Mikhail Katz: recent publications

year '18

Katz, K.; Katz, M.; Kerner, D.; Liokumovich. Y. "Determinantal variety and normal embedding." Journal of Topology and Analysis 10 (2018), no. 01, 27-34. See http://dx.doi.org/10.1142/S1793525318500073 and https://arxiv.org/abs/1602.01227 and https://mathscinet.ams.org/mathscinet-getitem?mr=3737507

year '17

17a. Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.; Katz, M.; Kudryk, T.; Kutateladze, S.; McGaffey, T.; Mormann, T.; Schaps, D.; Sherry, D. "Cauchy, infinitesimals and ghosts of departed quantifiers." Mat. Stud. 47 (2017), no. 2, 115-144. See https://arxiv.org/abs/1712.00226 and http://matstud.org.ua/texts/2017/47_2/115-144.pdf and http://dx.doi.org/10.15330/ms.47.2.115-144

17b. Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; McGaffey, T.; Reeder, P.; Schaps, D.; Sherry, D.; Shnider, S. "Interpreting the infinitesimal mathematics of Leibniz and Euler." Journal for General Philosophy of Science 48 (2017), no. 2, 195-238. See http://dx.doi.org/10.1007/s10838-016-9334-z and https://arxiv.org/abs/1605.00455 and http://www.ams.org/mathscinet-getitem?mr=3663035 Here we analyze Euler's approach to infinitesimal analysis and his proof of the infinite product decomposition for the sine function. We also examine Giovanni Ferraro's flawed historical scholarship and propose a sounder alternative.

17c. Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kudryk, T.; Mormann, T.; Sherry, D. "Is Leibnizian calculus embeddable in first order logic?" Foundations of Science 22 (2017), no. 4, 717-731. See http://dx.doi.org/10.1007/s10699-016-9495-6 and https://arxiv.org/abs/1605.03501 and https://mathscinet.ams.org/mathscinet-getitem?mr=3720412

17d. Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; Sherry, D. "Toward a history of mathematics focused on procedures." Foundations of Science 22 (2017), no. 4, 763-783. See http://dx.doi.org/10.1007/s10699-016-9498-3 and https://arxiv.org/abs/1609.04531 and https://mathscinet.ams.org/mathscinet-getitem?mr=3720415 Here we propose an approach to the history of mathematics that focuses in the procedures of the historical masters rather than set-theoretic ontology of the entities they use. We also examine Jeremy Gray's flawed historical scholarship and propose a sounder alternative.

17e. Błaszczyk, P.; Kanovei, V.; Katz, M.; Sherry, D. "Controversies in the foundations of analysis: Comments on Schubring's Conflicts." Foundations of Science 22 (2017), no. 1, 125-140. See http://dx.doi.org/10.1007/s10699-015-9473-4 and https://arxiv.org/abs/1601.00059 and http://www.ams.org/mathscinet-getitem?mr=3605125 See also Reception

17f. Fletcher, P.; Hrbacek, K.; Kanovei, V.; Katz, M.; Lobry, C.; Sanders, S. "Approaches to analysis with infinitesimals following Robinson, Nelson, and others." Real Analysis Exchange 42 (2017), no. 2, 193-252. See https://arxiv.org/abs/1703.00425 and http://msupress.org/journals/issue/?id=50-21D-61F and https://mathscinet.ams.org/mathscinet-getitem?mr=3721800 and eventually http://dx.doi.org/10.14321/realanalexch.41.1.0193

17g. Gutman, A.; Katz, M.; Kudryk, T.; Kutateladze, S. "The Mathematical Intelligencer Flunks the Olympics." Foundations of Science 22 (2017), no. 3, 539-555. See http://dx.doi.org/10.1007/s10699-016-9485-8 and https://arxiv.org/abs/1606.00160 and http://www.ams.org/mathscinet-getitem?mr=3696393 Here we examine Yaroslav Sergeyev's grossbit pathos.

