Presenter: Megan Kerr, Wellesley College
This Tuesday, the math department had its second lunch lecture of the semester. Megan Kerr of Wellesley College described the different ways one can measure the curvature of n-dimensional manifolds. She described her research in the field, which uses Algebra to find solutions to this Geometric problem. Her abstract follows:
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.
After the lecture, Professor Kerr talked to the Math 300 class about what it's like to be a working mathematician. She emphasized how happy she is to have found a workable balance between her professional and personal lives, and expressed optimism that it is becoming less necessary to delay having children until after getting tenure.