It is well-known that on a non-projective complex manifold, a
coherent sheaf may not have a resolution by a complex of holomorphic vector
bundles. Nevertheless, J. Block showed that such resolution always exists if
we allow anti-holomorphic flat superconnections which generalize complexes
of holomorphic vector bundles. Block's result makes it possible to study
coherent sheaves with differential geometric and analytic tools. For
example, in a joint work with J.M Bismut, S, Shen, and I, we give an
analytic proof of the Grothendieck-Riemann-Roch theorem for coherent sheaves
on complex manifolds. In this talk I will present the ideas and applications
of anti-holomorphic flat superconnections. I will also talk about analogous
constructions of superconnections in other areas of geometry.

There is a generalization of Lie theory from Lie algebras to differential graded Lie algebras. Ordinary Lie theory involves first order ordinary differential equations. Higher Lie theory may be understood as a non-linear generalization of the de Rham theorem on simplicial complexes (in Dupont's formulation), as against graphs. In this talk, we present an alternate approach to this theory, using the more elementary de Rham theorem on cubical complexes.

Along the way, we will need an interesting relationship between cubical and simplicial complexes, which has recently become better known due to its use in Lurie's theory of straightening for infinity categories.

Gravitational instantons are defined as non-compact hyperKahler
4-manifolds with L^2 curvature decay. They are all bubbling limits of K3
surfaces and thus serve as stepping stones for understanding the K3 metrics.
In this talk, we will focus on a special kind of them called
ALH*-gravitational instantons. We will explain the Torelli theorem, describe
their moduli spaces and some partial compactifications of the moduli spaces.
This talk is based on joint works with T. Collins, A. Jacob, R. Takahashi,
X. Zhu and S. Soundararajan.

Earlier time and joint seminar with Differential Geometry Seminar.

In this talk, we will show how six functors formalisms (which are central to algebraic geometry) may be used to define derived motivic measures (maps from the K-theory of varieties to other spectra).  In particular, we will use this to construct a derived motivic measurement which lifts the Gillet-Soulé motivic measure.  This addresses a conjecture of Campbell-Wolfson-Zakharevich (in fact, there are potentially several derived lifts).

The Borromean rings are three interlinked circles such that no two circles are linked: if we cut or remove one of the circles, the other two fall apart. Massey products are an algebraic manifestation of this phenomenon. Born as part of Algebraic Topology, they have now made a surprising appearance in Number Theory and Galois Cohomology. The Massey Vanishing Conjecture of Minac and Tan predicts that all Masseyproducts in the Galois cohomology of a field vanish as soon as they are defined. In this talk, I will give an informal introduction to Massey products in Topology and Galois Theory, and then describe recent progress on the Massey Vanishing Conjecture, joint with Alexander Merkurjev