Analysis Seminar

In this talk, we will describe how a connection between laminates and martingales can be used to construct examples of p-homogeneous rank-one convex integrands that are nowhere quasiconvex, answering a question of Kirchheim and Kristensen. We will also see how the method yields a homogeneous rank-one convex quartic polynomial on \mathbb{R}^{2\times3} that is nowhere quasiconvex. Together with Šverák’s 1992 result in \mathbb{R}^{3\times2}, this leaves only the 2\times2 case of Morrey’s problem open.

Wide neural networks provide an interesting meeting point between probability theory, analysis, and modern machine learning. As their width tends to infinity, randomly initialized networks converge to Gaussian processes, suggesting a relatively simple description of an otherwise highly nonlinear random system. But how accurate is this approximation, and what does it miss?

In this talk, I will discuss some recent mathematical results addressing these questions. Starting from Gaussian process limits and quantitative estimates based on optimal transport, we will explore the large-deviation regime, where atypical fluctuations reveal a richer, non-Gaussian structure governed by variational problems involving the internal kernels of the network.

We will also see how these two probabilistic regimes lead to different descriptions of Bayesian neural networks: quantitative convergence to Gaussian process posteriors on the one hand, and large-deviation principles that capture the interplay between prior probabilities and observations on the other.

I will discuss linearised gravity on a higher dimensional black hole known as the Schwarzschild black string. I will present some partial stability results and a proof of the Gregory—Laflamme instability which afflicts many black holes in higher dimensions. No prior knowledge of general relativity will be assumed.

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