We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders.
In this talk, I will present a geometric algorithm for determining whether a given set of elements in SO+(n,1) generates a discrete subgroup, as well as identifying the relators for the corresponding group presentation. The algorithm constructs certain hyperbolic manifolds that are always complete, a key condition for applying Poincaré Fundamental Polyhedron Theorem and ensuring the algorithm is valid.
Given a Legendrian link L in the contact 3-sphere, one can hope to classify the set of exact Lagrangian fillings of L, i.e. exact Lagrangian surfaces in the symplectic 4-ball with boundary equal to L. Much of the recent progress towards this classification relies on establishing a connection between sheaf-theoretic invariants of Legendrians and cluster algebras. In this talk, I will describe this connection and how these invariants behave with respect to certain symmetries of Legendrian links and their fillings. Parts of this are joint work with Agniva Roy.
In a recent note Francesco Lin showed that if a rational homology sphere Y admits a taut foliation then the Heegaard Floer module HF^-(Y) contains a copy of F[U]/U as a summand. This implies that either the L-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. In a recent paper in collaboration with Fraser Binns we verified that Lin's geography restriction holds for a wide class of rational homology spheres.
In this talk, I will present a complete coarse classification of strongly exceptional Legendrian realizations of the connected sum of two Hopf links in contact 3-spheres. This is joint work with Sinem Onaran.
An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which imitates and generalizes the behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation.
A codimension-1 submanifold embedded in a symplectic manifold is called “contact type” if it satisfies a certain convexity condition with respect to the symplectic structure. Given a symplectic manifold X it is natural to ask which manifolds Y can arise as contact type hypersurfaces. We consider this question in dimension 4, which appears much more constrained than higher dimensions; in particular we review evidence that no homology 3-sphere can arise as a contact type hypersurface in R^4 except the 3-sphere.
A basic question for any knot invariant asks which knots the invariant detects. For example, it is famously open whether the Jones polynomial detects the unknot. I'll focus in this talk on the detection question for knot invariants coming from Floer theory and the Khovanov--Rozansky link homology theories. I'll survey the progress made over the past twenty years, and will describe some of the topological ideas that go into my recent work with Sivek on these questions.
The uniformization theorem states that every Riemann surface is a quotient of some subset of the complex projective line by a group of Mobius transformations. However, a number of closely related questions regarding the structure of uniformization maps remain open. For example, it is unclear how one might associate a uniformizing map to a given Riemann surface. In this talk we will discuss an approach to this question due to Gunning by attaching a projective line bundle to a Riemann surface and studying its analytic properties.
