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Hyperbolic space is the classical non-Euclidean space introduced by Lobachevksy, Bolyai and Gauss, in which the sum of the angles of a triangle is less than \pi. Hyperbolic n-manifolds are spaces that have the local geometry of hyperbolic n-space. The study of these spaces, besides being natural from the viewpoint of classical geometry, has important connections with differential geometry, algebraic geometry, complex analysis, number theory and topology.
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In this talk, we present the key challenges in data-driven decision-making when faced with a complex data environment. Traditionally, most data-driven decision-support tools are developed in a static fashion, meaning they consider a set of fixed, well-structured data to derive inferences or decisions. Nevertheless, even though these decision-making tools are rigorous and mathematically sound, they have not often been effective due to the dynamic nature and underlying complexity of the data in practice.
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Much research has been recently been devoted to sparse signal recovery and image reconstruction from multiple measurement vectors (MMV).
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Given four random red lines in 3-space, how many blue lines touch all four red? The answer is two, and this is the first nontrivial question in "Schubert calculus". Hilbert's 15th problem was to give this theory a solid foundation, which we now see as the cohomology ring of the Grassmannian of 1-planes in 3-space (or k-planes in affine n-space). There are many variations, all of which are easy to study algebraically, but only a few of which are understood combinatorially.
What do the dust patterns on vibrating plates or the equilibrium shapes of phospholipid vesicles have in common? Both are governed by the bending energy W, the integral of the squared mean curvature over an immersed surface in 3-space. Although introduced over two centuries ago by Sophie Germain, W is now* named for Tom Willmore, who suggested the global problem of minimizing W for a fixed topological class of surfaces, and who proved round spheres minimize W among all closed surfaces. Willmore conjectured a particular torus is the W-minimizer among surfaces of genus
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