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.
Abstract: In this talk we will study the existence of contractible positive loops in groups of contactomorphisms. In particular, we will discuss the connection between such loops and the contact Hofer norm. This leads to an implicit criterion for the existence of positive loops in terms of open book decompositions and to new examples of non-orderable contact manifolds. The talk is based on joint work with Egor Shelukhin.
Abstract: This is a joint work with Jonathan Rosenberg.Let $(M,L)$ be a non-spin spin$^c$ manifold. A metric $g$ on $M$ and a connection $A$ on $L$ determine the associated spin$^c$ Dirac operator $D_L$ on $M$. The Lichnerowicz-Schr\"odinger formula takes the form $$ D_L^2 =\nabla^*\nabla+\frac{1}{4}R^{tw}_{(g,A)}, $$ where $R^{tw}_{(g,A)}=R_g+ 2ic(\Omega)$ is the zeroth order operator (which is called "twisted scalar curvature"). Here $R_g$ is the scalar curvature of $g$ and $2ic(\Omega)$ comes from the curvature $2$-form $\Omega$ of the connection $A$.
There is an affirmative existence result: a closed non-spin simply-connected spin$^c$-manifold $(M,L)$Â of dimension $n\geq 5$ admits a pair $(g,A)$ such that $R^{tw}_{(g,A)}$ is greater than $0$ if and only if the index $\alpha^c(M,L):=\textrm{ind} D_L$ vanishes in the complex K-theory $KU_n$.
It turns out there is also a scalar-valued "generalized scalar curvature" $R^{gen}_{(g,A)}$ with the property that the positivity condition of the operator $R^{tw}_{(g,A)}$ is equivalent to the positivity of the scalar function $R^{gen}_{(g,A)}$. I will present several interesting results concerning the curvature $R^{gen}_{(g,A)}$, in particular, on the topology of the relevant space of metrics/connections when this curvature is positive.
Abstract: One creates a fibered 3-manifold by thickening a surface by the interval and gluing its ends by a surface homeomorphism. In the finite-type setting, much is known about how the topological data of the gluing homeomorphism determine geometric information about the hyperbolic 3-manifold. Currently, there is a lot of research activity surrounding end-periodic homeomorphisms of infinite-type surfaces.
As an "infinite type" analogue to work of Minsky in the finite-type setting, we show that given a subsurface Y of S, the subsurface projections between the "positive" and "negative" Handel-Miller laminations provide bounds for the geodesic length of the boundary of Y as it resides in the end-periodic mapping torus.
In this talk, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic 3-manifolds, show these techniques may be used in the infinite-type setting, and how our main theorems return results to the closed, fibered setting.
Abstract: We will discuss two recent results on positive scalar curvature on circle bundles. In the context of trivial circle bundles over 4-manifolds, we consider two problems: Gromov's width inequality conjecture, and Rosenberg's S^1-stability conjecture. Both conjectures have counterexamples in dimension 4 based on Seiberg-Witten invariants. Nevertheless, we show that in the simply connected case both of these results are true upon considering 4-manifolds up to homeomorphism. We also obtain a result up to stabilization in the non-simply connected case. In the context of nontrivial bundles, we construct infinitely many examples of PSC circle bundles over enlargeable manifolds in all dimensions greater than 3. This answers a question of Gromov. We shall emphasize some analogies between symplectic geometry and positive scalar curvature that we encountered in the process of finding these results. Based on joint work with B. Sen.
Abstract:Â In 2021, Abouzaid, McLean and Smith achieved a breakthrough by constructing a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps in genus zero. In this talk, I will describe a generalisation of their construction for moduli spaces of stable maps of arbitrary genus and how this allows for a straightforward definition of Gromov-Witten invariants. Subsequently, I will outline a proof of a formula for the Gromov-Witten invariants of a product manifold in terms of the invariants of its factors. This is joint work with Mohan Swaminathan.
Abstract:Â The moduli space of holomorphic rank 2 bundles of odd degree and fixed determinant over a given Riemann surface is a symplectic manifold which has an interpretation as a certain PU(2) character variety. There is a homomorphism from a finite extension of the mapping class group of the surface to the symplectic mapping class group of this moduli space. When the genus is 2 or more, we prove that this homomorphism is injective. The proof uses Floer's instanton homology for 3-manifolds. This is joint work with Ali Daemi.
Abstract:Â The search for exotic manifolds - pairs or families of smooth manifolds that are homeomorphic but not diffeomorphic - has been a motivating theme in topology for decades, particularly in dimension 4. In this talk, I will discuss new methods for using Heegaard Floer homology to detect exotic, closed 4-manifolds with fundamental group Z or Z/2. Among other things, we provide the first known examples of (non-simply-connected) exotic 4-manifolds with negative definite intersection form, and the first known examples of exotic manifolds produced by Fintushel-Stern knot surgery using knots with trivial Alexander polynomial. This is joint work with Tye Lidman and Lisa Piccirillo.Â