This is the organizational meeting, where we will decide on the topics to be covered this semester.

A general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex plane. It is then demonstrated that the only stable fixed point (attractor) of the non-singular bubble dynamics corresponds precisely to the selected pattern. This thus shows that, contrary to the established theory, bubble selection in a Hele-Shaw cell does not require surface tension. The solutions reported here significantly extend previous results for a simply-connected geometry (finger) to a doubly-connected one (bubble). We conjecture that the same selection rule without surface tension holds for Hele-Shaw flows of arbitrary connectivity. We also believe that this mechanism can be found in other, similarly described, selection problems.

Introduction to Cluster Algebras: I will begin by introducing a class of projective varieties known as quiver Grassmannians, the points of these varieties correspond to subrepresentations with specified dimension vector inside a representation of a quiver. It turns out that quiver Grassmannians are very general, indeed Reineke has shown that every projective variety can be realized as a quiver Grassmannian. Cluster algebras were discovered by Fomin and Zelevinsky in their study of (dual) canonical bases and total positivity. They are generated by the cluster variables, a collection of recursively defined rational functions. The remarkable “Laurent Phenomenon” states that these recursions actually produce Laurent polynomials. The cluster variables can be seen as a tool for studying certain quiver Grassmannians, more precisely their Laurent expansions provide generating functions for the Euler-Poincare characteristic of some quiver Grassmannians. For simplicity I will restrict to introducing (coefficient-free) cluster algebras in rank 2 and explain the relationship between cluster variables and quiver Grassmannians in this setting

Theta basis is greedy basis Log Calabi-Yau varieties were discovered by Gross, Hacking, and Keel in their study of mirror symmetry. One construction involves gluing a collection of tori so the volume forms patch and the simplest among these are the cluster varieties, i.e. the spectra of the (upper) cluster algebras. The ring of global functions on a log Calabi-Yau variety comes with a canonical basis of functions called the theta basis. An original conjecture of Fomin and Zelevinsky (since proven in some generality) was the positivity of the Laurent expansions of cluster variables. Building on this Lee, Li, and Zelevinsky defined a basis with remarkable positivity properties in rank 2 cluster algebras known as the greedy basis. The greedy basis elements are naturally defined by a recursion which is not manifestly positive but is seen to be positive via a combinatorial construction involving the maximal Dyck path in a lattice rectangle. In this talk I will attempt to describe the geometry underlying the theta basis and why the theta basis coincides with the greedy basis. These results are from a joint project with several authors from the MRC Cluster Algebras workshop this summer.