Geometry and Analysis

Thin homotopy is an equivalence relation on paths which consist of two basic equivalences: reparametrizations and cancellation of retracings. This naturally arises in geometry when studying the invariances inherent in parallel transport on principal bundles. The path signature is the parallel transport map with respect to a “universal” translation-invariant connection on $R^n$, which has recently been used to develop the theory of rough paths. In fact, the path signature characterizes thin homotopy classes of paths.

In this talk, we discuss the generalization of this story to piecewise linear surfaces. We develop an algebraic model of piecewise linear paths and surfaces akin to the free group construction, and use this to study the relationship between thin homotopy of surfaces, and the notion of a “surface signature.” Based on joint work with Francis Bischoff.

NOTES: Online seminar, also streamed in RSLT11.
Coassociatives are four-dimensional calibrated submanifolds of $G_2$ manifolds, seven dimensional manifolds with holonomy $G_2$. There is especially rich geometry to be studied when two coassociatives intersect transversely inside the ambient $G_2$ manifold. Non-compact examples of this phenomenon involving the Harvey-Lawson $Sp(1)$ invariant coassociatives in $R^7$ have been studied by Lotay and Kapouleas, who use the $U(1)$ symmetry in these examples to show the transverse intersection can be resolved by gluing in a family of Lawlor necks. The topology of the resulting coassociative is that of a generalized connected sum: the connected sum of the original two coassociatives along their intersection circles. In this talk, I will report on progress towards a generalization of this gluing construction for compact coassociative submanifolds intersecting transversely in an arbitrary $G_2$ manifold. If time permits, I will also describe an invariant for certain types of such transverse coassociative pairs.

Gregory-Laflamme type instabilities seem to plague black holes in dimensions greater than 4. Gregory-Laflamme instabilities are exponentially growing solutions to the linearised Einstein vacuum equation and have been identified heuristically and numerically for black holes that have an event horizon that has one direction that is large compared to all others. In this talk, I will discuss a direct rigorous mathematical proof of the Gregory-Laflamme instability for the 5D Schwarzschild black string. The proof relies upon reducing the linearised vacuum Einstein equation to a Schrödinger equation to which direct variational methods can be applied.

The entropy of a continuous random variable behaves, in some ways, similarly to the logarithm of the volume of a set. In particular, the Entropy Power Inequality (EPI) is widely considered as an information-theoretic analogue of the Brunn-Minkowski inequality. In fact, a common proof of the two inequalities exists via a sharp form of Young's inequality.

After briefly reviewing this connection between convex geometry and information theory, we will present a new inequality for entropy, which improves the EPI under assumptions on the marginals, in the same way that Bonnesen's inequality improves the Brunn-Minkowski inequality under assumptions on the volume along some projection. We will characterize the equality case in the latter inequality. Furthermore, we will show how this inequality follows from a more general entropy inequality, which reduces to Bergström's inequality for determinants in the Gaussian case. We will also discuss a related inequality for the Fisher information.

This talk is based on joint work with Matthieu Fradelizi and Martin Rapaport.

We introduce an area formula for computing the spherical measure of an intrinsic graph of any codimension in an arbitrary homogeneous group. Our approach only assumes that the map generating the intrinsic graph is continuously intrinsically differentiable. The important novelty lies in the notion of Jacobian, which is built by the auxiliary Euclidean distance. The introduction of this Jacobian allows the spherical factor to appear in the area formula and enables explicit computations. This is joint work with Francesca Corni (University of Bologna).

The aim of the talk is to present recent developments of high frequency analysis for sub-elliptic operators and in sub-Riemannian geometry. I will start with discussing why these questions are closely related to many aspects of harmonic analysis. 

Magnetic monopoles were introduced by Dirac in 1931 to explain the quantisation of electric charges. In his model, they are singular solutions to an extension of Maxwell's equations allowing non-zero magnetic charges. An alternative model, developed by 't Hooft and Polyakov in the 1970s, is given, after a certain simplification, by smooth solutions to a different set of equations, the Bogomolny equations, whose moduli space of solutions has connected components Mk indexed by positive integers k (the "total magnetic charge"). These moduli spaces, which are non-compact manifolds, have an interpretation in terms of rational self-maps of CP1 due to Donaldson and their stable homotopy types may be described in terms of braid groups by a result of F. Cohen, R. Cohen, Mann and Milgram. A partial compactification of Mk has recently been constructed by Kottke and Singer, whose boundary strata may be called "ideal" or "asymptotic" monopole moduli spaces. I will describe joint work with Ulrike Tillmann in which we prove the existence of stability patterns in the homology of these spaces.

Is there a curvature identity that holds on any Riemannian manifold? Through the Chern-Gauss-Bonnet theorem, we can derive curvature identities that apply to 4-dimensional or 6-dimensional Riemannian manifolds. As an application of curvature identities, we prove Lichnerowicz’s conjecture in 4 dimensions under a slightly more general setting. Furthermore, we explore weakly Einstein manifolds, which arise as a generalization of 4-dimensional Einstein manifolds through the application of curvature identities. We also investigate the existence and non-existence of weakly Einstein metrics on certain Lie groups in recent studies, and propose a conjecture based on these results. (This is joint work with Y. Euh, S. Kim and Nikolayevsky.)

Nearly Kähler manifolds are Riemannian 6-manifolds admitting real Killing spinors. They are the cross-sections of Riemannian cones with holonomy G2. Like the Einstein equation, the nearly Kähler condition has a variational interpretation in terms of volume functionals, first introduced by Hitchin in 2001.

The existence problem for nearly Kähler manifolds is poorly understood, and the only currently known inhomogeneous examples were found in 2017 by Foscolo and Haskins using cohomogeneity one methods. For one of their examples, we establish non-trivial bounds on the coindex of the Hitchin-type and Einstein functionals. We do this by analysing the eigenvalue problem for the Laplacian on coclosed primitive (1,1)-forms under a cohomogeneity-one symmetry assumption.

Constant mean curvature (CMC) surfaces are special geometric variational objects, closely related to minimal surfaces. The key properties of a CMC surface are its area, mean curvature, genus, and index. The index of a CMC surface measures its stability: the index counts how many ways one can perturb the surface to decrease the area while keeping the enclosed volume constant. In this talk we discuss relationships between these key properties. In particular we present recent joint work with Ben Sharp, where we bound the index of CMC surfaces linearly from above by genus and the correct scale-invariant quantity involving mean curvature and area.