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This talk will be a tour through the combinatorics of reduced
decompositions in the symmetric group (and, if time permits, the affine
symmetric and hyperoctahedral groups). This subject has connections to
many areas of mathematics. For instance, it is closely tied to the
geometry of the Grassmannian and of the flag variety, through the theory
of Schubert polynomials. Also there are also some very surprising (and,
apparently, very difficult) analytic and probabilistic questions about
large reduced decompositions.
A garbled circuit (GC) is a way of encrypting a computation in a way that reveals only the final output and none of the intermediate state. In this talk, I'll describe Yao's classical GC construction (1987), and its application to secure two-party computation protocols. Starting in the early 2000s, interest gained in actually implementing secure computation protocols. As a result, garbled circuits became the focus of an interesting line of combinatorial optimizations, which I will present. These works culminated in a recent GC construction that can be proven optimal in some sense.
Newton's third law of motion says, basically, that if you push on me then I push back automatically, even if I do not want to. Fulfilling this condition can be a problem when a discontinuous Galerkin numerical method is applied to a layered model of ocean circulation. The dependent variables can be discontinuous across cell edges, and in particular the one-sided limits of the pressure forcing at an edge need not be equal. One way to define values of the solution at a cell edge is to employ a Riemann problem, in which the dynamics of the PDE are used to interpolate between two state