Nets Katz: Difference between revisions
mNo edit summary |
m typo(s) fixed: Mathematics → mathematics, 2011-2012 → 2011–2012 |
||
(23 intermediate revisions by 18 users not shown) | |||
Line 1: | Line 1: | ||
'''Nets Hawk Katz''' is |
'''Nets Hawk Katz''' is the W.L. Moody Professor of Mathematics at [[Rice University]]. He was a professor of mathematics at [[Indiana University Bloomington]] until March 2013 and the IBM Professor of Mathematics at the [[California Institute of Technology]] until 2023. |
||
Katz earned a B.A. in mathematics from [[Rice University]] in 1990 at the age of 17. He received his Ph.D. in 1993 under [[Dennis DeTurck]] at the [[University of Pennsylvania]], with a dissertation titled "Noncommutative Determinants and Applications".<ref>{{mathgenealogy|name=Nets Hawk Katz|id=23071}}.</ref> |
Katz earned a B.A. in mathematics from [[Rice University]] in 1990 at the age of 17. He received his Ph.D. in 1993 under [[Dennis DeTurck]] at the [[University of Pennsylvania]], with a dissertation titled "Noncommutative Determinants and Applications".<ref>{{mathgenealogy|name=Nets Hawk Katz|id=23071}}.</ref> |
||
He is the author of |
He is the author of several important results in [[combinatorics]] (especially [[additive combinatorics]]), [[harmonic analysis]] and other areas. In 2003, jointly with [[Jean Bourgain]] and [[Terence Tao]], he proved that any subset of <math>\Z/p\Z</math> grows substantially under either addition or multiplication. More precisely, if <math>A</math> is a set such that <math> \max(|A \cdot A|, |A+A|) \leq K|A| </math>, then <math>A</math> has size at most <math>K^C</math> or at least <math>p/K^C</math> where <math>C</math> is a constant that depends on <math>A</math>. This result was followed by the subsequent work of Bourgain, [[Sergei Konyagin]] and Glibichuk, establishing that every approximate field is almost a field. |
||
Somewhat earlier he was involved in establishing new bounds in connection with the dimension of [[Kakeya set]]s. Jointly with |
Somewhat earlier he was involved in establishing new bounds in connection with the dimension of [[Kakeya set]]s. Jointly with [[Izabella Łaba]] and [[Terence Tao]] he proved that the upper Minkowski dimension of Kakeya sets in 3 dimensions is strictly greater than 5/2, and jointly with [[Terence Tao]] he established new bounds in large dimensions. |
||
In 2010, |
In 2010, Katz along with [[Larry Guth]] published the results of their collaborative effort to solve the [[Erdős distinct distances problem]], in which they found a "near-optimal" result, proving that a set of <math>N</math> points in the plane has at least <math>cN / \log N </math> distinct distances.<ref>{{cite journal |
||
| first1=Larry | last1=Guth | authorlink1=Larry Guth |
|||
| first2= Nets Hawk | last2=Katz |
|||
| journal=[[Annals of Mathematics]] |
|||
⚫ | |||
| doi=10.4007/annals.2015.181.1.2 |
|||
| title=On the Erdős distinct distances problem in the plane |
|||
| year=2015 |
|||
| pages=155–190 |
|||
| volume=181 |
|||
| issue=1 |
|||
| zbl=1310.52019 |
|||
| arxiv=1011.4105 |
|||
| mr=3272924| s2cid=43051852 }}</ref> |
|||
⚫ | |||
In early 2011, in joint work with Michael Bateman, he improved the best known bounds in the [[cap set]] problem: if A is a subset of (Z/ |
In early 2011, in joint work with Michael Bateman, he improved the best known bounds in the [[cap set]] problem: if <math>A</math> is a subset of <math>(\Z/3\Z)^n</math> of cardinality at least <math>3^n/n^{1+c}</math>, where <math>c > 0</math>, then <math>A</math> contains three elements in a line. |
||
