We identify conditions for which a Dirichlet space (a metric measure space with diffusion) admitting a sub-Gaussian heat kernel would be in the Da Prato-Debussche regime of the $\Phi^{n+1}$ equation. For this purpose, we use heat kernel based Besov spaces, where regularity of Schwartz-type distributions is measured using the small time behavior of the heat kernel. In the process, we show how many nontrivial parts of the solution theory such as construction of paraproducts and energy estimates are made more difficult by the roughness of the underlying geometry. These difficulties in fact produce a more restrictive regime than one may first expect by typical scaling heuristics.

Large language models (LLM) have impressive performance on hard tasks, but also exhibit brittleness in simple tasks. We describe an experiment on a basic ``sub-reasoning'' task: deciding if an element belongs to a set. The results give a comprehensive picture of the various types of errors that can occur. In the second part of the talk we give a brief overview of the mathematical challenges posed by the goal of understanding how a neural network works, including understanding what an LLM ``knows''. Joint work with Gabor Berend, Lea Hergert, Mark Jelasity and Mario Szegedy.

It is well known that asset returns usually do not follow a normal distribution, rather, they have long and fat tails. This paper focuses on the quantile portfolio methodology, which considers the whole distribution of asset returns and employs expected loss as a risk measurement. In particular, we explore statistical properties of tau risk and propose related theories of quantile portfolio optimization. We also introduce portfolio performance terms for the quantile portfolio framework.