Abstract: Gromov used convex integration to prove that any closed orientable three-manifold equipped with a volume form admits three divergence-free vector fields which are linearly independent at every point. We provide an alternative proof of this using geometric properties of eigenspinors in three dimensions. In fact, our proof shows that for any Riemannian metric, one can find three divergence-free vector fields such that at every point they are orthogonal and have the same non-zero length.
Abstract: The space of quasi-Fuchsian surface group representations into PSL(2,C) has been largely studied in recent years, also due to the direct relation with hyperbolic geometry. In this framework, Bers' simultaneous uniformization theorem shows that it is parameterized by two copies of Teichmüller space of the closed surface. In this seminar, we are interested in surface group representations into SL(3,C) acting naturally on a homogeneous space called the bi-complex hyperbolic space. After briefly describing its main features, we will define minimal Lagrangian surface immersions and their structural equations. If they are equivariant under the action of a representation into SL(3,C), we will see that their embedding data provide a parameterization of the space of SL(3,C)-quasi-Fuchsian representations by two copies of the bundle of holomorphic cubic differentials over Teichmüller space. This is a joint work with Andrea Tamburelli.
Abstract: Punctured holomorphic curves have been a key tool in symplectic topology and dynamics since their introduction by Hofer in 1993. I’ll describe a new compactness theorem for sequences of punctured holomorphic curves with actions converging to 0 and with bounded topology, without the traditional assumption of bounded Hofer energy. In the limit, one obtains a family of compact invariant subsets of the underlying Hamiltonian flow. Such sequences turn out to be abundant in low-dimensional symplectic dynamics. Dynamical applications include (1) a generalization of the Le Calvez-Yoccoz theorem to higher-genus surfaces and three-manifolds and (2) a generalization of a recent theorem by Fish-Hofer on four-dimensional autonomous Hamiltonian flows. This talk is based on arXiv preprints 2401.14445 (joint with Dan Cristofaro-Gardiner) and 2405.01106.
Abstract: Khovanov homology is a refinement of the Jones polynomial of a knot. Recently, there have been a number of exciting applications of Khovanov homology to 4-dimensional topology. In this talk, we will use an indirect approach to re-prove one of these results, that Khovanov homology distinguishes some pairs of disks in the 4-ball. Our proof uses a relationship between the Khovanov homology of a strongly invertible knot and its quotient, coming from a stable homotopy refinement of Khovanov homology. This is joint work with Sucharit Sarkar. Most of the talk should be fairly broadly accessible; in particular, it will not assume any knowledge of Khovanov homology.
Abstract: The cosmetic surgery conjecture predicts that for a non-trivial knot in the three-sphere, performing two Dehn surgeries with different surgery coefficients r and r' results in distinct oriented three-manifolds. Earlier works on the conjecture implies that one needs to consider only the case r'=-r and r=2 or 1/n for some integer n. In this talk, I'll explain how one can remove the case of r=1/n, reducing the conjecture to the case r=2. Furthermore, in the case r=2, the Alxander polynomial of the knot is trivial. The key tools in the proof are a persistence module associated to any integer homology sphere using instanton gauge theory and an exact triangle relating instanton Floer homology groups of 1/n-surgeries on a knot. This talk is based on a joint work with Lidman and Miller Eismeier.
Abstract: The Heegaard Floer homology of a 3-manifold with torus boundary can be understood very concretely in terms of curves in the punctured torus. A manifold with many torus boundary components gives us an object in the Fukaya category of a product of punctured tori. Such objects can be complicated, but in many cases they can be profitably understood via the induced functor on the Fukaya category of the torus.
Abstract: Seiberg-Witten theory has an analogue for 3- and 4-manifolds with involutions called real Seiberg-Witten theory. This theory can be used to construct invariants of links and embedded surfaces by passing to double branched covers. This talk will focus on a framed version of real Seiberg-Witten-Floer homology. It turns out this invariant of links has rather surprising properties not seen in ordinary Seiberg-Witten theory. I will explain why it is special and how it is related to some recent developments.
Abstract: Given a $3$-dimensional manifold $X$, and a positive real number $k$, we define a $k$-surface in $X$ to be an immersed surface of constant extrinsic curvature equal to $k$ which is complete with respect to the sum of its first and third fundamental forms. The theory of such surfaces was revolutionized by François Labourie in a series of papers from the late 80's to the early 2000's which revealed how Gromov's theory of pseudo-holomorphic curves may be fruitfully applied to their study. In this talk, I will discuss my own recent contributions.
Abstract: The problem of classifying smooth, closed, highly-connected n-manifolds (for n \geq 5) up to diffeomorphism has a long history in the study of geometric topology—in the 1950s, Milnor's study of this problem led to his discovery of exotic spheres. In this talk, I will survey this classical problem and discuss its solution, which builds on the work of many people, including Wall and Stolz. This solution, obtained in joint works with Burklund, Hahn and Zhang, relies on modern techniques in the higher algebra of E_\infty-rings. Time permitting, I will also explain how to generalize this classification from highly-connected manifolds to the wider class of metastably-connected manifolds.