In 2012, he was named a [[Guggenheim Fellowship|Guggenheim fellow]].<ref>{{cite web|title=2012 Fellows by field in the United States and Canada|url=http://www.gf.org/news-events/2012-Fellows-by-field-in-the-United-States-and-Canada/|publisher=John Simon Guggenheim Memorial Foundation|accessdate=5 June 2012}}</ref> During |
In 2012, he was named a [[Guggenheim Fellowship|Guggenheim fellow]].<ref>{{cite web|title=2012 Fellows by field in the United States and Canada|url=http://www.gf.org/news-events/2012-Fellows-by-field-in-the-United-States-and-Canada/|publisher=John Simon Guggenheim Memorial Foundation|accessdate=5 June 2012|url-status=dead|archiveurl=https://web.archive.org/web/20120618123358/http://www.gf.org/news-events/2012-Fellows-by-field-in-the-United-States-and-Canada|archivedate=18 June 2012}}</ref> During 2011–2012, he was the managing editor of the [[Indiana University Mathematics Journal]].<ref>{{cite web|title=Editorial Board|url=http://www.iumj.indiana.edu/Editorial/editorialb.php|publisher=Indiana University Mathematics Journal|accessdate=5 June 2012}}</ref><ref>{{cite web|title=Nets Katz|url=http://www.gf.org/fellows/17244-nets-katz|publisher=John Simon Guggenheim Memorial Foundation|accessdate=5 June 2012|url-status=dead|archiveurl=https://web.archive.org/web/20120511024921/http://www.gf.org/fellows/17244-nets-katz|archivedate=11 May 2012}}</ref> In 2014, he was an [[invited speaker at the International Congress of Mathematicians]] at Seoul and gave a talk ''The flecnode polynomial: a central object in incidence geometry''.<ref>{{cite arXiv|eprint=1404.3412|author=Katz, Nets Hawk|title=The flecnode polynomial: a central object in incidence geometry|date=13 April 2013|class=math.CO}}</ref> In 2015, he received the [[Clay Research Award]].<ref>[http://www.claymath.org/events/news/2015-clay-research-award Clay Research Award 2015]</ref> |
||
==Work== |
==Work== |
||
* {{cite journal | last1=Katz | first1=Nets Hawk| last2=Tao | first2=Terence |
* {{cite journal | last1=Katz | first1=Nets Hawk| last2=Tao | first2=Terence |
||
| title = New bounds for Kakeya problems | mr=1945284 |
| title = New bounds for Kakeya problems | mr=1945284 |
||
| journal = |
| journal = [[Journal d'Analyse Mathématique]] | volume = 87 | pages = 231–263 | year = 2002 | doi=10.1007/BF02868476 | doi-access=free |
||
| arxiv=math/0102135| s2cid=119644987}} |
|||
}} |
|||
* {{cite journal | last1=Bourgain | first1=Jean | last2=Katz | first2=Nets Hawk | last3=Tao | first3=Terence |
* {{cite journal | last1=Bourgain | first1=Jean | last2=Katz | first2=Nets Hawk | last3=Tao | first3=Terence |
||
Line 25: | Line 37: | ||
| journal=Geometric and Functional Analysis |
| journal=Geometric and Functional Analysis |
||
| volume=14 | issue=1 | year=2004 | pages=27–57 |
| volume=14 | issue=1 | year=2004 | pages=27–57 |
||
| arxiv=math/0301343 | mr=2053599}} |
| arxiv=math/0301343 | mr=2053599| s2cid=14097626 }} |
||
⚫ | |||
⚫ | |||
==References== |
==References== |
||
{{reflist}} |
{{reflist}} |
||
⚫ | |||
⚫ | |||
{{authority control}} |
|||
{{DEFAULTSORT:Katz, Nets}} |
{{DEFAULTSORT:Katz, Nets}} |
||
Line 39: | Line 53: | ||
[[Category:Indiana University faculty]] |
[[Category:Indiana University faculty]] |
||
[[Category:University of Pennsylvania alumni]] |
[[Category:University of Pennsylvania alumni]] |
||
[[Category: |
[[Category:California Institute of Technology faculty]] |
Revision as of 14:56, 24 December 2023
Nets Hawk Katz is the W.L. Moody Professor of Mathematics at Rice University. He was a professor of mathematics at Indiana University Bloomington until March 2013 and the IBM Professor of Mathematics at the California Institute of Technology until 2023.
