Research Talk: Old and New Trends in Systolic Geometry

Let MM be a non-simply connected Riemannian manifold. The least length of a closed curve on MM that is not contractible to a point is called the systole of MM. Yu. Burago and V. Zalgaller and, independently, J. Hebda proved that the systole of any closed Riemannian surface MM does not exceed 2Area(M)−−−−−−−−√2Area(M).  In 1983 M. Gromov discovered that for a special class of essential manifolds the systole does not exceed c(n)volume(M)1nc(n)volume(M)1n, where nn denotes the dimension. His paper established connections between the systolic inequality and higher codimension isoperimetric inequalities in Banach spaces, introduced new natural metric invariants of Riemannian manifolds and became a starting point for development of the area of systolic geometry. We are going to sketch essential elements of Gromov's proof and then review some newer developments in the study of systoles including a much simpler proof of Gromov's result recently found by P. Papazoglou. Other topics include isoperimetric inequalities for Hausdorff contents and upper bounds for the systole in terms of Hausdorff contents discovered by Y. Liokumovich, B. Lishak, R. Rotman and the speaker.

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