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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When the complex evolution of multi-physics systems is treated as a monolithic object, the time step selection is governed by the most rapidly varying component. However, the appropriate analysis can often reveal a splitting that allows rapid, efficient, and accurate simulation of the full system, by carefully coordinating the uncoupled computation of each subsystem.
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The Hitchin system is an integrable system depending on a choice of a smooth complex Riemann surface and a positive integer. It was introduced by Nigel Hitchin in 1987. It received a huge amount of attention, partly because many classically known integrable systems can be embedded into the Hitchin system, partly because the system is related to many areas of mathematics such as algebraic geometry, Langlands program, and mathematical physics.
There are still completely open fundamental questions about one-variable polynomials. One example is Hilbert’s 13th Problem, which concerns formulas for the roots of a polynomial in terms of its coefficients.
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