Katz earned a B.A. in mathematics from Rice University in 1990 at the age of 17. He received his Ph.D. in 1993 under Dennis DeTurck at the University of Pennsylvania, with a dissertation titled "Noncommutative Determinants and Applications".[1]
He is the author of several important results in combinatorics (especially additive combinatorics), harmonic analysis and other areas. In 2003, jointly with Jean Bourgain and Terence Tao, he proved that any subset of grows substantially under either addition or multiplication. More precisely, if is a set such that , then has size at most or at least where is a constant that depends on . This result was followed by the subsequent work of Bourgain, Sergei Konyagin and Glibichuk, establishing that every approximate field is almost a field.
Somewhat earlier he was involved in establishing new bounds in connection with the dimension of Kakeya sets. Jointly with Izabella Łaba and Terence Tao he proved that the upper Minkowski dimension of Kakeya sets in 3 dimensions is strictly greater than 5/2, and jointly with Terence Tao he established new bounds in large dimensions.
In 2010, Katz along with Larry Guth published the results of their collaborative effort to solve the Erdős distinct distances problem, in which they found a "near-optimal" result, proving that a set of points in the plane has at least distinct distances.[2] [3]
In early 2011, in joint work with Michael Bateman, he improved the best known bounds in the cap set problem: if is a subset of of cardinality at least , where , then contains three elements in a line.
In 2012, he was named a Guggenheim fellow.[4] During 2011–2012, he was the managing editor of the Indiana University Mathematics Journal.[5][6] In 2014, he was an invited speaker at the International Congress of Mathematicians at Seoul and gave a talk The flecnode polynomial: a central object in incidence geometry.[7] In 2015, he received the Clay Research Award.[8]
Work
- Katz, Nets Hawk; Tao, Terence (2002). "New bounds for Kakeya problems". Journal d'Analyse Mathématique. 87: 231–263. arXiv:math/0102135. doi:10.1007/BF02868476. MR 1945284. S2CID 119644987.
- Bourgain, Jean; Katz, Nets Hawk; Tao, Terence (2004). "A sum-product estimate in finite fields, and applications". Geometric and Functional Analysis. 14 (1): 27–57. arXiv:math/0301343. doi:10.1007/s00039-004-0451-1. MR 2053599. S2CID 14097626.
References
- ^ Nets Hawk Katz at the Mathematics Genealogy Project.
- ^ Guth, Larry; Katz, Nets Hawk (2015). "On the Erdős distinct distances problem in the plane". Annals of Mathematics. 181 (1): 155–190. arXiv:1011.4105. doi:10.4007/annals.2015.181.1.2. MR 3272924. S2CID 43051852. Zbl 1310.52019.
- ^ Tao, Terence (20 Nov 2010), The Guth-Katz bound on the Erdős distance problem, retrieved 3 Apr 2012
- ^ "2012 Fellows by field in the United States and Canada". John Simon Guggenheim Memorial Foundation. Archived from the original on 18 June 2012. Retrieved 5 June 2012.
- ^ "Editorial Board". Indiana University Mathematics Journal. Retrieved 5 June 2012.
- ^ "Nets Katz". John Simon Guggenheim Memorial Foundation. Archived from the original on 11 May 2012. Retrieved 5 June 2012.
- ^ Katz, Nets Hawk (13 April 2013). "The flecnode polynomial: a central object in incidence geometry". arXiv:1404.3412 [math.CO].
- ^ Clay Research Award 2015