Title: The stability-compactness method and qualitative properties of nonlinear elliptic PDEs

Speaker: Henri Berestycki (University of Maryland, College Park & EHESS, Paris)

Date, Time, Location: Wed. October 16th @ 4:30 PM in 520 Math Hall

Abstract: Nonlinear elliptic equations describe the stationary states of numerous systems in physics, biology and medicine. Qualitative properties such as monotonicity, symmetry, stability or uniqueness are essential features of their study. In this talk, I will present a new general framework to approach this kind of questions, focusing on uniqueness. It rests on decomposing the domain into one region with a certain compactness feature and another supporting a form of spectral stability. This approach has proved to be unexpectedly versatile and in fact encompasses past works on the subject such as the general moving plane method originating in the study of minimal surfaces by Alexandrov. The results concern positive solutions of nonlinear elliptic PDEs in general unbounded domains with Dirichlet boundary conditions and for various types of reaction terms. This represents a series of joint works with Cole Graham and with Cole Graham and Jun-Cheng Wei.

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Title: Matrix Rigidity

Speaker: Josh Alman (Columbia CS)

Date, Time, Location: Wed. Sept 25 @ 4:30 PM in 520 Math Hall

Abstract: A matrix is called rigid if one must change many of its entries before it becomes a low-rank matrix. Leslie Valiant introduced the notion in 1977 as a tool to prove lower bounds on the number of arithmetic operations needed to compute linear transformations like the discrete Fourier transform. Since then, many connections have been demonstrated between matrix rigidity and other topics in the theory of computation.

Unfortunately, proving that matrices of interest are rigid has shown to be a major challenge. In fact, it remains an open problem to prove that any explicit family of matrices is rigid. By contrast, it is known that a random matrix is rigid with high probability.

In this talk, I’ll give a brief overview of matrix rigidity and its uses in computer science. I’ll then discuss some recent results, including proofs that matrices like Hadamard and Fourier matrices, which were previously conjectured to be rigid, are in fact not rigid. The talk will not assume the audience has a background in computer science.

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Please join us in congratulating Professor Richard Hamilton on being awarded the Chinese Basic Science Lifetime Award in Mathematics by the International Congress of Basic Science.  (more)

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