Mikhail Katz | |
---|---|

Born | 1958 |

Nationality | Israeli |

Education | Harvard University Columbia University |

Scientific career | |

Fields | Mathematics |

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] }

This section may contain indiscriminate, excessive, or irrelevant examples.(October 2022) |

- Bair, Jacques; Błaszczyk, Piotr; Ely, Robert; Henry, Valérie; Kanovei, Vladimir; Katz, Karin; Katz, Mikhail; Kutateladze, Semen; McGaffey, Thomas; Schaps, David; Sherry, David; Shnider, Steve (2013), "Is mathematical history written by the victors?" (PDF),
*Notices of the American Mathematical Society*,**60**(7): 886–904, arXiv: 1306.5973 , doi:10.1090/noti1001 . - Katz, Mikhail G.; Sherry, David (2012), "Leibniz's laws of continuity and homogeneity",
*Notices of the American Mathematical Society*,**59**(11): 1550–1558, arXiv: 1211.7188 , Bibcode:2012arXiv1211.7188K, doi:10.1090/noti921, S2CID 42631313 . - Katz, Mikhail G.; Schaps, David; Shnider, Steve (2013), "Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond",
*Perspectives on Science*,**21**(3): 283–324, arXiv: 1210.7750 , Bibcode:2012arXiv1210.7750K, doi:10.1162/POSC_a_00101, S2CID 57569974 . - Borovik, Alexandre; Jin, Renling; Katz, Mikhail G. (2012), "An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals",
*Notre Dame Journal of Formal Logic*,**53**(4): 557–570, arXiv: 1210.7475 , Bibcode:2012arXiv1210.7475B, doi:10.1215/00294527-1722755, S2CID 14850847 . - Kanovei, Vladimir; Katz, Mikhail G.; Mormann, Thomas (2013), "Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics",
*Foundations of Science*,**18**(2): 259–296, arXiv: 1211.0244 , doi:10.1007/s10699-012-9316-5, S2CID 7631073 . - Katz, Mikhail; Tall, David (2012), "A Cauchy-Dirac delta function",
*Foundations of Science*,**18**: 107–123, arXiv: 1206.0119 , Bibcode:2012arXiv1206.0119K, doi:10.1007/s10699-012-9289-4, S2CID 119167714 . - Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond",
*Erkenntnis*,**78**(3): 571–625, arXiv: 1205.0174 , Bibcode:2012arXiv1205.0174K, doi:10.1007/s10670-012-9370-y, S2CID 119329569 . - Błaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking",
*Foundations of Science*,**18**: 43–74, arXiv: 1202.4153 , Bibcode:2012arXiv1202.4153B, doi:10.1007/s10699-012-9285-8, S2CID 119134151 . - Katz, Mikhail; Tall, David (2012),
*Tension between Intuitive Infinitesimals and Formal Mathematical Analysis*, Bharath Sriraman, Editor. Crossroads in the History of Mathematics and Mathematics Education. The Montana Mathematics Enthusiast Monographs in Mathematics Education 12, Information Age Publishing, Inc., Charlotte, NC, pp. 71–89, arXiv: 1110.5747 , Bibcode:2011arXiv1110.5747K . - Katz, Karin Usadi; Katz, Mikhail G. (2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?",
*Intellectica*,**56**(2): 223–302, arXiv: 1110.5456 , Bibcode:2011arXiv1110.5456U . - Borovik, Alexandre; Katz, Mikhail G. (2012), "Who gave you the Cauchy—Weierstrass tale? The dual history of rigorous calculus",
*Foundations of Science*,**17**(3): 245–276, arXiv: 1108.2885 , doi:10.1007/s10699-011-9235-x, S2CID 119320059 . - Katz, Karin Usadi; Katz, Mikhail G. (2011), "Cauchy's continuum",
*Perspectives on Science*,**19**(4): 426–452, arXiv: 1108.4201 , doi:10.1162/POSC_a_00047, MR 2884218, S2CID 57565752 . - Katz, Karin Usadi; Katz, Mikhail G.; Sabourau, Stéphane; Shnider, Steven; Weinberger, Shmuel (2011), "Relative systoles of relative-essential 2-complexes",
*Algebraic & Geometric Topology*,**11**(1): 197–217, arXiv: 0911.4265 , doi:10.2140/agt.2011.11.197, MR 2764040, S2CID 20087785 . - Katz, Karin Usadi; Katz, Mikhail G. (2012), "Stevin numbers and reality",
*Foundations of Science*,**17**(2): 109–123, arXiv: 1107.3688 , doi:10.1007/s10699-011-9228-9, S2CID 119167692 . - Katz, Karin Usadi; Katz, Mikhail G. (2012), "A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography",
*Foundations of Science*,**17**(1): 51–89, arXiv: 1104.0375 , doi:10.1007/s10699-011-9223-1, MR 2896999, S2CID 119250310 . - Katz, Mikhail G. (2007),
*Systolic geometry and topology*, Mathematical Surveys and Monographs, vol. 137, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4177-8, MR 2292367 . With an appendix by J. Solomon. - Katz, Karin Usadi; Katz, Mikhail G. (2010), "When is .999... less than 1?",
*The Montana Mathematics Enthusiast*,**7**(1): 3–30, arXiv: 1007.3018 , Bibcode:2010arXiv1007.3018U, doi:10.54870/1551-3440.1381, S2CID 11544878, archived from the original on 2011-07-20. - Katz, Karin Usadi; Katz, Mikhail G. (2010), "Zooming in on infinitesimal 1–.9.. in a post-triumvirate era",
*Educational Studies in Mathematics*,**74**(3): 259–273, arXiv: 1003.1501 , Bibcode:2010arXiv1003.1501K, doi:10.1007/s10649-010-9239-4, S2CID 115168622 . - Bangert, Victor; Katz, Mikhail G. (2003), "Stable systolic inequalities and cohomology products",
*Communications on Pure and Applied Mathematics*,**56**(7): 979–997, arXiv: math/0204181 , doi:10.1002/cpa.10082, MR 1990484, S2CID 14485627 . - Katz, Mikhail G.; Rudyak, Yuli B. (2006), "Lusternik–Schnirelmann category and systolic category of low-dimensional manifolds",
*Communications on Pure and Applied Mathematics*,**59**(10): 1433–1456, arXiv: dg-ga/9708007 , doi:10.1002/cpa.20146, MR 2248895, S2CID 15470409 . - Bangert, Victor; Katz, Mikhail G.; Shnider, Steven; Weinberger, Shmuel (2009), "E
_{7}, Wirtinger inequalities, Cayley 4-form, and homotopy",*Duke Mathematical Journal*,**146**(1): 35–70, arXiv: math.DG/0608006 , doi:10.1215/00127094-2008-061, MR 2475399, S2CID 2575584 . - Croke, Christopher B.; Katz, Mikhail G. (2003), "Universal volume bounds in Riemannian manifolds", in Yau, S. T. (ed.),
*Surveys in Differential Geometry VIII, Lectures on Geometry and Topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, May 3–5, 2002*, Int. Press, Somerville, MA, pp. 109–137, arXiv: math.DG/0302248 , MR 2039987 . - Katz, Mikhail G. (1983), "The filling radius of two-point homogeneous spaces",
*Journal of Differential Geometry*,**18**(3): 505–511, doi: 10.4310/jdg/1214437785 , MR 0723814 .

