the relevant polynomials will become higher. Here the multiplicity of the biggest absolute value of the cross-correlation is no more than one-sixth of the multiplicity corresponding the smallest absolute value.

cs.IT↗
arXiv2005

A Lie Algebra Method for Rational Parametrization of Severi-Brauer Surfaces

It is well-known that a Severi-Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi-Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane, or such a rational point, does exist; and to construct such an isomorphism or such a point in the affirmative case. We give an algorithm using Lie algebra techniques. The algorithm has been implemented in Magma.

math.AG↗