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.
Heegaard Floer homology is a package of invariants for 3 manifolds introduced by Ozsváth and Szabó, which is a symplectic alternative to more gauge theoretic invariants such as monopole Floer homology. A variation of this theory, called knot Floer homology, defines an invariant for knots in 3-manifolds. It was developed independently by Ozsváth and Szabó, and by Rasmussen. In this talk, we will outline the construction, some properties and applications of these invariants. If time permits, I will discuss my recent project to compute the knot Floer homology for a large class of satellite knots. This is joint work with Ian Zemke and Hugo Zhou.
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.
When transitioning from studying Euclidean space to more Riemannian manifolds, one must first unlearn many special properties of the flat world. The same is true in physics: while one can make sense of classical physics on an arbitrary curved background space, many seemingly foundational concepts (like the center of mass) turn out to have no place in the general theory. Freed from the constraints such properties induce, classical physics on a curved background space has many surprises in store. In this talk I will share some stories related to joint work with Brian Day and Sabetta Matsumoto on understanding and simulating such situations, focusing on hyperbolic space when convenient. To give a taste, here are two such surprises:
(1) there is no Galilean relativity: inside a sealed box in hyperbolic geometry it is possible to perform an experiment which detects your precise velocity. And (2): it's possible to ‘swim’ in the vacuum in hyperbolic space - to move your arms and legs in a specific pattern that causes you to translate along a geodesic with no external forces. The arguments for the former are readily accessible to beginning graduate students in geometry, and the latter illustrates a use of gauge theory in classical mechanics, following work of Wilczek and Montgomery.
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.
The study of real planar curves dates back to antiquity, where the ancient Greeks studied curves defined on the plane cut out by polynomials of two variables. We’ll provide a friendly overview to beautiful formulas of Plücker which govern the “shape” of planar curves. We will discuss the Shapiro—Shapiro conjecture and connections to the real Schubert calculus, and end by presenting some new conjectures and computational evidence joint with Frank Sottile.