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Suppose we are given a globally hyperbolic spacetime (M,g) solving the Einstein vacuum equations, and a timelike geodesic in M. I will explain how to construct, on any compact subset of M, a solution g_\epsilon of the Einstein vacuum equations which is approximately equal to g far from the…
Title: Cubic Weyl sums in 2-D from the Fourier restriction perspective
Abstract: We consider cubic Weyl sums of the form \sum_{n=N}^{2N} e(x\cdot(n,n^3)), where x lies in the unit square [0,1]^2. It is expected that for many x, such sums exhibit square root cancellation behavior, but…
We consider cubic Weyl sums of the form \sum_{n=N}^{2N} e(x\cdot(n,n^3)), where x lies in the unit square [0,1]^2. It is expected that for many x, such sums exhibit square root cancellation behavior, but this remains far out of reach: the classical argument for quadratic Gauss sums does not…
The dimer model refers to the study of random dimer covers (or perfect matchings) of a bipartite graph. A remarkable feature of these models is the emergence of limit shapes: in large periodic graphs, a random matching concentrates around a deterministic shape. Although general dimer models…
Any reasonable exotic phenomena in simply connected 4-manifolds are unstable. It is an open question if there is a universal upper bound to the number of stabilizations needed. The case of 1 stabilization was proven in works of Lin and Guth-K., but whether we need more than two stabilizations…