MARK KAC SEMINAR

Studying $\mathbb Z^d$-extensions (i.e., skew-product with $\mathbb Z^d$ as fiber space) are more difficult to study than their base dynamics; they are non-compact and their invariant measures are infinite. But they model important systems such as Ehrenfest's wind-tree model and flows on infinite surfaces. In this talk I want to present some result concerning their ergodicity and recurrence, based on joint papers with Olga Lukina, and with Charles Fourgeron, Davide Ravotti, Dalia Terhesiu.

Membranes in living cells adapt a wide variety of continually evolving shapes closely related to their function. These shapes are regulated by curvature-inducing proteins, which also interact via the membrane deformations they impose. We study such membrane-mediated interactions in the globally curved and crowded setting of membranes inside the living cell. To do so, we rely heavily on tools from differential geometry to describe the shape and evolution of the membranes. In this talk, I will focus on the application of these mathematical tools to our biological system.

I will start with introducing the necessary framework of two-dimensional surfaces embedded in three-dimensional space. We will discuss how we can build complicated networks of membrane tubes and sheets by coupling global membrane properties to locally induced curvature. We find that the collective action of many proteins can change the overall membrane shape, and lead to the formation of many striking patterns that can be tested both in vivo and in biomimetic soft matter systems. Moreover, by coupling to active components like molecular motors or a growing and shrinking cytoskeleton, we can make the membrane dynamical, adapting its shape in response to a varying environment. These membrane dynamics are the basis of many biological functions, and by studying them we can eventually understand not only what a cell does, but also how it manages to do just that.