17h. Katz, M.; Polev, L. "From Pythagoreans and Weierstrassians to true infinitesimal calculus." Journal of Humanistic Mathematics 7 (2017), no. 1, 87-104. See http://dx.doi.org/10.5642/jhummath.201701.07 and https://arxiv.org/abs/1701.05187

17i. Kanovei, V.; Katz, M. "A positive function with vanishing Lebesgue integral in Zermelo-Fraenkel set theory." Real Analysis Exchange 42(2), 2017, 385-390. See https://arxiv.org/abs/1705.00493 and http://msupress.org/journals/issue/?id=50-21D-61F and https://mathscinet.ams.org/mathscinet-getitem?mr=3721807

year '16

Bascelli, T.; Błaszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Schaps, D.; Sherry, D. "Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania." HOPOS: The Journal of the International Society for the History of Philosophy of Science 6 (2016), no. 1, 117-147. See http://dx.doi.org/10.1086/685645 and http://arxiv.org/abs/1603.07209

Błaszczyk, P.; Borovik, A.; Kanovei, V.; Katz, M.; Kudryk, T.; Kutateladze, S.; Sherry, D. "A non-standard analysis of a cultural icon: The case of Paul Halmos." Logica Universalis 10 (2016), no. 4, 393-405. http://dx.doi.org/10.1007/s11787-016-0153-0 and http://arxiv.org/abs/1607.00149 and http://www.ams.org/mathscinet-getitem?mr=3566230

Kanovei, V.; Katz, K.; Katz, M.; Nowik, T. "Small oscillations of the pendulum, Euler's method, and adequality." Quantum Studies: Mathematics and Foundations 3 (2016), no. 3, 231-236. See http://dx.doi.org/10.1007/s40509-016-0074-x and http://arxiv.org/abs/1604.06663 and http://www.ams.org/mathscinet-getitem?mr=3531864

Katz, K.; Katz, M.; Schein, M.; Vishne, U. "Bolza quaternion order and asymptotics of systoles along congruence subgroups." Experimental Mathematics 25 (2016), no. 4, 399-415. See http://dx.doi.org/10.1080/10586458.2015.1073642 and http://www.ams.org/mathscinet-getitem?mr=3499706

year '15

Kanovei, V.; Katz, K.; Katz, M.; Sherry, D. "Euler's lute and Edwards' oud." The Mathematical Intelligencer 37 (2015), 48-51. See http://arxiv.org/abs/1506.02586 and http://dx.doi.org/10.1007/s00283-015-9565-6

Kanovei, V.; Katz, K.; Katz, M.; Schaps, M. "Proofs and Retributions, Or: Why Sarah Can't Take Limits." Foundations of Science 20 (2015), no. 1, 1-25. See http://dx.doi.org/10.1007/s10699-013-9340-0 and http://www.ams.org/mathscinet-getitem?mr=3312498

Katz, M.; Kutateladze, S. "Edward Nelson (1932-2014)." The Review of Symbolic Logic 8 (2015), no. 3, 607-610. See http://dx.doi.org/10.1017/S1755020315000015 and http://arxiv.org/abs/1506.01570

Katz, M.; Sabourau, S. "Dyck's surfaces, systoles, and capacities." Transactions of the American Mathematical Society 367 (2015), no. 6, 4483-4504. See http://dx.doi.org/10.1090/S0002-9947-2014-06216-8 and http://arxiv.org/abs/1205.0188 and http://www.ams.org/mathscinet-getitem?mr=3324936

Nowik, T; Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1-44. See http://dx.doi.org/10.4115/jla.2015.7.5 and http://arxiv.org/abs/1405.0984

year '14

Bascelli, T.; Bottazzi, E.; Herzberg, F.; Kanovei, V.; Katz, K.; Katz, M.; Nowik, T.; Sherry, D.; Shnider, S. "Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow." Notices of the American Mathematical Society 61 (2014), no. 8, 848-864. See http://www.ams.org/notices/201408/rnoti-p848.pdf and http://arxiv.org/abs/1407.0233

