K3 surfaces are compact complex surfaces whose canonical bundle and first cohomology group are trivial. In this talk we will see how these properties allow for a much easier analysis than with more general complex surfaces, especially due to how they simplify Serre Duality and the Hirzebruch-Riemann-Roch theorem. These will allow us to completely describe the Hodge diamond, as well as giving some general facts about divisors on K3 surfaces.
Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on the preprint arXiv:2301.13227.
Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on the preprint arXiv:2301.13227.
Abstract (Levi Lorenzo): K-theory is the generalized cohomology theory that has proven incredibly effective at classifying C*- algebras. In this talk, we discuss its dual homology theory, K-homology. K-homology and K-theory admit a bilinear pairing to the integers, called the index pairing, which allows one to learn about K-theory using K-homology and vice versa. In general, this pairing is very difficult to compute; however, if a C*algebra admits a dense subalgebra on which its K-homology is finitely summable, Connes has equipped us with a more computable formula. Given a smooth manifold, M, of dimension n, the K-homology of is finitely summable on for In this way, the summability also sees the dimension of the manifold. It is natural to ask whether similar results hold in the noncommutative setting. Towards this end, Connes showed that the irrational rotation algebras have uniformly summable K-homology. The irrational rotation algebras are examples of crossed product algebras arising from minimal actions on a topological space. In my work, I study other noncommutative algebras and ask the same question. For example, we make the space a Cantor Set and examine the K-homology of crossed products arising from minimal actions on it. In this talk, we discuss progress in such directions.
Abstract (Maggie Reardon): Matui’s HK-conjecture proposes a relationship between the homology groups of a nice enough groupoid and the -theory of the associated reduced -algebra. The conjecture is not true in general and there are a number of counterexamples. However, the conjecture holds for certain classes of groupoids including AF groupoids.
Putnam introduced a new class of groupoids in the paper titled “Some classifiable groupoid -algebras with prescribed -theory”. These new groupoids are related to AF groupoids and this prompts a natural question: does this new class of groupoids satisfy the HK-conjecture?
Finitely Summable K-homology and the HK-conjecture for certain groupoids constructed by Putnam
Introduction to K3 surfaces