# Past Geometry and Analysis Seminar

Noncommutative projective geometry is the study of quantum versions of projective space and other projective varieties. Starting with the celebrated work of Artin, Tate and Van den Bergh on noncommutative projective planes, a substantial theory of noncommutative curves and surfaces has been developed, but the classification of noncommutative versions of projective three-space remains unknown. I will explain how a portion of this classification can be obtained, via deformation quantization, from a corresponding classification of holomorphic foliations due to Cerveau and Lins Neto. In algebraic terms, the result is an explicit description of the deformations of the polynomial ring in four variables as a graded Calabi--Yau algebra.

We produce new families of steady and expanding Ricci solitons

that are not of Kahler type. In the steady case, the asymptotics are

a mixture of the Hamilton cigar and the Bryant soliton paraboloid

asymptotics. We obtain some examples of Ricci solitons on homeomorphic

but non-diffeomorphic spaces. We also find numerical evidence of solitons

with more complicated topology.

I will explain a new formulation of Einstein’s equations in 4-dimensions using the language of gauge theory. This was also discovered independently, and with advances, by Kirill Krasnov. I will discuss the advantages and disadvantages of this new point of view over the traditional "Einstein-Hilbert" description of Einstein manifolds. In particular, it leads to natural "sphere conjectures" and also suggests ways to find new Einstein 4-manifolds. I will describe some first steps in these directions. Time permitting, I will explain how this set-up can also be seen via 6-dimensional symplectic topology and the additional benefits that brings.