Katz, K.; Katz, M.; Kudryk, T. "Toward a clarity of the extreme value theorem." Logica Universalis 8 (2014), no. 2, 193-214. See http://dx.doi.org/10.1007/s11787-014-0102-8 and http://arxiv.org/abs/1404.5658 and http://www.ams.org/mathscinet-getitem?mr=3210286

Sherry, D.; Katz, M. "Infinitesimals, imaginaries, ideals, and fictions." Studia Leibnitiana 44 (2012), no. 2, 166-192. See http://arxiv.org/abs/1304.2137

Tall, D.; Katz, M. "A cognitive analysis of Cauchy's conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus." Educational Studies in Mathematics 86 (2014), no. 1, 97-124. See http://dx.doi.org/10.1007/s10649-014-9531-9 and http://arxiv.org/abs/1401.1468

year '13

Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.; Katz, M.; Kutateladze, S.; McGaffey, T.; Schaps, D.; Sherry, D.; Shnider, S. "Is mathematical history written by the victors?" Notices of the American Mathematical Society 60 (2013) no. 7, 886-904. Accessible here, http://www.ams.org/notices/201307/rnoti-p886.pdf, http://www.ams.org/mathscinet-getitem?mr=3086638, and http://arxiv.org/abs/1306.5973

Błaszczyk, P.; Katz, M.; Sherry, D. "Ten misconceptions from the history of analysis and their debunking." Foundations of Science 18 (2013), no. 1, 43-74. See http://dx.doi.org/10.1007/s10699-012-9285-8, http://www.ams.org/mathscinet-getitem?mr=3031794, and http://arxiv.org/abs/1202.4153

Kanovei, V.; Katz, M.; Mormann, T. "Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics." Foundations of Science 18 (2013), no. 2, 259--296. See http://dx.doi.org/10.1007/s10699-012-9316-5, http://www.ams.org/mathscinet-getitem?mr=3064607, and http://arxiv.org/abs/1211.0244

Katz, M.; Leichtnam, E. "Commuting and noncommuting infinitesimals." American Mathematical Monthly 120 (2013), no. 7, 631-641. See http://dx.doi.org/10.4169/amer.math.monthly.120.07.631, http://www.ams.org/mathscinet-getitem?mr=3096469, and http://arxiv.org/abs/1304.0583

Katz, M.; Schaps, D.; Shnider, D. "Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond." Perspectives on Science 21 (2013), no. 3, 283-324. See http://www.mitpressjournals.org/doi/abs/10.1162/POSC_a_00101, http://www.ams.org/mathscinet-getitem?mr=3114421, and http://arxiv.org/abs/1210.7750

Katz, M.; Sherry, D. "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond." Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174

Katz, M.; Tall, D. "A Cauchy-Dirac delta function." Foundations of Science, 18 (2013), no. 1, 107-123. See http://dx.doi.org/10.1007/s10699-012-9289-4, http://www.ams.org/mathscinet-getitem?mr=3031797, and http://arxiv.org/abs/1206.0119

Mormann, T.; Katz, M. "Infinitesimals as an issue of neo-Kantian philosophy of science." HOPOS: The Journal of the International Society for the History of Philosophy of Science 3 (2013), no. 2, 236-280. See http://www.jstor.org/stable/10.1086/671348 and http://arxiv.org/abs/1304.1027

year '12

Mikhail G. Katz and David Sherry. "Leibniz's laws of continuity and homogeneity." Notices of the American Mathematical Society 59 (2012), no. 11, 1550-1558. See http://www.ams.org/notices/201211/rtx121101550p.pdf, http://arxiv.org/abs/1211.7188, and http://u.cs.biu.ac.il/~katzmik/straw2.html

Borovik, A.; Jin, R.; Katz, M. "An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals." Notre Dame Journal of Formal Logic 53 (2012), no. 4, 557-570. See http://arxiv.org/abs/1210.7475 and http://dx.doi.org/10.1215/00294527-1722755

Katz, M.; Sabourau, S. "Hyperellipticity and systoles of Klein surfaces." Geometriae Dedicata 159 (2012), no. 1, 277-293. See http://dx.doi.org/10.1007/s10711-011-9659-z and http://arxiv.org/abs/1201.0361 and http://www.ams.org/mathscinet-getitem?mr=2944532

