Caucher Birkar (Kurdish: کۆچەر بیرکار) is a Kurdish mathematician and a British citizen. He is a professor at the University of Cambridge. Birkar is a main 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)
Birkar was born in 1978 in Marivan, Kurdistan Province, Iran where he spent his school years. He studied mathematics at the University of Tehran where he received his bachelor's degree. After moving to the UK, in 2001–2004 Birkar was a PhD student at the University of Nottingham. In 2003 he was awarded the London Mathematical Society Prize as the most promising PhD student.
Birkar's main area of interest is algebraic geometry, in particular, higher dimensional birational geometry. He studied fundamental aspects of key problems in modern mathematics such as minimal models, Fano varieties, singularities, and linear systems. His theories provided solutions of various long-standing conjectures.
Together with Paolo Cascini, Christopher Hacon and James McKernan he settled several important 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. He also showed that the minimal model conjecture follows from the abundance conjecture and established links between the former conjecture and various other notions such as log canonical thresholds and Zariski decompositions.
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 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. This direction is viewed to have fundamental applications in birational geometry. Birkar answered a question of Gang Tian on alpha-invariants and answered a question of Jean-Pierre Serre on Jordan property of Cremona groups building on the work of Yuri Prokhorov and Constantin Shramov.
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.Leverhulme Prize in mathematics and statistics for "his outstanding contributions to fundamental research in algebraic geometry"
Prize of the Fondation Sciences Mathématiques de Paris
AMS Moore Prize