The objective of this lecture series is to discuss the techniques and results that can be used to explore the classification of objects such as vector bundles in algebraic geometry and beyond. In the first lecture, we'll discuss some basic examples and constructions with a view towards some of the main topics to come.
Lecture 2: The Bogomolov-Gieseker inequality and geography of moduli of vector bundles - 2 PM, 11 February 2020
We'll start by introducing some of the main concepts in the moduli theory of stable vector bundles. Over higher dimensional varieties, the Bogomolov-Gieseker inequality constrains strongly the existence and properties of moduli spaces of stable vector bundles. A recent theme is exploration of the moduli spaces for intermediate values of c_2.
Lecture 3: Moduli of local systems: the Dolbeault approach - 2 PM, 12 February 2020
The moduli spaces of local systems over compact Riemann surfaces are loaded with structure given by Hitchin's equations relating them to moduli spaces of Higgs bundles. Recent topics include the Donagi-Pantev approach to the Geometric Langlands correspondence. Over higher-dimensional varieties, the classification of local systems remains mysterious.
Lecture 4: Study of the nonabelian Hodge correspondence at infinity - 2 PM, 13 February 2020
An important recent theme in the study of local systems is the structure of the noncompact moduli spaces, and their various correspondences, at infinity. We'll discuss some of the geometry and topology, the geometric P=W conjecture, and the relationship with harmonic mappings. This leads to the relationship with Gaiotto-Moore-Neitzke spectral networks and eventually to Fukaya categories and stabiity conditions.
Lecture 5: The potential of AI, illustrated in the classification of finite algebraic structures - 2 PM, 14 February 2020
The classification of finite associative operations, for example finite categories, is a topic that shares a great formal ressemblance with the moduli problem in algebraic geometry. The combinatorial nature of the question makes it amenable to experiments on the utilisation of AI to go farther in the search for complicated structures. We'll discuss some current attempts in this direction.
This lecture series is part of the program Moduli of bundles and related structures.