I will report on a research program which uses ideas from stochastic analysis in the context of constructive Euclidean quantum field theory. Stochastic analysis is the study of measures on path spaces via push-forward from Gaussian measures. The foundational example is the map, introduced by Itô, which sends Brownian motion to a diffusion process solution to a stochastic differential equation. Parisi–Wu's stochastic quantisation is the stochastic analysis of an Euclidean quantum field, in the above sense. In this introductory talk, I will put these ideas in context and illustrate various stochastic quantisation procedures and some of the rigorous results one can obtain from them.

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object that depends only on a poset, yet it governs the asymptotic behaviour of a natural gradient flow in the space of metrics of the object. 

In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects, thus generalising the HKKP filtration to other moduli problems of non-linear origin. In particular, we will make sense of the notion of a filtration labelled by sequence of numbers for a point of an algebraic stack.

We give an introduction to the subject of higher geometry, by giving many examples of higher geometric objects, and looking at their properties. These include examples of 2-rings, 2-vector spaces, and 2-vector bundles. We show how these concepts help solve problems in ordinary geometry, as one of the many motivations of the subject. We assume no prerequisites on the subject, and the talk should be applicable to both differential and algebraic geometry.