SR2

We will describe the space of Bridgeland stability conditions

of the derived category of some CY3 algebras of quivers drawn on the

Riemann sphere. We give a biholomorphic map from the upper-half plane to

the space of stability conditions lifting the period map of a meromorphic

differential on a 1-dimensional family of elliptic curves. The map is

equivariant with respect to the actions of a subgroup of $\mathrm{PSL}(2,\mathbb Z)$ on the

left by monodromy of the rational elliptic surface and on the right by

autoequivalences of the derived category.

The complement of a divisor in the rational elliptic surface can be

identified with Hitchin's moduli space of connections on the projective

line with prescribed poles of a certain order at marked points. This is

the space of initial conditions of one of the Painleve equations whose

solutions describe isomonodromic deformations of these connections.

A factorability structure on a group G is a specification of normal forms

of group elements as words over a fixed generating set. There is a chain

complex computing the (co)homology of G. In contrast to the well-known bar

resolution, there are much less generators in each dimension of the chain

complex. Although it is often difficult to understand the differential,

there are examples where the differential is particularly simple, allowing

computations by hand. This leads to the cohomology ring of hv-groups,

which I define at the end of the talk in terms of so called "horizontal"

and "vertical" generators.

An S-act over a monoid S is a representation of a monoid by tranformations of a set, analogous to the notion of a G-act over a group G being a representation of G by bijections of a set. An S-poset is the corresponding notion for an ordered monoid S.

Persistent homology is a relatively new tool to analyse the topology of data sets.

We will give a brief introduction and tutorial as preparation for the third talk in the afternoon.

We'll survey some of the ways that hyperbolic groups have been studied

using analysis on their boundaries at infinity.