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Symmetries in geometry:
Exploring (different) constant curvature spaces
Geometers study shapes: shapes of surfaces.
Differential geometry has applications to a wide arena of problems,
from cosmology (e.g. the shape of the universe) to biomechanics
(e.g. the shape of red blood cells).
The curvature of a surface measures the shape, determined by a metric.
For example, the curvature of a small round sphere is greater than that
of a big round sphere --- a sphere with a very large radius looks flat
(zero curvature).
Just as there are infinitely many ways to bend and stretch a surface
without making holes or creases,
there are infinitely many metrics on a surface.
What are the best metrics? For a two-dimensional surface,
where there is only one notion of curvature,
the metrics of constant curvature are the nicest.
For a higher dimensional surface, called a manifold,
we need to generalize our concept of curvature.
No single measurement of curvature tells the whole story, even at one point.
Sectional curvature assigns a value to each two-dimensional subspace
(called a section) of the tangent space at a point.
Ricci curvature assigns a value to each tangent vector,
by averaging sectional curvatures.
Scalar curvature assigns a value to each point,
by averaging the Ricci curvatures.
I consider a special class of manifolds
with a high degree of symmetry.
Happily, these symmetries arise naturally.
They not only represent beautiful geometry,
but also carry additional algebraic structure.
I will talk about what happens when we vary the shape of a given manifold,
controlling the variations so that the symmetries---or most of them---remain.
The goal is to find new examples with special curvature constraints.
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