After receiving his Ph.D. from Princeton University in 2001, he was awarded the American Institute of Mathematics five-year fellowship.[4] He is a recipient of the Sloan fellowship (2004)[5] and of the Simons fellowship (2016).[6] He was awarded a Moscow Mathematical Society Prize (2001) and a Barcelona Prize in Dynamical Systems (2019).[7] In 2020 he received a gold medal from the International Consortium of Chinese Mathematics (ICCM).[8]
From 2006 to 2018 he was an editor of Inventiones mathematicae. He is a member of the editorial boards of Advances in Mathematics, Analysis & PDE, Revista Matemática Iberoamericana, and Ergodic Theory and Dynamical Systems.
In 2020 he was elected to Academia Europaea (the Academy of Europe). [13]
In 2023 he was elected to the European Academy of Sciences and Arts.
Recently he received the Frontier of Science Award [14]