A very useful strategy in studying topological manifolds is to factor them into “smaller" pieces. An open book decomposition of an n-manifold (the open book) is a fibration that helps us study our manifold in terms of its (n-1)-dimensional fibers (the pages) and (n-2)-dimensional boundary of these submanifolds (the binding). Open books provide a natural framework for studying topological properties of certain geometric structures on smooth manifolds such as "contact structures". Thanks to open books, contact manifolds, odd dimensional smooth manifolds carrying these geometric structures, can be studied from an entirely topological viewpoint. For example, every contact 3-manifold can be presented as an open book whose pages are surfaces and binding is a knot/link. In this talk, we will talk about higher-dimensional contact manifolds and provide a setting where we study these manifolds in terms of 3D open books. We also present various results along with examples concerning geometric and topological aspects of contact and symplectic manifolds along with upcoming work concerning these special fibrations.

Sheaves have long been classical tools for studying the topology of manifolds. Symplectic geometry, which encodes topological information about a manifold via its cotangent bundle, has revealed a profound connection to sheaf theory through the microlocal framework developed by Kashiwara and Schapira. Remarkably, many important symplectic invariants can now be computed using sheaves. In this talk, I will survey several well-known applications of sheaf theory in symplectic geometry and also consider the reverse perspective: how symplectic geometry provides constructions and insights that deepen our understanding of sheaf theory. This latter viewpoint is central to obtaining a global version of the microlocal Riemann-Hilbert correspondence in joint work with Côté, Nadler, and Shende.

Vertex operator algebras (VOAs) and their modules define sheaves of conformal blocks over the moduli space of stable curves, generalizing sheaves of conformal blocks attached to Lie algebras. In this talk I will discuss how these sheaves are constructed and which properties they satisfy. I will in particular describe conditions that guarantee that these sheaves are actually vector bundles of finite rank and related open questions. This is based on a joint work with A. Gibney, N. Tarasca and D. Krashen.

I will explain something of the theory of homogeneous Einstein metrics and why certain generalizations of this equation occur naturally in the study of homogeneous spaces.

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.