Mikhail G. Katz

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We explore M. Gromov's counterexamples to systolic inequalities. Does the manifold S 2 × S 2 admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit area? This question is still open, but the answer is affirmative for its analogue in the case of S '~ × S n, n > 3. Our point of departure is M. Gromov's(More)
We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous(More)
In 1972, Marcel Berger defined a metric invariant that captures the 'size' of ^-dimensional homology of a Riemannian manifold. This invariant came to be called the fc-dimensional systole. He asked if the systoles can be constrained by the volume, in the spirit of the 1949 theorem of C. Loewner. We construct metrics, inspired by M. Gromov's 1993 example,(More)
We show that the geometry of a Riemannian manifold (M,G) is sensitive to the apparently purely homotopy-theoretic invariant of M known as the Lusternik-Schnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M . It is denoted catsys(M), and defined in terms of the existence of systolic(More)