Kolloquium Wilhelm Killing

If we complete the rational numbers $\mathbb{Q}$ with respect to the standard absolute value, we obtain the reals $\mathbb{R}$, wich is a connected topological space. The completion $\mathbb{Q}_p$ of $\mathbb{Q}$ with respect to the $p$-adic valuation for a prime $p$, however, is totally disconnected. So it seems that the concept of a path connecting two points in the $p$-adic numbers $\mathbb{Q}_p$ (or $\mathbb{Q}_p^n$) does not make sense. In fact, the space $\mathbb{Q}_p$ itself is not quite suitable for doing geometry. Instead one can consider the affine line $\mathbb{A}_{\mathbb{Q}_p}^1$ as an adic space. It contains $\mathbb{Q}_p$ as so called \emph{classical points} but has many more points. In this talk we will convince ourselves that $\mathbb{A}_{\mathbb{Q}_p}^1$ is indeed path connected if we use the right notion of a path.

Let X be a proper, smooth rigid space and G a commutative rigid group. We study the relationship between G-representations of the fundamental group of X and G-Higgs bundles on X. This is joint work with Ben Heuer and Mingjia Zhang.

The mathematical core of deep learning is function approximation by neural networks trained on data using stochastic gradient descent.

I will present a collection of results on training dynamics for the deep linear network (DLN). The DLN is a phenomenological model of deep learning for linear functions that was introduced by computer scientists. It allows a comprehensive analysis that reveals interesting ties with several areas of mathematics and several lessons for 'true' deep learning.

This is joint work with several co-authors: Nadav Cohen (Tel Aviv), Kathryn Lindsey (Boston College), Alan Chen, Zsolt Veraszto and Tianmin Yu (Brown).

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