L5

It is a classical fact of rational homotopy theory that the $E_\infty$-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still $E_n$-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of $E_n$-algebras, closely related to Koszul duality for the $E_n$-operads. We will also see that these results are essentially sharp: if the R-valued cochains of $S^n$ have square-zero $E_{n+1}$-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.

Minimal surfaces, which are critical points of the area functional, have long been a source of fruitful problems in geometry. In this talk, I will introduce a new approach, primarily coming from a recent paper of M. Struwe, to constructing free boundary minimal discs using a gradient flow of a suitable energy functional. I will discuss the uniqueness of solutions to the gradient flow, including recent work on the uniqueness of weak solutions, and also what is known about the qualitative behaviour of the flow, especially regarding the interpretation of singularities which arise. Time permitting, I will also mention ongoing joint work with M. Rupflin and M. Struwe on extending this theory to general surfaces with boundary.

to follow

Single-cell sequencing data consists of a point cloud where the points are cells, with coordinates RNA expression levels in each gene. Since the tissue is destroyed by the sequencing procedure, the dynamics of gene expression must be inferred from the structure and geometry of the point cloud. In this talk, we will build a biological interpretation of the one-dimensional cohomology classes in hallmark gene subsets as models for transient biological processes. Such processes include the cell-cycle, but more generally model homeostatic negative feedback loops. Our procedure uses persistent cohomology to identify features, and integration of differential forms to estimate the cascade of genes associated with the underlying dynamics of gene expression.

This is joint work with Markus Youssef and Tâm Nguyen at EPFL.