Conferences and Meetings
Department of Mathematics and Statistics
Texas Tech University
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We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties of such billiards are studied. The novelty, caused by reflex angles on boundary, induces invariant leaves of higher genera and dynamical behavior different from Liouville--Arnold's Theorem. The billiard flow generates a measurable foliation defined by a closed 1-form w. Using the closed form, a transformation of the given billiard table to a rectangular cylinder is constructed and a trajectory equivalence between corresponding billiards has been established. A local version of Poncelet Theorem is formulated and necessary algebro-geometric conditions for periodicity are presented. It is proved that the dynamics depends on arithmetic of rotation numbers, but not on geometry of a given confocal pencil of conics. Examples of billiard trajectories having a fixed circle concentric with the boundary semicircles as the caustic, such that the rotation numbers with respect to the half-circles are different pairs of numbers r1 and r2 respectively, are presented. Are such billiard trajectories periodic, and what are all possible periods for given r1 and r2? This presentation is based on joint works with Milena Radnovic.
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What does it mean to know mathematics? Usually we associate math with equations and think of it as the epitome of symbolic reasoning. Yet all around us simple organisms and objects are performing extraordinary pieces of mathematics. Sea slugs and corals realize in their anatomies hyperbolic geometry, a form that human mathematicians spent hundreds of years trying to prove impossible. Swooping towards a prey, a peregrine falcon executes a logarithmic spiral. Bouncing about a room, sounds waves enact the complexity of a Fourier Transform. Meanwhile African artisans discovered fractals centuries before European mathematicians and have long incorporated them into designs of cloth, pottery and village architecture. Islamic mosaicists, masters of geometrical construction, discovered aperiodic tilings more than 500 years before modern mathematicians. In this multidisciplinary talk, Margaret Wertheim will argue that mathematics isn't just something we do with our minds, it can also be engaged with through the process of material exploration and play.
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When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. In this talk we will survey two cases in which progress has been made focusing primarily on the case of surfaces with boundary. We will describe the geometric structure of the critical metrics which turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces.The minisymposium is addressed to all graduate students and post-docs in the area of applied mathematics.