Emancipating Lincoln: The Proclamation in Text, Context, and Memory

Putting Emancipation into Perspective This volume gathers expanded versions of the Nathan I. Huggins lectures delivered in October 2010 by Harold Holzer. The title of the book has a double meaning: it refers to Lincoln’s actions as emancipator and seeks to emancipate Lincoln from the jud...

Ergodic directions for billiards in a strip with periodically located obstacles

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<\lambda<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $\lambda/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.Comment: The article is complementary to arXiv:1109.458

Competition and Congestion in Trademark Law

Trademark law exists to promote competition. If consumers know which companies make which products, they can more easily find the products they actually want to purchase. Trademark law has long treated “source significance”—the fact that a particular trademark is identified with a particular producer—as both necessary and sufficient for establishing a valid trademark. That is, trademark law has traditionally viewed source significance as the only necessary precondition for a trademark being pro-competitive. In this paper, we argue that this equation of source significance and pro-competitiveness is misguided. Some marks use words that are so closely connected with the product being branded that giving just one firm a monopoly over those words provides that firm with a meaningful competitive advantage—an artificial advantage granted by the state. This problem becomes worse as the number of firms producing (and branding) a type of product increases. The more words cordoned off by trademark law, the more trouble a new entrant will have in describing or attracting attention to its product. Trademark law is thus being hijacked by strategic firms for anti-competitive purposes. Traditional doctrinal tools are inadequate to address this problem because the goal should be to limit the number of such trademarks rather than eliminate them completely. However, costly screens could be used to impose a form of congestion pricing on trademarks, eliminating them in all but the most worthwhile cases. In this paper, we develop a theory of the anti-competitive nature of certain trademark rules. We then propose a series of overlapping doctrinal rules and costly screens to address the problem of rampant anticompetitive trademark

Competition and Congestion in Trademark Law

Trademark law exists to promote competition. If consumers know which companies make which products, they can more easily find the products they actually want to purchase. Trademark law has long treated “source significance”—the fact that a particular trademark is identified with a particular producer—as both necessary and sufficient for establishing a valid trademark. That is, trademark law has traditionally viewed source significance as the only necessary precondition for a trademark being pro-competitive. In this Article, we argue that this equation of source significance and pro-competitiveness is misguided. Some marks use words that are so closely connected with the product being branded that giving just one firm a monopoly over those words provides that firm with a meaningful competitive advantage—an artificial advantage granted by the state. This problem becomes worse as the number of firms producing (and branding) a type of product increases. The more words cordoned off by trademark law, the more trouble a new entrant will have in describing or attracting attention to its product. Trademark law is thus being hijacked by strategic firms for anticompetitive purposes. Traditional doctrinal tools are inadequate to address this problem because the goal should be to limit the number of such trademarks rather than eliminate them completely. However, costly screens could be used to impose a form of congestion pricing on trademarks, eliminating them in all but the most worthwhile cases. In this Article, we develop a theory of the anticompetitive nature of certain trademark rules. We then propose a series of overlapping doctrinal rules and costly screens to address the problem of rampant anticompetitive trademarks

A random tunnel number one 3-manifold does not fiber over the circle

We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem. The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1.Comment: This is the version published by Geometry & Topology on 15 December 200

Diffractive orbits in isospectral billiards

Isospectral domains are non-isometric regions of space for which the spectra of the Laplace-Beltrami operator coincide. In the two-dimensional Euclidean space, instances of such domains have been given. It has been proved for these examples that the length spectrum, that is the set of the lengths of all periodic trajectories, coincides as well. However there is no one-to-one correspondence between the diffractive trajectories. It will be shown here how the diffractive contributions to the Green functions match nevertheless in a ''one-to-three'' correspondence.Comment: 20 pages, 6 figure

Periodic Orbits and Spectral Statistics of Pseudointegrable Billiards

We demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than $l$ increases as $\pi b_0l^2/\langle a(l) \rangle$, where $b_0$ is a constant and $\langle a(l) \rangle$ is the average area occupied by these families. We also find that $\langle a(l) \rangle$ increases with $l$ before saturating. Finally, we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies.Comment: 13 pages in RevTex including 5 figure

A Combination Theorem for Metric Bundles

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added. Characterization of convex cocompact subgroups of mapping class groups of surfaces with punctures in terms of relative hyperbolicity given v4: Final version incorporating referee comments: 63 pages 5 figures. To appear in Geom. Funct. Ana