# Past Geometry and Analysis Seminar

I will discuss theta-stability, a framework for analyzing moduli problems in algebraic geometry by finding a special kind of stratification called a theta-stratification, a notion which generalizes the Kempf-Ness stratification in geometric invariant theory and the Harder-Narasimhan-Shatz stratification of the moduli of vector bundles on a Riemann surface.

The famous Greek astronomer Ptolemy created his well-known table of chords in order to aid his astronomical observations. This table was based on the renowned relation between the four sides and the two diagonals of a quadrilateral whose vertices lie on a common circle.

In 2002, the mathematicians Fomin and Zelevinsky generalised this relation to introduce a new structure called cluster algebra. This is a set of clusters, each cluster made of n numbers called cluster variables. All clusters are obtained from some initial cluster by a sequence of transformations called mutations. Cluster algebras appear in a variety of topics, including total positivity, number theory, Teichm\”uller theory and computer graphics. A quantisation procedure for cluster algebras was proposed by Berenstein and Zelevinsky in 2005.

After introducing the basics about cluster algebras, in this talk we will link cluster algebras to the theory of Painlevé equations. This link will provide the foundations to introduce a new class of cluster algebras of geometric type. We will show that the quantisation of these new cluster algebras provide a geometric setting for the Berenstein–Zelevinsky construction.

Milnor has shown that three-dimensional Lie groups with left invariant Riemannian metrics furnish examples of 3-manifolds with principal Ricci curvatures of fixed signature --- except for the signatures (-,+,+), (0,+,-), and (0,+,+). We examine these three cases on a Riemannian 3-manifold, and prove global obstructions in certain cases. For example, if the manifold is closed, then the signature (-,+,+) is not globally possible if it is of the form -µ,f,f, with µ a positive constant and f a smooth function that never takes the values 0,-µ (this generalizes a result by Yamato '91). Similar obstructions for the other cases will also be discussed. Our methods of proof rely upon frame techniques inspired by the Newman-Penrose formalism. Thus, we will close by turning our attention to four dimensions and Lorentzian geometry, to uncover a relation between null vector fields and exact symplectic forms, with relations to Weinstein structures.

From a complex analytic perspective, both Teichmüller spaces and

symmetric spaces can be realised as contractible bounded domains, that

have several features in common but also exhibit many differences. In

this talk we will study isometric maps between these two important

classes of bounded domains equipped with their intrinsic Kobayashi metric.

I will discuss some properties of Hermitian metrics on compact complex manifolds, having constant Chern scalar curvature, focusing on the existence problem in fixed Hermitian conformal classes (the "Chern-Yamabe problem"). This is joint work with Daniele Angella and Simone Calamai.