Geometry and Analysis Seminar

A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving half a million hyperbolic 3-manifolds, including new or at least improved techniques for computing each of these properties.
 

We give new obstructions to the existence of planar open books on contact structures, in terms of the homology of their fillings. I will talk about applications to links of surface singularities, Seifert fibred spaces, and integer homology spheres. No prior knowledge of contact or symplectic topology will be assumed. This is joint work with Paolo Ghiggini and Olga Plamenevskaya.
 

A Morse function (and more generally a Morse-Bott function) on a compact manifold M has associated Morse inequalities. The aim of this
talk is to explain how we can associate Morse inequalities to any smooth function on M (reporting on work of/with G Penington).

A generic surface homeomorphism (up to isotopy) is what we call it pseudo-Anosov. These maps come equipped with an algebraic integer that measures
how much the map stretches/shrinks in different direction, called the stretch factor. Given a surface homeomorsphism, one can ask if it is the lift (by a branched or unbranched cover) of another homeomorphism on a simpler surface possibly of small genus. Farb conjectured that if the algebraic degree of the stretch factor is bounded above, then the map can be obtained by lifting another homeomorphism on a surface of bounded genus.
This was known to be true for quadratic algebraic integers by a Theorem of Franks-Rykken. We construct counterexamples to Farb's conjecture.

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature k<=0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k < 0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

While many cubic fourfolds are known to be rational, it is expected that the very general cubic fourfold is irrational (although none have been
proven to be so). There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett)
or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a `noncommutative K3 surface' to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is `geometric', i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to  Batyrev-Borisov). In contrast to the algebraic side of the story, the mirror is always `geometric': i.e., it is always just an honest K3 surface equipped with an appropriate Kähler form. After explaining this background, I will state a theorem: homological mirror symmetry holds in this context (joint work with Ivan Smith).

G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as 7-dimensional analogues of Calabi-Yau 3-folds.
In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds. The metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in Calabi-Yau geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.

Associated to a finite graph without loops is the Kac-Moody Lie algebra for the Cartan matrix whose off diagonal entries are (minus) the adjacency matrix for the graph.  Two famous conjectures of Kac, proved by Hausel, Letellier and Villegas, hint that there may be some larger cohomologically graded algebra associated to the graph (even if there are loops), providing "higher" Kac moody Lie algebras, or at least their positive halves.  Using work with Sven Meinhardt, I will give a geometric construction of the (full) Kac-Moody algebra for a general finite graph, using cohomological DT theory.  Along the way we'll see a proof of the positivity conjecture for the modified Kac polynomials of Bozec, Schiffmann and Vasserot counting various types of representations of quivers.

In this talk two different methods for constructing complete steady and expanding Ricci solitons of cohomogeneity one will be discussed. The first is based on an estimate on the growth of the soliton potential and holds for large classes of cohomogeneity one manifolds. The second approach is specific to the two summands case and uses a Lyapunov function. This method also carries over to the Einstein case and as an application, a simplified construction of B\"ohm's Einstein metrics of positive scalar curvature on spheres will be explained.

The theory of connections on curves and Hitchin systems is something like a “global theory of Lie groups”, where one works over a Riemann surface rather than just at a point. We’ll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of “Dynkin diagrams” as a step towards classifying some examples of such objects.