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.

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.

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.

**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/sys^{2}. The *systolic ratio* SR is the reciprocal quantity sys^{2}/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

**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.

**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.

**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 2 3 Curriculum vitae
^{[ permanent dead link ]}, retrieved 2011-05-23. - ↑ Kalogeropoulos, Nikolaos (2017). "Systolic aspects of black hole entropy". arXiv: 1711.09963 [gr-qc].
- ↑ Riemannian Geometry: A Modern Introduction, by Isaac Chavel, pg. 235 https://books.google.com/books?id=3Gjp4vQ_mPkC&pg=PA235
- ↑ "Clara Katz, a Soviet émigré who saved her ailing granddaughter, dies at 85 – The Boston Globe".
*archive.boston.com*. Retrieved 2018-01-10. - ↑ "Grandmother bucked the Soviet system – Obituaries – smh.com.au".
*www.smh.com.au*. 12 October 2006. Retrieved 2018-01-10. - ↑ Mikhail Katz at the Mathematics Genealogy Project
- ↑ 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 - ↑ Berger, M.: What is... a Systole?
*Notices of the AMS*55 (2008), no. 3, 374–376. - ↑ Katz & Katz (2010).
- ↑ Stewart, I. (2009)
*Professor Stewart's Hoard of Mathematical Treasures*, Profile Books, p. 174.

- Mikhail Katz's home page
- Mikhail Katz publications indexed by Google Scholar

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