Mikhail Katz

Mikhail Katz is a mathematician known for work in differential geometry, systolic geometry, and historical and philosophical studies of infinitesimal concepts. He has developed results connecting Riemannian geometry with global invariants and contributed to discussions involving non-standard analysis and the foundations of calculus.

Early life and education

Katz was born in Lviv and educated in the Soviet Union before emigrating to Israel, where he attended Tel Aviv University and pursued graduate study influenced by scholars associated with Moscow State University and the Steklov Institute of Mathematics. His formative years involved interaction with traditions stemming from Israel Gelfand, Andrey Kolmogorov, Ludwig Faddeev, and networks linked to Nikolai Chebotaryov and Igor Shafarevich. During this period Katz encountered work by Bernhard Riemann, Henri Poincaré, Élie Cartan, and the legacy of David Hilbert that shaped his orientation toward global geometric problems.

Academic career

Katz held faculty positions at Bar-Ilan University and visiting appointments at Johns Hopkins University, Princeton University, Massachusetts Institute of Technology, and the Steklov Institute of Mathematics. He collaborated with researchers at ETH Zurich, University of Oxford, University of Cambridge, Harvard University, and the Institute for Advanced Study. Katz supervised graduate students who went on to positions at institutions such as University of Chicago, Columbia University, UC Berkeley, Tel Aviv University, and Weizmann Institute of Science. He participated in conferences organized by Mathematical Sciences Research Institute, Centre International de Rencontres Mathématiques, and the International Congress of Mathematicians.

Research and contributions

Katz's research spans interactions among systolic geometry, Riemannian geometry, and coarse geometric invariants. He has proved and refined inequalities related to the systole and applied methods tracing to Morse theory, Geometric Group Theory, and Algebraic Topology. Katz developed comparisons invoking ideas from Gromov, Loewner, Pu, and Hebda and examined relationships with minimal surface theory and the Calabi-Yau context. He produced work engaging non-standard analysis frameworks associated with Abraham Robinson and explored connections to historical texts by Augustin-Louis Cauchy, Bernard Bolzano, George Berkeley, and Augustin Cauchy. Katz analyzed conceptual issues raised by Gottfried Wilhelm Leibniz and Isaac Newton concerning infinitesimals, bringing perspectives informed by Kurt Gödel-informed foundational debates and modern treatments inspired by Errett Bishop and Ulrich Moerdijk.

In collaboration with colleagues from University of California, Berkeley, Tel Aviv University, University of Bonn, Technion – Israel Institute of Technology, and Rutgers University, Katz addressed quantitative systolic inequalities for classes of manifolds influenced by results of Mikhail Gromov, John Milnor, and William Thurston. His work engages techniques from spectral geometry, Floer homology, and variational approaches linked to Richard Hamilton and Grigori Perelman. Katz contributed to bridging classical analysis with categorical and sheaf-theoretic tools prominent in studies by Alexander Grothendieck and Jean-Pierre Serre.

Awards and honors

Katz received recognition from institutions including Bar-Ilan University and research fellowships affiliated with European Research Council programs and national funding agencies like Israel Science Foundation and National Science Foundation. He was invited to speak at meetings held by the American Mathematical Society, the European Mathematical Society, and plenary sessions at regional symposia such as those organized by Hebrew University of Jerusalem, Weizmann Institute of Science, and Technion. Fellowships and visiting appointments included terms at Institut des Hautes Études Scientifiques, Clay Mathematics Institute, and grants from foundations associated with Simons Foundation and Alexander von Humboldt Foundation.

Selected publications

- Katz authored and coauthored monographs and articles published in journals and presses associated with Springer, Cambridge University Press, and journals such as Annals of Mathematics, Journal of Differential Geometry, Inventiones Mathematicae, Proceedings of the National Academy of Sciences, Geometry & Topology, Duke Mathematical Journal, Communications in Mathematical Physics, and Mathematical Intelligencer. - Representative papers address systolic inequalities, relations between volume and systole, stability results for geometric structures, and historical-philosophical analyses of infinitesimals with publications in venues connected to Bulletin of the American Mathematical Society and Archive for History of Exact Sciences. - Katz contributed chapters to edited volumes appearing under series by Springer-Verlag, Cambridge University Press, and collections from conferences at Banff International Research Station and Newton Institute.

Category:Mathematicians Category:Differential geometers