Jeffrey Lagarias

Jeffrey Clark Lagarias (born November 16, 1949 in Pittsburgh ) is an American mathematician.

Lagarias was a Putnam Fellow in 1970 (as the winner of the competition) and studied at the Massachusetts Institute of Technology (MIT), where he received his doctorate from Harold Stark in 1974 ( The 4-part of the class group of a quadratic field ). From 1975 he was at the ATT Bell Laboratories , where he became a Distinguished Member of the Technical Staff. Since 1995 he has been a technology consultant at ATT Research Laboratories. In 2002 he became a professor at the University of Michigan .

Lagarias worked on number theory, complexity theory, cryptography, mathematical physics, dynamic systems, low-dimensional topology (knot theory), linear optimization and discrete geometry (such as circular packings, quasicrystals). In 1992 he found a counter-example to the Keller conjecture in Peter Shor . He also proved that the following elementary conjecture is equivalent to the Riemann conjecture :

For all true         and the equal sign is only for . ${\displaystyle n>0}$ ${\displaystyle \sigma (n)\leq H_{n}+e^{H_{n}}\ln {H_{n}}}$ ${\displaystyle n=1}$

It is the sum of the divider of and the th harmonic number . ${\displaystyle \sigma (n)}$ ${\displaystyle n}$ ${\displaystyle H_{n}=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\ldots +{\tfrac {1}{n}}}$ ${\displaystyle n}$

Working with Ronald Graham , Allan Wilks and others in the 2000s, he investigated number-theoretic aspects of Apollonian circular packings (developed by Alex Kontorovich , Hee Oh ).

With Joel Hass and Nicholas Pippenger , he investigated the unknot problem and showed that it belongs to the complexity class NP. In 2001, Hass and Lagarias also estimated the number of Reidemeister movements for untying the knot.

He also worked on the Collatz problem . Lagarias was guest editor with Gábor Fejes Tóth of the special issue of Discrete & Computational Geometry, which published the proof of the Kepler conjecture . Lagarias was involved in the review of the proof by Thomas C. Hales and Samuel P. Ferguson , which took place in the form of a week-long workshop in 1999 at the Institute for Advanced Study and summarized the structure of the proof in an essay that appeared in Discrete & Computational Geometry in 2002 . As of 2003, Lagarias was also actively involved in peer reviewing Hales and Ferguson's papers on the problem.

He is a member of the National Academy of Sciences (2001) and a Fellow of the American Mathematical Society (2012). In 1986 and 2007 he received the Lester Randolph Ford Award , for 2015 he was awarded the Levi L. Conant Prize together with Chuanming Zong .

Fonts

• The 3x + 1 problem and its generalizations , American Mathematical Monthly 92, 1985, pp. 3-23
• Editor with Michael J. Todd : Mathematical developments arising from linear programming , Contemporary Mathematics 114, American Mathematical Society 1990
• Point Lattices , in Handbook of Combinatorics, Elsevier 1995, pp. 919-966
• An elementary problem equivalent to the Riemann hypothesis , American Mathematical Monthly 109, 2002, pp. 534-543.
• Bounds for local density of sphere packings and the Kepler conjecture , Discrete & Computational Geometry, Volume 27, 2002, 165–193 (also printed in the book he edited about the Kepler conjecture)
• Hilbert spaces of entire functions and Dirichlet L-functions , in Pierre Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove (Eds.): Frontiers in Number Theory, Geometry and Physics , Volume 1, Springer Verlag, 2006
• Jeffrey C. Lagarias (Ed.): The ultimate challenge: The 3x + 1 problem. Amer. Math. Soc., Providence RI 2010, ISBN 0-8218-4940-9 .
• as editor: The Kepler conjecture. The Hales-Ferguson proof , Springer Verlag 2011 (with Thomas C. Hales, Samuel P. Ferguson, the introductory chapters are from Lagarias), ISBN 978-1-4614-1128-4 .
• with Chuanming Zong: Mysteries in packing regular tetrahedra , Notices AMS, December 2012,
• Euler's constant: Euler's work and modern developments , Bulletin AMS, Volume 50, 2013, pp. 527-628
• with Ronald L. Graham, Colin L. Mallows, Allan R. Wilks, Catherine H. Yan: Apollonian circle packings: number theory, J. Number Theory, Volume 100, 2003, pp. 1-45
• with Ronald L. Graham, Colin L. Mallows, Allan R. Wilks, Catherine H. Yan: Apollonian circle packings: geometry and group theory. I. The Apollonian group, Discrete Comput. Geom., Vol. 34, 2005, pp. 547-585

Web links

• Homepage of Lagarias at the University of Michigan

Individual evidence

1. ^ Hass, Lagarias, Pippenger: The computational complexity of knot and link problems, Journal of the ACM, Volume 46, 1999, pp. 185-211, Arxiv
2. ^ Hass, Lagarias, The number of Reidemeister moves needed for unknotting, Journal of the American Mathematical Society, Volume 14, 2001, pp. 399-428, Arxiv