János Kollár Explained

János Kollár
Birth Date:	7 June 1956
Birth Place:	Budapest
Nationality:	Hungarian
Fields:	Mathematics
Workplaces:	Princeton University University of Utah
Alma Mater:	Brandeis University Eötvös University
Doctoral Advisor:	Teruhisa Matsusaka
Doctoral Students:	Carolina Araujo Alessio Corti Chenyang Xu

János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry.

Professional career

Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 under the direction of Teruhisa Matsusaka with a thesis on canonical threefolds. He was Junior Fellow at Harvard University from 1984 to 1987 and professor at the University of Utah from 1987 until 1999. Currently, he is professor at Princeton University.^[1]

Contributions

Kollár is known for his contributions to the minimal model program for threefolds and hence the compactification of moduli of algebraic surfaces, for pioneering the notion of rational connectedness (i.e. extending the theory of rationally connected varieties for varieties over the complex field to varieties over local fields), and finding counterexamples to a conjecture of John Nash. (In 1952 Nash conjectured a converse to a famous theorem he proved,^[2] and Kollár was able to provide many 3-dimensional counterexamples from an important new structure theory for a class of 3-dimensional algebraic varieties.) ^[3]

Kollár also gave the first algebraic proof of effective Nullstellensatz: let

f _{1,\ldots,f} _{m} be polynomials of degree at most d \ge 3 in n\ge 2 variables; if they have no common zero, then the equation g _{1} f _{1+ …} + g _{m} f _{m=1} has a solution such that each polynomial g _{j} has degree at most d ^{n} - d .

Awards and honors

Kollár is a member of the National Academy of Sciences since 2005 and received the Cole Prize in 2006.^[4] He is an external member of the Hungarian Academy of Sciences since 1995.^[5] In 2012 he became a fellow of the American Mathematical Society.^[6] In 2016 he became a fellow of the American Academy of Arts and Sciences. In 2017 he received the Shaw Prize in Mathematical Sciences.^[7]

In 1990 he was an invited speaker at the International Congress of Mathematicians (ICM) in Kyōto. In 1996 he gave one of the plenary addresses at the European Mathematical Congress in Budapest (Low degree polynomial equations: arithmetic, geometry and topology). He was also selected as a plenary speaker at the ICM held in 2014 in Seoul.

As a high school student, Kollár represented Hungary and won Gold medals at both the 1973 and 1974 International Mathematical Olympiads.

Works

• Book: Kollár, János . Shafarevich maps and automorphic forms . Princeton, New Jersey . 1995 . 978-1-4008-6419-5 . 889251457.
• Book: Kollár, János . Rational curves on algebraic varieties . Springer . Berlin . 1996 . 3-540-60168-6 . 33243194. ^[8]
• Book: Kollár, János . Mori . Shigefumi . Clemens . C. Herbert . Birational geometry of algebraic varieties . Cambridge University Press . Cambridge . 1998 . 0-521-63277-3 . 39147883. ^[9] (Japanese by Iwanami Shoten).
• Book: Kollár, János . Lectures on Resolution of Singularities (AM-166) . Princeton University Press . 31 December 2009 . 978-1-4008-2780-0 . 10.1515/9781400827800. ^[10]
• Book: Kollár, János . Kovács . Sándor . Singularities of the Minimal Model Program . Cambridge University Press . 21 February 2013 . 978-1-107-03534-8 . 10.1017/cbo9781139547895.

External links

• • Homepage in Princeton

Notes and References

1. Web site: Mathematics Department Directory. Princeton University. 23 January 2010.
2. Real algebraic manifolds . . 1952 . 405–21 . 56 . 10.2307/1969649. 1969649 . Nash . John . 3 ., . See Proc. Internat. Congr. Math . AMS . 1952 . 516–17.
3. Kollár, János. The Nash conjecture for threefolds. Electron. Res. Announc. Amer. Math. Soc.. 4. 1998. 10. 63–73 (electronic). 1641168. 10.1090/s1079-6762-98-00049-3. free.
4. https://www.ams.org/notices/200604/comm-cole.pdf Notices AMS on Winner of the Cole Prize 2006, pdf-data file
5. Web site: HAS: Members of HAS. Hungarian Academy of Sciences. 23 January 2010. https://web.archive.org/web/20091203233044/http://www.mta.hu/index.php?id=members. 3 December 2009. dead.
6. https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society
7. https://www.shawprize.org/laureates/mathematical-sciences/2017 Shaw Prize 2017
8. Reid, Miles. Miles Reid. Review: Rational curves on algebraic varieties, by János Kollár. Bull. Amer. Math. Soc. (N.S.). 2000. 38. 1. 109–115. 10.1090/s0273-0979-00-00889-2. free.
9. Kawamata, Yujiro. Yujiro Kawamata. Review: Birational geometry of algebraic varieties, by János Kollár and Shigefumi Mori. Bull. Amer. Math. Soc. (N.S.). 2001. 38. 2. 267–272. 10.1090/s0273-0979-01-00910-7. free.
10. Abramovich, Dan. Review: Resolution of singularities by Steven Dale Cutkovsky and Lectures on resolution of singularities by János Kollár. Bull. Amer. Math. Soc. (N.S.). 48. 1. 115–122. 10.1090/s0273-0979-10-01301-7. free.