Nearly $G_2$-structures in dimension seven are, up to scaling, critical points of a geometric flow called (modified) Laplacian co-flow. Moreover, since nearly $G_2$-structures define Einstein metrics, they can also be associated to critical points of the volume-normalised Ricci flow. In this talk, we will discuss a recent joint work with Jason Lotay, showing that many of these nearly $G_2$ critical points are unstable for the modified co-flow, and giving a lower bound on the index.

Entanglement entropy has long served as a key diagnostic of topological order in (2+1) dimensions. In particular, the topological entanglement entropy captures a universal quantity (the total quantum dimension) of the underlying topological order. However, this information alone does not uniquely determine which topological order is realized, indicating the need for more refined probes. In this talk, I will present a family of quantities formulated as multi-entropy measures, including examples such as reflected entropy and the modular commutator. Unlike the conventional bipartite setting of topological entanglement entropy, these multi-entropy measures are defined for tripartite partitions of the Hilbert space and capture genuinely multipartite entanglement. I will discuss how these measures encode additional universal data characterizing topologically ordered ground states.

In this talk, I will introduce a new framework for working with moduli stacks in enumerative geometry, aimed at generalizing existing theories of enumerative invariants counting objects in linear categories, such as Donaldson–Thomas theory, to general, non-linear moduli stacks. This involves a combinatorial object called the component lattice, which is a globalization of the cocharacter lattice and the Weyl group of an algebraic group.

Several important results and constructions known in linear enumerative geometry can be extended to general stacks using this framework. For example, Donaldson–Thomas invariants can be defined for a general class of stacks, not only linear ones such as moduli stacks of sheaves. As another application, under certain assumptions, the cohomology of a stack, which is often infinite-dimensional, decomposes into finite-dimensional pieces carrying enumerative information, called BPS cohomology, generalizing a result of Davison–Meinhardt in the linear case.

This talk is based on joint works with Ben Davison, Daniel Halpern-Leistner, Andrés Ibáñez Núñez, Tasuki Kinjo, and Tudor Pădurariu.