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.
Kollár memulai studinya di Universitas Eötvös di Budapest dan kemudian menerima gelar Ph.D. di Universitas Brandeis pada 1984 di bawah bimbingan Teruhisa Matsusaka dengan tesis mengenai lipat tiga kanonis.
He was Junior Fellow at Harvard University from 1984 to 1987 and professor at the University of Utah from 1987 until 1999.
Ia merupakan Anggota Junior di Universitas Harvard dari 1984 hingga 1987 dan profesor di Universitas Utah dari 1987 hingga 1999.
Currently, he is professor at Princeton University.
Saat ini, ia merupakan profesor di Universitas Princeton.
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.
Kollár dikenal atas kontribusinya pada program model minimal untuk lipat tiga dan pengompakan moduli permukaan aljabar, mempelopori gagasan keterhubungan rasional (seperti memperluas teori varietas terhubung secara rasional untuk varietas di atas medan kompleks ke varietas di atas medan lokal), serta menemukan contoh kebalikan dari konjektur John Nash (pada 1952, Nash membuat konjektur mengenai teorema terkenal yang ia buktikan, dan Kollár dapat memberikan banyak contoh 3-dimensi sebaliknya dari teori struktur baru yang penting untuk kelas vaietas aljabar tiga dimensi).
Kollár also gave the first algebraic proof of effective Nullstellensatz: let f 1 , … , f m ,\ldots ,f_{m}} be polynomials of degree at most d ≥ 3 in n ≥ 2 variables; if they have no common zero, then the equation g 1 f 1 + ⋯ + g m f m1 f_{1}+\cdots +g_{m}f_{m}=1} has a solution such that each polynomial g j } has degree at most d n − d -d} .
Kollár juga memberikan bukti aljabar pertama dari Nullstellensatz efektif: misal f 1 , … , f m ,\ldots ,f_{m}} merupakan polinom dengan derajat tertinggi d ≥ 3 dalam variabel n ≥ 2 ; jika polinom-polinom ini tidak memiliki akar bersama, maka persamaan g 1 f 1 + ⋯ + g m f m1 f_{1}+\cdots +g_{m}f_{m}=1} memiliki solusi sedemikian sehingga setiap polinom g j } memiliki derajat tertinggi d n − d -d} .
He is an external member of the Hungarian Academy of Sciences since 1995.
Ia merupakan anggota luar Akademi Ilmu Pengetahuan Hungaria sejak 1995.
In 2016 he became a fellow of the American Academy of Arts and Sciences.
Pada 2016, ia menjadi anggota Akademi Seni dan Ilmu Pengetahuan Amerika Serikat.
In 2017 he received the Shaw Prize in Mathematical Sciences.
Pada 2017, ia menerima Penghargaan Shaw dalam Ilmi Matematika.
In 1990 he was an invited speaker at the International Congress of Mathematicians (ICM) in Kyōto.
Pada 1990, ia merupakan pembicara tamu di Kongres Matematikawan Internasional (ICM) di 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).
Pada 1996, ia mengisi salah satu sesi paripurna di Kongres Matematika Eropa di Budapest berjudul "Low degree polynomial equations: arithmetic, geometry and topology" (bahasa Indonesia: "Persamaan polinom berderajat rendah:aritmetika, geometri, dan topologi"code:
He was also selected as a plenary speaker at the ICM held in 2014 in Seoul.
Ia juga terpilih sebagai pembicara sesi paripurna di ICM yang diadakan pada 2014 di Seoul.
As a high school student, Kollár represented Hungary and won Gold medals at both the 1973 and 1974 International Mathematical Olympiads.
Saat menjadi siswa sekolah menengah, Kollár mewakili Hungaria dan memenankan medali emas Olimpiade Matematika Internasional pada 1973 dan 1974.
Lectures on Resolution of Singularities (AM-166).
Lectures on Resolution of Singularities.
Princeton University Press.
Princeton University Press 2007.
Singularities of the Minimal Model Program.
Singularities of the Minimal Model Program (with contributions by Sándor Kovács).