Beyond Part III

Young Researchers in Mathematics 2009

16-18 April 2009

Centre for Mathematical Sciences, Cambridge

Differential Geometry/Topology Session

Talks will be in MR13, except for Thursday between 4.30 and 6 (Pavillion E committee room).

Thursday

Friday

Riemannian holonomy groups and calibrated geometry - Dominic Joyce (keynote speaker)

Differential geometry is the study of manifolds -- multidimensional smooth spaces, often with nontrivial topology -- and of geometric structures upon them. One of the most basic kinds of geometric structure is a Riemannian metric, which tells you the lengths of smooth paths in the manifold. I will explain the definition of the holonomy group of a Riemannian metric, and Berger's classification of holonomy groups. The holonomy group measures what extra geometric structures are compatible with the Riemannian metric. So, Berger's classification is a list of the most interesting geometric structures in Riemannian geometry. They include Kahler manifolds, hyperkahler manifolds, Calabi-Yau manifolds, G2 and Spin(7) manifolds. All these are of interest in String Theory in theoretical physics.

In the last part of the talk, I will discuss calibrated submanifolds, which are special classes of submanifolds with minimal volume in Riemannian manifolds with special holonomy.

The Cobordism Category of Complex Annuli and Its Variations - George Raptis

I will present a result about the homotopy type of the cobordism categories of topological and complex annuli (that were introduced by Costello to describe the passage from the moduli space of curves to the moduli space of stable curves). I will also discuss how the arguments apply to other cobordism categories of surfaces and relate to various stable transfer maps.

Sub-Riemannian geometry - Jan Cristina

Suppose we are in some sort of topological space, where we can reasonably assign a notion of length to paths. If we consider some subfamily of the space of continuous paths, the members of which are locally rectifiable with respect to our notion of length, we can define a new (infinity-pseudo-)metric (and possibly new topology) on our space by considering the infimum of lengths of paths between points. As a simple non-trivial example of this, we can consider the set of continuously differentiable Legendrian curves in a contact 3-manifold, with a metric defined by the contact form and an almost complex structure. It is well known that any two points on a connected contact manifold can be joined by such a curve, and that this gives a true metric structure on the space. This metric gives a wholly different geometry to the object in question, as can be seen by calculating the Hausdorff-dimension of our 3 dimensional contact manifold which is 4.

A Sub-Riemannian geometry is a smooth manifold with whose curve family is given by a distribution of hyperplanes at each point specifying the directions that curves are allowed to take. Surprisingly they retain many useful analytical properties which I hope to discuss in my talk.

(Pseudo-) Nearly Kähler Geometry - Felix Ketelaar

Nearly Kähler geometry is a special type of non-integrable almost Hermitian geometries which is characterized by $\nabla J$ being a (non-zero) three-form. In the Riemannian case this condition turns out to be rather rigid, so for example up to Kählerian factors any complete Nearly Kähler manifold is compact and Einstein. However, no full classification of complete simply connected strict Nearly Kähler manifolds has been achieved so far. The first part of the talk will give a review-style account on some established results in this area including relations to Spin geometry and manifolds with holonomy $G_2$. In the pseudo-Riemannian case, most results from the Riemannian case do not hold any longer as we lack of de Rham's splitting theorem as well as Myers' compactness theorem. Although this should in principle improve the prospects of finding pseudo-Nearly Kähler structures, only few examples are known and the only case which is well-understood so far is the flat one. The second part of the talk will mainly deal with an attempt to construct new examples of pseudo-Nearly Kähler structures on tangent bundles of certain classes of manifolds.

This talk will be largely accessible to anyone who has taken a Part-III-course in Differential Geometry or has equivalent preknowledge.

Can one hear the shape of a broken drum? (accessible talk) - Peter Herbrich

Marc Kac's celebrated article "Can one hear the shape of a drum" marks the starting point of Inverse Spectral Geometry which has since become an independent branch of Differential Geometry. If a drum is viewed as a planar domain with Dirichlet boundary, then the eigenfrequencies of the drum correspond to the eigenvalues of the Laplace-Beltrami operator on the domain. In other words, Kac asked whether these eigenvalues already encode the entire geometry of the domain. Since the question was raised in 1966, various non-smooth counterexamples have been constructed using a technique called transplantation of eigenfunctions. The talk will focus on its extension to mixed boundary conditions which may be interpreted as drums with a partly unattached membrane. The systematic search which involved graph theory and the usage of computer algebra software has confirmed several topological and geometrical properties that are not spectrally determined such as the fundamental group and orientability. The illustrations and animations should make the talk both understandable and enjoyable to students at a graduate level.

A nice extension property - Fiontann Roukema

In the framework of long knots, starting with an extension of a finite type invariant to a finite type function on virtual knot diagrams, we conclude Goussarov’s theorem about the existence of Gauss diagram formulas, and explain, in elementary terms, why this is interesting. We look at natural choices for such extensions, learning something from each attempt, and with the correct choice we ask the question of whether we have virtual invariance? The talk is not high-tech, and the few definitions necessary will be presented. All arguments, definitions, and algorithms are expressed visually, which should make the talk accessible to anyone who can understand what a Z-module of formal linear combinations of objects is, and what my drawings are supposed to represent!

Polar and Legendre Duality in Geometric Flows - Edward White

In recent years geometric flows have emerged as powerful tools for studying geometric inequalities for convex bodies. The theory of polar and Legendre duality for convex bodies makes it possible, through the use of support and gauge functions, to relate convex bodies to their polar duals. We explore this theory, and show that its key results even if the boundary of a convex body is not smooth. When applied to the study of geometric evolution equations on convex bodies this theory has interesting ramifications, and potential application to the study of geometric inequalities. This talk should be accessible to Cambridge Part II and Part III students with introductory knowledge of differential equations and geometry.

Lagrangian knots and pseudoholomorphic curves - Luis Haug

Consider two Lagrangian embeddings of some surface into a symplectic 4-manifold. One can ask a number of different questions: Are they smoothly isotopic? Isotopic through Lagrangian embeddings? Is there even a Hamiltonian isotopy connecting them? This is known as the "Problem of Lagrangian knots in symplectic 4-manifolds", and it is not very well understood in general. Even in the case of linear symplectic 4-space $(\mathbb{C}^2,\omega_0)$, only partial results are known. Roughly speaking, it is hard for a Lagrangian torus to be smoothly knotted, in the sense of not being smoothly isotopic to the Clifford torus (which may be regarded as the "unknot"). However, things are different in the symplectic category: There is at least one example of an exotic Hamiltonian isotopy class.

After giving a brief introduction to the problem, I will explain a possible new line of attack. The idea is to look at Lagrangian immersions into $\mathbb{C}^2$ as pseudoholomorphic curves and to use the language of moduli spaces.

The talk will not assume any particular knowledge about symplectic topology.