Lecture 1: Representations of Fuchsian groups, parahoric group schemes.
Lecture 2: Parahoric torsors, parabolic bundles and applications.
The mini course will be on representations of Fuchsian groups, parahoric group schemes and Parabolic bundles on Riemann Surfaces. The course will introduce parahoric groups and Bruhat-Tits group schemes and torsors and relate them to representations of Fuchsian groups. I will close the course with new developments involving these themes towards the study of torsors on stable curves.
Higgs bundles and higher Teichmüller components
In these lectures, we first briefly review the non-abelian Hodge correspondence between the moduli space of G-Higgs bundles over a compact Riemann surface and the moduli space of representations of the fundamental group of the surface in G. We then introduce the Hitchin components, corresponding to the case when G is a split real group, the maximal Toledo components, corresponding to G being a real group of Hermitian type, and certain special components when G=SO(p,q). These are all examples of higher Teichmüller components. As for the classical Teichmüller space (a component of the moduli space for G=PSL(2,R)), such components consist entirely of discrete and faithful representations. We will finish describing a general construction that is believed to produce all the possible Teichmüller components (including some existing for certain exceptional real forms) in terms of sl_2-triples and the Cayley transform.
Lecture 1: Beyond geometric invariant theory 1: Harder-Narasimhan theory.
I will describe a recent approach to studying the structure of moduli problems using the language of algebraic stacks. The idea is that a moduli stack which does not have a moduli space might nevertheless admit a Theta-stratification, a structure analogous to the Harder-Narasimhan stratification of the moduli of principal G-bundles on a curve. Such a stratification can be specified by a “numerical invariant,” generalizing the Hilbert-Mumford numerical invariant from GIT. I will present necessary and sufficient conditions for a numerical invariant to define a Theta-stratification. One example where things work nicely, developed jointly with Eduardo Gonzalez and Pablo Solis, is the moduli of “gauged maps” from a curve to a projective G-variety.
Lecture 2: Beyond geometric invariant theory 2: Good moduli spaces, and applications.
One of the main results of geometric invariant theory is the construction of a moduli space of semistable orbits for a reductive group G on a projective variety. In modern terms, this can be understood as constructing a “good moduli space” for the stack obtained as the quotient of the locus of semistable points by G. For a general algebraic stack with a numerical invariant, one can define the locus of semistable points and ask if this admits a good moduli space. I will present necessary and sufficient conditions for an algebraic stack to admit a good moduli space, developed jointly with Jarod Alper and Jochen Heinloth. I will also describe conditions on a numerical invariant which guarantee that the semistable locus admits a good moduli space. Time permitting, I will discuss some recent applications this has had to studying wall-crossing of moduli spaces of Bridgeland-semistable complexes on a K3-surface.
The Hodge conjecture for moduli spaces of stable sheaves over a nodal curve
The Hodge conjecture is known for the Jacobian variety of a general, smooth, projective curve. Balaji-King-Newstead used this to prove the conjecture for the moduli space of rank 2 stable sheaves with fixed odd degree determinant over a general, smooth, projective curve of genus g ≥ 2. In this talk I will discuss an analogous result when the underlying curve is general, irreducible nodal and show why techniques from the smooth case fail. This is joint work with A. Dan.
Poisson and symplectic geometry of the moduli spaces of Higgs bundles
I will talk about some natural Poisson and symplectic proprties of the moduli spaces of Higgs bundles when the sum extra structure, such as a framing, is added. This is an overview of various works with I. Biswas, J. Martens, A. Peón-Nieto and S. Szabó.
Non commutative K3 surfaces, with application to Hyperkäler and Fano manifolds
The aim of this minicourse is to explain the relation between polarized Hyperkähler manifolds, non-commutative K3 surfaces, and certain Fano manifolds.
In the first talk, I will introduce non-commutative K3 surfaces and moduli spaces of objects in them. The second talk will be centered on examples, including non-commutative K3 surfaces associated to cubic fourfolds and Gushel-Mukai manifolds; finally, we will discuss possible ways to associate a non-commutative K3 surface to a polarized Hyperkähler fourfold, and possible applications to Chow groups.
Classification of obstructed bundles over a very general sextic surface and Mestrano-Simpson Conjecture.
Let S be a very general sextic surface over complex numbers. Let M(H,c2) be the moduli space of rank 2 stable bundles on S with fixed first Chern class H and second Chern class c2. In this talk we will introduce a new approach using Alexander-Hirschowitz Theorem to classify the obstructed bundles in M(H,c2). We will apply this classification to proof Mestrano -Simpson conjecture on number of irreducible components of M(H,11).
Lecture series title: "Exploring Moduli"
Lecture 1: Exploring Moduli: basic constructions and examples
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
We'll start by introducing some of the main concepts in the moduli theory of stable vector bundles. Over higher dimensional varieties, the Bogomolov-Giesekerinequality 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
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
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
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.