ICTS

Let $S$ be a very general sextic surface over complex numbers. Let $\mathcal{M}(H, c_2)$ be the moduli space of rank $2$ stable bundles on $S$ with fixed first Chern class $H$ and second Chern class $c_2$. In this talk we will introduce a new approach using Alexander-Hirschowitz Theorem to classify the obstructed bundles in $ \mathcal{M}(H, c_2)$. We will apply this classification to proof Mestrano -Simpson conjecture on number of irreducible components of $\mathcal{M}(H, 11)$.

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.

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.

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.

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.

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.

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.

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.
 

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.