Katz, K.; Katz, M. "Stevin numbers and reality." Foundations of Science 17 (2012), no. 2, 109-123. See http://dx.doi.org/10.1007/s10699-011-9228-9 and http://arxiv.org/abs/1107.3688

Karin Usadi Katz and Mikhail G. Katz. "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography." Foundations of Science 17 (2012), no. 1, 51-89. See http://dx.doi.org/10.1007/s10699-011-9223-1 See http://arxiv.org/abs/1104.0375

Borovik, A.; Katz, M. "Who gave you the Cauchy--Weierstrass tale? The dual history of rigorous calculus." Foundations of Science 17 (2012), no. 3, 245-276. see http://dx.doi.org/10.1007/s10699-011-9235-x and http://arxiv.org/abs/1108.2885

Mikhail G. Katz, David Tall. "Tension between Intuitive Infinitesimals and Formal Mathematical Analysis." Chapter in: 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, 2012, pp. 71-89. See http://arxiv.org/abs/1110.5747

year '11

Karin Usadi Katz, Mikhail G. Katz: Meaning in Classical Mathematics: Is it at Odds with Intuitionism? Intellectica 56 (2011), no. 2, 223-302. See http://arxiv.org/abs/1110.5456

Katz, K.; Katz, M.: Cauchy's continuum. Perspectives on Science 19 (2011), no. 4, 426-452. See http://dx.doi.org/10.1162/POSC_a_00047 and http://arxiv.org/abs/1108.4201

Katz, M.; Schaps, M.; Vishne, U.: Hurwitz quaternion order and arithmetic Riemann surfaces. Geometriae Dedicata 155 (2011), no. 1, 151-161, see http://arxiv.org/abs/math/0701137 and http://dx.doi.org/10.1007/s10711-011-9582-3

Katz, K.; Katz, M.; Sabourau, S.; Shnider, S.; Weinberger, Sh.: Relative systoles of relative-essential 2-complexes. Algebraic & Geometric Topology 11 (2011), 197-217. See arXiv:0911.4265 and online version at journal

Dranishnikov, A.; Katz, M.; Rudyak, Y.: Cohomological dimension, self-linking, and systolic geometry. Israel Journal of Math. 184 (2011), no. 1, 437--453. See arXiv:0807.5040

Ambrosio, L.; Katz, M.: Flat currents modulo p in metric spaces and filling radius inequalities, Commentarii Math. Helvetici 86 (2011), no. 3, 557--591. DOI: 10.4171/CMH/234 See arXiv:1004.1374

Katz, Karin Usadi; Katz, M.: Bi-Lipschitz approximation by finite-dimensional imbeddings. Geometriae Dedicata 150 (2011) no. 1, 131--136. See arXiv:0902.3126 and online version at journal

year '10

Katz, K.; Katz, M.: Zooming in on infinitesimal 1-.9.. in a post-triumvirate era. Educational Studies in Mathematics 74 (2010), no. 3, 259-273. See arXiv:1003.1501.

Katz, M.; Shnider, S.: Cayley 4-form, comass, and triality isomorphisms. Israel J. Math. 178 (2010), 187-208. See https://link.springer.com/article/10.1007%2Fs11856-010-0062-5 and arXiv:0801.0283

Katz, K.; Katz, M.: When is .999... less than 1? The Montana Mathematics Enthusiast, Vol. 7 (2010), No. 1, pp. 3--30. See arXiv:1007.3018

year '09

Horowitz, C.; Katz, Karin Usadi; Katz, M.: Loewner's torus inequality with isosystolic defect. Journal of Geometric Analysis 19 (2009), no. 4, 796-808. See arXiv:0803.0690

Balacheff, F.; Croke C.; Katz, M.: A Zoll counterexample to a geodesic length conjecture. Geometric and Functional Analysis (GAFA), 19 ('09), no. 1, 1-10. arXiv:0711.1229.

