|Residence||Cambridge, United Kingdom|
|Fields||Higher-dimensional and birational algebraic geometry|
|Institutions||University of Cambridge|
|Doctoral advisor||Ivan Fesenko and Vyacheslav Shokurov|
|Known for||flips, minimal models, finite generation, pluricanonical systems, boundedness of Fano varieties, char p geometry|
|Notable awards||Leverhulme prize, Prize of the Fondation Sciences Mathématiques de Paris, AMS Moore Prize|
Caucher Birkar (Kurdish: کۆچەر بیرکار) is a Kurdish mathematician who is currently a professor at the University of Cambridge. 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)
Early years and education
Caucher Birkar was born in 1978 in Marivan, Kurdistan, Iran where he spent his school years. He studied mathematics at the University of Tehran where he received his bachelor's degree. Birkar then did his PhD at the University of Nottingham, after moving to the United Kingdom.
His main area of interest is algebraic geometry, in particular, higher dimensional birational geometry. He studied fundamental problems in the field such as minimal models, Fano varieties, singularities, and linear systems, proving various long-standing conjectures.
Birkar together with Paolo Cascini, Christopher Hacon and James McKernan 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. He 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 
- , some of his papers
- The Mathematics Genealogy Project Birkar on the Mathematics Genealogy Project
- Birkar on Journalogy
- C. Birkar, P. Cascini, C. Hacon, J. McKernan Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405-468.
- C. Birkar, Existence of log canonical flips and a special LMMP, Pub. Math. IHES 115 (2012), Issue 1, 325-368.
- C. Birkar, D.-Q. Zhang, Effectivity of Iitaka ﬁbrations and pluricanonical systems of polarized pairs. To appear in Pub. Math IHES.
- C. Birkar, Anti-pluricanonical systems on Fano varieties. arXiv:1603.05765
- C. Birkar, Singularities of linear systems and boundedness of Fano varieties. arXiv:1609.05543.
- C. Birkar, Existence of ﬂips and minimal models for 3-folds in char p. Annales scientiﬁques de l’ENS 49 (2016), 169-212.