Coagulation Trajectories, Gelation, and Large Deviations in the Marcus–Lushnikov Process

This seminar concerns a path-space description of the Marcus–Lushnikov coagulation process, with particular emphasis on large deviations and gelation. Rather than focusing only on the empirical distribution of cluster sizes at a fixed time, the analysis is formulated in terms of the empirical measure of coagulation trajectories, that is, the microscopic histories through which individual clusters are formed. This perspective retains genealogical information and provides a finer description of the stochastic dynamics.

In the non-gelating regime, the trajectory-based empirical process admits a representation in terms of a Poissonian reference measure combined with an interaction term encoding the mutual exclusion between trajectories. This structure leads naturally to a Gibbs-type variational formulation and to a large deviation principle for the empirical measure on path space. The corresponding law of large numbers can then be interpreted through the associated variational problem.

The second part of the seminar addresses the onset of gelation. Mathematically, gelation corresponds to the emergence of clusters of macroscopic size, so that a positive fraction of the total mass is no longer captured by the empirical measure on finite coagulation trajectories. This shows that the original path-space framework, while complete in the non-gelating regime, becomes insufficient once gelation occurs. The seminar will therefore also discuss the conceptual and mathematical difficulties involved in extending the trajectory description beyond the purely non-gelating setting, and will outline possible directions for incorporating gel-forming trajectories into a path-space large deviation framework.

Massey products and Borromean rings

Singular cohomology groups provide one of the most basic examples of homotopy invariants of a topological space. However, they also carry additional layers of algebraic structure that are often overlooked. In this short talk, we will discuss two simple instances of this phenomenon, showing how increasingly rich algebraic structures on singular cohomology can capture finer and finer features of the homotopy type of the underlying space. In the first introductory example, we will see how the ring structure induced by the cup product detects the Hopf link. In the second, we will see how Massey products detect the ternary linking relation in the famous example of the Borromean rings, where the ring structure alone fails to do so.