Bangert, V; Katz, M.; Shnider, S.; Weinberger, S.: E7, Wirtinger inequalities, Cayley 4-form, and homotopy. Duke Math. J. 146 ('09), no. 1, 35-70. See https://projecteuclid.org/euclid.dmj/1229530284 and arXiv:math.DG/0608006

year '08

Dranishnikov, A.; Katz, M.; Rudyak, Y.: Small values of the Lusternik-Schnirelmann category for manifolds. Geometry and Topology 12 ('08), 1711-1727. See arXiv:0805.1527.

Katz, M.; Rudyak, Y.: Bounding volume by systoles of 3-manifolds. Journal of London Math. Society 78 ('08), no 2, 407-417. See arXiv:math.DG/0504008.

Katz, M.: Systolic inequalities and Massey products in simply-connected manifolds. Israel J. Math. 164 ('08), 381-395. See arXiv:math.DG/0604012.

year '07

Katz, M.: Systolic geometry and topology. With an appendix by J. Solomon. Mathematical surveys and monographs, volume 137, American Mathematical Society, '07.

Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 ('07), no. 3, 399-422. See arXiv:math.DG/0505007.

Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Boundary case of equality in optimal Loewner-type inequalities. Trans. Amer. Math. Soc. 359 ('07), no. 1, 1--17. See arXiv:math.DG/0406008.

year '06

Katz, M.; Rudyak, Y.: Sabourau, S.: Systoles of 2-complexes, Reeb graph, and Grushko decomposition. Int. Math. Res. Not. 2006 ('06). Art. ID 54936, pp. 1--30. See arXiv:math.DG/0602009.

Katz, M.; Sabourau, S.: An optimal systolic inequality for CAT(0) metrics in genus two. Pacific J. Math. 227 ('06), no. 1, 95-107. See arXiv:math.DG/0501017.

Katz, M.; Rudyak, Y.: Lusternik-Schnirelmann category and systolic category of low dimensional manifolds. Communications on Pure and Applied Mathematics 59 ('06), no. 10, 1433-1456. See arXiv:math.DG/0410456.

Katz, M.; Sabourau, S.: Hyperelliptic surfaces are Loewner. Proc. Amer. Math. Soc. 134 ('06), no. 4, 1189-1195. See arXiv:math.DG/0407009.

year '05

Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory and Dynamical Systems, 25 ('05) no. 4, 1209-1220. See arXiv:math.DG/0410312.

Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Filling area conjecture and ovalless real hyperelliptic surfaces, Geometric and Functional Analysis (GAFA) 15 ('05), no. 3, 577-597. See arXiv:math.DG/0405583.

Katz, M.; Lescop, C.: Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers. Proceedings of conference and workshop in memory of R. Brooks, held at the Technion, IMCP, Contemporary Mathematics 387 ('05), 181-200. Amer. Math. Soc., Providence, R.I. See arXiv:math.DG/0412011.

year '04

Bangert, V; Katz, M.: An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm. Comm. Anal. Geom. 12 ('04), number 3, 703-732. See arXiv:math.DG/0304494.

Ivanov, S.; Katz, M.: Generalized degree and optimal Loewner-type inequalities. Israel J. Math. 141 ('04), 221-233. See arXiv:math.DG/0405019.

year '03

Bangert, V.; Katz, M.: Stable systolic inequalities and cohomology products. Comm. Pure Appl. Math. 56 ('03), 979--997. See arXiv:math.DG/0204181.

Croke, C.; Katz, M.: Universal volume bounds in Riemannian manifolds. 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, 02, edited by S.T. Yau (Somerville, MA: International Press, '03.) pp. 109 - 137. See arXiv:math.DG/0302248.

Katz, M.: Four-manifold systoles and surjectivity of period map. Comment. Math. Helv. 78 ('03), 772-876. See arXiv:math.DG/0302306.

year '02

Katz, M. Local calibration of mass and systolic geometry. Geom. Funct. Anal. (GAFA) 12, issue 3 ('02), 598-621. See arXiv:math.DG/0204182.

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