- Friday, September 27, 2019, 12:00pm, Physics 119, Mathematics Colloquium Seminar
*Model theory and complexity*

Maryanthe Malliaris (University of Chicago)- The ultraproduct construction in model theory gives a way of averaging an infinite sequence of mathematical structures, such as fields, graphs, or linear orders. The talk will be about the strength of such a construction.

- Friday, October 4, 2019, 12:00pm, 119 Physics, Mathematics Colloquium Seminar
*Intrinsic complexity: from approximation of random vectors and random fields to solutions of PDEs*

Hongkai Zhao (University of California, Irvine)- We characterize the intrinsic complexity of a set in a metric space by the least dimension of a linear space that can approximate the set to a given tolerance. This is dual to the characterization of the set using Kolmogorov n-width, the distance from the set to the best n-dimensional linear space. In this talk I will start with the intrinsic complexity of a set of random vectors (via principal component analysis) and random fields (via Karhunen–Loève expansion) and then characterize solutions to partial differential equations of various type. Our study provides a mathematical understanding of the complexity/richness and its mechanism of the underlying problem independent of representation basis. In practice, our study is directly related to the question of whether there is a low rank approximation to the associated (discretized) linear system, which is essential for dimension reduction and developing fast algorithms.

- Wednesday, October 30, 2019, 3:15pm, Physics 119, Mathematics Colloquium Seminar
*TBA*

Sarah Koch (University of Michigan)

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