We will introduce some ideas from geometric representation theory, including basic D-modules, (a realisation of) infinity categories, and the Ran space, and then go on to define chiral and factorisation algebras.
This talk has no prerequisites.
We will explore the many guises under which groups of 2x2 matrices appear, such as isometries of the hyperbolic plane, mapping class groups and the modular group. Along the way we will learn some interesting and perhaps surprising facts.
In 1924, James W. Alexander constructed a 2-sphere in R3 that is not ambiently homeomorphic to the standard 2-sphere, which demonstrated the failure of the Schoenflies theorem in higher dimensions. I will describe the construction of the Alexander horned sphere and the Antoine necklace and describe some of their properties.
The computation of multidimensional persistent homology is one of the major open problems in topological data analysis.
One can define r-dimensional persistent homology to be a functor from the poset category N^r, where N is the poset of natural numbers, to the category of modules over a commutative ring with identity. While 1-dimensional persistent homology is theoretically well-understood and has been successfully applied to many real-world problems, the theory of r-dimensional persistent homology is much harder, as it amounts to understanding representations of quivers of wild type.
In this talk I will introduce persistent homology, give some motivation for how it is related to the study of data, and present recent results related to the classification of multidimensional persistent homology.