Abstract: The question of whether a symplectic manifold embeds into another is central in Symplectic topology. Since the Gromov nonsqueezing theorem, it is known that this is a different problem from volume preserving embeddings. Embedded contact homology (ECH) has been shown to be very useful to obtain obstructions for symplectic embeddings in dimension 4. In this talk we will recall some results in dimension 4 due to Mcduff-Schlenk, Hutchings, Frenkel-Müller, Cristofaro-Gardiner, and Ramos. Furthermore, we will explain some recent results about embeddings into the disk cotangent bundle of the round sphere and of some spheres of revolution. This is a joint work with Vinicius Ramos and Alejandro Vicente.
Abstract: Closed manifolds with topology N = M x S^1 do not admit metrics of positive Ricci curvature by the theorem of Bonnet-Myers, while the the resolution of the Geroch conjecture implies that the torus T^n does not admit a metric of positive scalar curvature. In this talk we explain a non-existence result for metrics of positive m-intermediate curvature (a notion of curvature reducing to Ricci curvature for m = 1, and scalar curvature for m = n-1) on closed manifolds with topology N^n = M^{n-m} x T^m for n <= 7. Our proof uses minimization of weighted areas, the associated stability inequality, and delicate estimates on the second fundamental form. This is joint work with Simon Brendle and Sven Hirsch.
Abstract: Pleated surfaces are an important tool introduced by Thurston to study hyperbolic 3-manifolds and can be described as piecewise totally geodesic surfaces, bent along a geodesic lamination lambda. Bonahon generalized this notion to representations of surface groups in PSL_2(C) and described a holomorphic parametrization of the resulting open charts of the character variety in term of shear-bend cocycles.
In this talk I will discuss joint work with Martone, Mazzoli and Zhang, where we generalize this theory to representations in PSL_d(C). In particular, I will discuss the notion of d-pleated surfaces, and their holomorphic parametrization.
Abstract: Given a flow on a manifold, an old question is whether it is possible to perturb it in order to create a periodic orbit passing through a given region. Statements of this kind are called "closing lemmas". An interesting relation between this problem and pseudo-holomorphic curves was found by Irie in dimension 3. I will discuss such a relation in general dimensions, including a proof of a conjecture of Irie regarding certain flows on the boundaries of ellipsoids. All symplectic preliminaries will be explained. This is a joint work with Julian Chaidez, Ipsita Datta and Rohil Prasad.
Abstract: In 1950 Kuiper classified complete affine structures on 2-tori. Later Baues showed that the deformation space (the analog of Teichmuller space), is homeomorphic to R^2, and in particular is Hausdorff. The mapping group action is the usual LINEAR action of GL(2,Z) on R^2 whose quotient (the analog of the Riemann moduli space) is thus an intractable non-Hausdorff mess. Last year Deligne observed the deformation space naturally identifies with the cone on the twisted cubic curve in real projective $3$-space. In my talk I will describe several perspectives on this striking result.
Abstract: de Rham regularization is a tool to find an isomorphism between the smooth de Rham cohomology of a smooth manifold and the de Rham cohomology using forms with coefficients in some Banach space of functions of finite regularity. Flat Wieler solenoids are inverse limit spaces with the local product structure of a Euclidean space and a Cantor set. In this talk I will discuss how one can obtain a regularization-type of result for these spaces and how understanding transversal Hölder regularity is the key to make this work. Time permitting, I will mention some applications.