Caucher Birkar, Fields Medal 2018

Caucher Birkar FRS (Kurdish: کۆچەر بیرکار/Koçer Bîrkar‎, lit. 'migrant mathematician'; born Fereydoun Derakhshani (Persian: فریدون درخشانی‎); July 1978, in Marivan County, Kurdistan Province, Iran) is a UK-based Iranian Kurdish mathematician and a professor at the University of Cambridge.

Birkar is an important contributor to modern birational geometry. In 2010 he received the Leverhulme Prize in mathematics and statistics for his contributions to algebraic geometry, and in 2016, the AMS Moore Prize for the article "Existence of minimal models for varieties of log general type", Journal of the AMS (2010) (joint with P. Cascini, C. Hacon and J. McKernan). He was awarded the Fields Medal in 2018, "for his proof of boundedness of Fano varieties and contributions to the minimal model program".

Early life and education

Birkar is an ethnic Kurd, and was born in 1978 in Marivan County, Kurdistan Province, Iran, on a subsistence farm, and raised during the Iran-Iraq War. He studied mathematics at the University of Tehran where he received his bachelor's degree. He was awarded the third prize in the International Mathematics Competition for University Students in 2000 and, shortly after, while still studying in the University, relocated to the UK as a refugee and asked for political asylum. In 2001–2004 Birkar was a PhD student at the University of Nottingham. In 2003 he was awarded the Cecil King Travel Scholarship by the London Mathematical Society as the most promising PhD student. Upon emigrating to the UK he changed his name to Caucher Birkar, which means "migrant mathematician" in Kurdish.

Research and career

Together with Paolo Cascini, Christopher Hacon and James McKernan, Birkar settled several conjectures including existence of log flips, finite generation of log canonical rings, and existence of minimal models for varieties of log general type, building upon earlier work of Vyacheslav Shokurov and of Hacon and McKernan.

In the setting of log canonical singularities, he proved existence of log flips along with key cases of the minimal model and abundance conjectures. (This was also proved independently by Hacon and Chenyang Xu.)

In a different direction, he studied the old problem of Iitaka on effectivity of Iitaka fibrations induced by pluri-canonical systems on varieties of non-negative Kodaira dimension. The problem consists of two halves: one related to general fibres of the fibration and one related to the base of the fibration. Birkar and Zhang co-solved the second half of the problem, hence essentially reducing Iitaka's problem to the special case of Kodaira dimension zero.

In more recent work, Birkar studied Fano varieties and singularities of linear systems. He proved several fundamental problems such as Shokurov's conjecture on boundedness of complements and Borisov–Alexeev–Borisov conjecture on boundedness of Fano varieties. In 2018, Birkar was given the Fields Medal for his Fano varieties and his other contributions the minimal model problem.[9] In a video made available by the Simons Foundation, Birkar expressed hope that his Fields Medal will put “just a little smile on the lips” of the world's estimated 40 million Kurds. Birkar's Fields Medal was stolen on the same day it was awarded to him. In a special ceremony at ICM 2018, Birkar was presented with a replacement medal.

Birkar is also active in the field of birational geometry over fields of positive characteristic. His work together with work of Hacon-Xu nearly completes the minimal model program for 3-folds over fields of characteristic at least 7.

Awards and honours

• 2010 Leverhulme Prize in mathematics and statistics
• 2010 Prize of the Fondation Sciences Mathématiques de Paris
• 2016 AMS Moore Prize
• 2018 Fields Medal.
• 2019 Honorary Doctorate of the Salahuddin University, Erbil