Properties of Enhanced Branching Structures (26w5560)

The Institute for Advanced Study in Mathematics will host the "Properties of Enhanced Branching Structures" workshop in Hangzhou, China from June 7 to June 12, 2026.

In lay terms, a branching structure may be thought of as any object which is tree-like, either geometrically, e.g. a naive diagram of a tree on a piece of paper, or in a more abstract sense, e.g. the way a computer may sequentially store incoming data. Branching structures appear naturally in a huge range of other phenomena, such as genealogies in evolutionary biology, spread and survival of populations in ecology, river formation in hydrology, to name but a few.

Wherever there is recursion, there is a tree, and so branching structures also turn up in an astonishing variety of abstract mathematical contexts as disparate as geometry and the analysis of algorithmic complexity. Nowhere is this more the case than for random branching structures, which are firmly embedded in the mathematical DNA of probability theory. For example the fine structure of the extrema of the two-dimensional Gaussian free field has recently been understood thanks to a hidden connection with spatial branching processes; in the context of statistical mechanical models on lattices, trees may be used to provide an approximation to the geometry of the base graph in high dimensions, which is good enough to capture key phenomena; and certain non-linear travelling waves of partial differential equations can be fully characterised using branching Brownian motion.

The development over the last 20 years of various limit theories for random graph-type objects has also given a new impetus to the study of random trees and their limits. For example, spectacular progress has been made in understanding the scaling limits of random planar maps, which may be thought of as random 2-dimensional surfaces, with continuum random trees playing a fundamental role in their construction and analysis. Similar methods have been brought to bear on various models of critical random graphs, giving rise to a detailed description of their scaling limits as random fractals with surprisingly explicit geometric properties.

We have reached the point where there is a need to push the theory of branching structures further, in terms of the complexity of the mathematical problems that we can use them to solve, and deeper, in terms of our understanding of the theory itself. Researchers from around the globe, will unite to examine emerging theories of enhanced branching structures. That is branching structures which contain an extra layer of information that interacts with the branching dynamics. This extra information might represent a spatial location, an evolutionary fitness parameter, a community in a social network, or even some more complex, high-dimensional quantity. Depending on the choice of extra information and the stochastic rules for how it interacts with the branching, this extension affords vastly increased modelling power but also entails significantly greater mathematical complexity. We will explore theoretical considerations of enhanced branching structures that play a role solving next-generation problems in stochastic genetics, random graph theory, PDE, SDE and SPDE theory, among others, which currently lie out of reach.

The Institute for Advanced Study in Mathematics (IASM) in Hangzhou, China, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF) and Alberta Technology and Innovation.