Mikhail Katz

Last updated
Mikhail Katz
Education Harvard University
Columbia University
Scientific career
Institutions Bar-Ilan University
Thesis Jung's Theorem in Complex Projective Geometry
Doctoral advisor Troels Jørgensen
Mikhail Gromov
Website http://u.cs.biu.ac.il/~katzmik/

Mikhail "Mischa" Gershevich Katz (born 1958, in Chișinău) [1] is an Israeli mathematician, a professor of mathematics at Bar-Ilan University. His main interests are differential geometry, geometric topology and mathematics education; he is the author of the book Systolic Geometry and Topology, which is mainly about systolic geometry. The Katz–Sabourau inequality is named after him and Stéphane Sabourau. [2] [3]



Mikhail Katz was born in Chișinău in 1958. His mother was Clara Katz (née Landman). In 1976, he moved with his mother to the United States. [4] [5]

Katz earned a bachelor's degree in 1980 from Harvard University. [1] He did his graduate studies at Columbia University, receiving his Ph.D. in 1984 under the joint supervision of Troels Jørgensen and Mikhael Gromov. [6] His thesis title is Jung's Theorem in Complex Projective Geometry.

He moved to Bar-Ilan University in 1999, after previously holding positions at the University of Maryland, College Park, Stony Brook University, Indiana University Bloomington, the Institut des Hautes Études Scientifiques, the University of Rennes 1, Henri Poincaré University, and Tel Aviv University. [1]


Katz has performed research in systolic geometry in collaboration with Luigi Ambrosio, Victor Bangert, Mikhail Gromov, Steve Shnider, Shmuel Weinberger, and others. He has authored research publications appearing in journals including Communications on Pure and Applied Mathematics, Duke Mathematical Journal, Geometric and Functional Analysis , and Journal of Differential Geometry . Along with these papers, Katz was a contributor to the book "Metric Structures for Riemannian and Non-Riemannian Spaces". [7] Marcel Berger in his article "What is... a Systole?" [8] lists the book (Katz, 2007) as one of two books he cites in systolic geometry.

More recently Katz also contributed to the study of mathematics education [9] including work that provides an alternative interpretation of the number 0.999.... [10]

Selected publications

Related Research Articles

In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

<span class="mw-page-title-main">Systolic geometry</span> Minimum length of a noncontractible closed curve in a metric space

In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry.

<span class="mw-page-title-main">Loewner's torus inequality</span>

In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.

In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality

In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983; it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane.

In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.

In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form.

The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite". Kepler used the law of continuity to calculate the area of the circle by representing it as an infinite-sided polygon with infinitesimal sides, and adding the areas of infinitely many triangles with infinitesimal bases. Leibniz used the principle to extend concepts such as arithmetic operations from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus. The transfer principle provides a mathematical implementation of the law of continuity in the context of the hyperreal numbers.

<span class="mw-page-title-main">Introduction to systolic geometry</span> Non-technical introduction to systolic geometry

Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for A: there is no interesting lower bound just in terms of the length.

In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949. Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to be the ratio area/sys2. The systolic ratio SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry.

The mathematician Shmuel Aaron Weinberger is an American topologist. He completed a PhD in mathematics in 1982 at New York University under the direction of Sylvain Cappell. Weinberger was, from 1994 to 1996, the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, and he is currently the Andrew MacLeish Professor of Mathematics and chair of the Mathematics department at the University of Chicago.

Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.

Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover.

<span class="mw-page-title-main">Victor Bangert</span> German mathematician

Victor Bangert is Professor of Mathematics at the Mathematisches Institut in Freiburg, Germany. His main interests are differential geometry and dynamical systems theory. He specialises in the theory of closed geodesics, wherein one of his significant results, combined with another one due to John Franks, implies that every Riemannian 2-sphere possesses infinitely many closed geodesics. He also made important contributions to Aubry–Mather theory.

Steve Shnider is a retired professor of mathematics at Bar Ilan University. He received a PhD in Mathematics from Harvard University in 1972, under Shlomo Sternberg. His main interests are in the differential geometry of fiber bundles; algebraic methods in the theory of deformation of geometric structures; symplectic geometry; supersymmetry; operads; and Hopf algebras. He retired in 2014.

<span class="mw-page-title-main">Simon Antoine Jean L'Huilier</span> Swiss mathematician (1750–1840)

Simon Antoine Jean L'Huilier was a Swiss mathematician of French Huguenot descent. He is known for his work in mathematical analysis and topology, and in particular the generalization of Euler's formula for planar graphs.

Lawrence David Guth is a professor of mathematics at the Massachusetts Institute of Technology.

Alexandre V. Borovik is a Professor of Pure Mathematics at the University of Manchester, United Kingdom. He was born in Russia and graduated from Novosibirsk State University in 1978. His principal research lies in algebra, model theory, and combinatorics—topics on which he published several monographs and a number of papers. He also has an interest in mathematical practice: his book Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice examines a mathematician's outlook on psychophysiological and cognitive issues in mathematics.

<span class="mw-page-title-main">Semën Samsonovich Kutateladze</span>

Semën Samsonovich Kutateladze is a mathematician. He is known for contributions to functional analysis and its applications to vector lattices and optimization. In particular, he has made contributions to the calculus of subdifferentials for vector-lattice valued functions, to whose study he introduced methods of Boolean-valued models and infinitesimals.

David M. Sherry is a philosopher and professor at Northern Arizona University in Flagstaff, Arizona. He teaches History of Philosophy, History of Logic, as well as Philosophy of Mathematics. He has published on Logic, Philosophy of Mathematics and Philosophy of Science.


  1. 1 2 3 Curriculum vitae [ permanent dead link ], retrieved 2011-05-23.
  2. Kalogeropoulos, Nikolaos (2017). "Systolic aspects of black hole entropy". arXiv: 1711.09963 [gr-qc].
  3. Riemannian Geometry: A Modern Introduction, by Isaac Chavel, pg. 235 https://books.google.com/books?id=3Gjp4vQ_mPkC&pg=PA235
  4. "Clara Katz, a Soviet émigré who saved her ailing granddaughter, dies at 85 – The Boston Globe". archive.boston.com. Retrieved 2018-01-10.
  5. "Grandmother bucked the Soviet system – Obituaries – smh.com.au". www.smh.com.au. 12 October 2006. Retrieved 2018-01-10.
  6. Mikhail Katz at the Mathematics Genealogy Project
  7. Gromov, Misha: Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp.  ISBN   0-8176-3898-9
  8. Berger, M.: What is... a Systole? Notices of the AMS 55 (2008), no. 3, 374–376.
  9. Katz & Katz (2010).
  10. Stewart, I. (2009) Professor Stewart's Hoard of Mathematical Treasures, Profile Books, p. 174.