Schedule for: 23w5062 - Women in Geometry 3

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Gathering at BIRS Lounge (PDC 2nd Floor).

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Symplectic geometry arose as a way of encoding Hamiltonian mechanics. Today symplectic geometry (and its odd dimensional cousin contact geometry) are vibrant areas of research with connections to low-dimensional topology, algebraic geometry, and dynamics. We will start with an introduction to the origins of symplectic geometry. Then we will describe some of the key notions appearing in our project and some motivation behind the questions we are working on.

Simply connected solvmanifolds form a class of noncompact homogeneous spaces that include symmetric spaces of noncompact type and all simply connected homogeneous spaces with nonpositive sectional curvature. In this talk, we will discuss a sampling of results on the geometry of solvmanifolds, while introducing some of the basic tools used in the field. We will illustrate these notions concretely in the special case of the complex hyperbolic space, which can be viewed as an explicit extension of the well known Heisenberg group.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet at the Vistas Dining Room for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In particle cosmology, a cosmic string is a vacuum defect that can occur if the quantum vacuum has nontrivial topological structure. To describe their cosmological impact, a "zero width" limit is usually taken, where the string is a codimension 2 submanifold. However, in the literature there are mathematical results that seem to throw into question this approach, indicating that you cannot take a "delta function" type of limit for a codimension two object, and that a codimension 2 surface sourcing a conical singularity has to be totally geodesic (which rules out most of the cosmic string trajectories). Our aim in this workshop is to rigorously explore what kind of metric can describe the string as a concentrated source, and whether an appropriate generalised function or distribution framework can be used to describe non-straight strings.

Holonomy groups are a useful tool in the study of semi-Riemannian manifolds. Those of simply-connected Riemannian manifolds are classified and their relation to special geometries is well studied. One of the possible holonomy groups of a 7-dimensional Riemannan manifold is G2. The associated geometric structures are called torsionfree G2-structures. The study of pseudo-Riemannian holonomy groups is much more complicated since the holonomy representation, i.e., the natural representation of a holonomy group on the tangent space can have isotropic invariant subspaces and is not necessarily completely reducible. A complete classification is only known for Lorentzian manifolds. Here we want to consider pseudo-Riemannian analogues to G2-structures. These structures live on manifolds of signature (4,3). They are characterised by the fact that their holonomy is contained in the non-compact group of type G2. Such holonomy groups were classified by Fino and Kath. Our aim for WiG is to describe these groups by their associated geometric structures and to find (at least in some cases) homogeneous models. In the talk we recall basic properties of the compact and the non-compact G2. We give a short introduction to the classification of holonomy groups contained in the non-compact G2 and explain what is known about the realisation by `globally defined' metrics and about associated geometric structures.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The class of manifolds with constant {\sl sectional} curvature is very restrictive: up to universal covering, we get only spheres $S^n$, Euclidean space $R^n$, and hyperbolic space $H^n$. The class of manifolds with constant {\sl scalar} curvature is not very restrictive at all: in dimension $n \geq 3$, every compact $M^n$ carries a metric of constant scalar curvature. The class of manifolds with constant {\sl Ricci} curvature, called Einstein spaces, lies between these extremes. They have further natural interest, as Einstein metrics are precisely the critical points of what is called the total scalar curvature functional: $$T(g) = \int_M s_g(x)\, d{\rm vol}_g.$$ How plentiful are these? We know of no topological obstructions when dimension 5 and above. We have a lot of examples, but no broad existence theorems. Our approach is to add symmetries: we suppose $M = G/H$. That is, we have a group $G$ of isometries acting transitively on $M$. The subgroup $H$ of $G$ fixing a point of $M$ is called the isotropy subgroup (unique up to conjugacy). In the homogeneous setting, scalar curvature is always constant on $M$, and moreover, homogeneous Einstein metrics are precisely the critical points of scalar curvature restricted to the space of homogeneous metrics. Existence and non-existence questions abound. For certain families of homogeneous spaces, can we classify the invariant Einstein metrics? The first half of the talk will be a survey of what is known. In the second half we will outline the question we are focusing on here and our methods.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

For variational problems it is natural to ask whether maps which have nearly minimal energy must be close to a minimising state and whether maps for which the derivative of the energy is very small must be close to a stationary state of the system. In practice it is important to understand these questions not only at a qualitative level but to establish quantitative control in the form of quantitative stability estimates, which bound the distance to minimisers in terms of the energy defect $E(u)-\min E$, respectively Lojasiewicz estimates, which bound the distance of $u$ and $E(u)$ to critical points and values in terms of $\norm{d E(u)}$. In this talk we will discuss some aspects of quantitative estimates for classical geometric variational problems, such as (relative) isoperimetric problems, and the close relation between such estimates and geometric flows.

Motivated by isoperimetric problems such as estimates for low eigenvalues of elliptic operators, we consider the possible shape of a solution to an elliptic problem on a convex domain. How many critical points can such a solution have? Are the super-level sets convex? We are particularly interested in the case that our domain lies in hyperbolic space, because here we know counterexamples--- this suggests that we need to impose much stronger conditions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.