János Kollár
Encyclopedia
János Kollár is a Hungarian mathematician
, specializing in algebraic geometry
. He is a member of the National Academy of Sciences
since 2005 and received the Cole Prize
in 2006.
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 from 1984 to 1987 and Professor
at the University of Utah
from 1987 until 1999. Currently, he is professor at Princeton University
. He is an external member of the Hungarian Academy of Sciences
(1995).
Kollár is known for his contributions to the minimal model program
for threefolds and hence the compactification
of moduli
of surfaces, for pioneering the notion of rational
connectedness
and finding counterexamples to a conjecture of John Nash.
Kollár also gave the following effective version of Hilbert's Nullstellensatz
. Let f1,...,fm be polynomials of degree at most d≥3 in n≥2 variables. If they have no common zero, then g1f1+...+gmfm=1 has a solution such that each gj has degree at most dn-d. The value dn-d is sharp.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, specializing in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. He is a member of the National Academy of Sciences
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
since 2005 and received the Cole Prize
Cole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...
in 2006.
Kollár began his studies at the Eötvös University in Budapest and later received his PhD
PHD
PHD may refer to:*Ph.D., a doctorate of philosophy*Ph.D. , a 1980s British group*PHD finger, a protein sequence*PHD Mountain Software, an outdoor clothing and equipment company*PhD Docbook renderer, an XML renderer...
at Brandeis University
Brandeis University
Brandeis University is an American private research university with a liberal arts focus. It is located in the southwestern corner of Waltham, Massachusetts, nine miles west of Boston. The University has an enrollment of approximately 3,200 undergraduate and 2,100 graduate students. In 2011, it...
in 1984 under the direction of Teruhisa Matsusaka with a thesis on canonical threefolds. He was Junior Fellow at Harvard from 1984 to 1987 and Professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
at the University of Utah
University of Utah
The University of Utah, also known as the U or the U of U, is a public, coeducational research university in Salt Lake City, Utah, United States. The university was established in 1850 as the University of Deseret by the General Assembly of the provisional State of Deseret, making it Utah's oldest...
from 1987 until 1999. Currently, he is professor at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. He is an external member of the Hungarian Academy of Sciences
Hungarian Academy of Sciences
The Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
(1995).
Kollár is known for his contributions to the minimal model program
Minimal model (birational geometry)
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible...
for threefolds and hence the compactification
Compactification
Compactification may refer to:* Compactification , making a topological space compact* Compactification , the "curling up" of extra dimensions in string theory* Compaction...
of moduli
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of surfaces, for pioneering the notion of rational
Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K...
connectedness
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
and finding counterexamples to a conjecture of John Nash.
Kollár also gave the following effective version of Hilbert's Nullstellensatz
Hilbert's Nullstellensatz
Hilbert's Nullstellensatz is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...
. Let f1,...,fm be polynomials of degree at most d≥3 in n≥2 variables. If they have no common zero, then g1f1+...+gmfm=1 has a solution such that each gj has degree at most dn-d. The value dn-d is sharp.