Excellence in Supporting Teaching and Learning award. (2023), Invited speaker for ICM (2022), President's Awards for Excellence in Teaching (2017), Rector’s and FoNS Awards for Excellence in Teaching (2012), Senior Berwick Prize (2008), Whitehead Prize (2002), Smith Prize (1993)

New Horizons in Mathematics Prize (2023), Invited speaker for ICM (2022), Leverhulme Prize in Mathematics and Statistics (2020), Fellow of the American Mathematical Society (2020), European Mathematical Society prize winner (2020), Imperial President's award for Outstanding Early Career Research (2019), London Mathematical Society Whitehead Prize (2018)

Imperial President's Award for Outstanding Early Career Research (2019), American Mathematical Society Moore Prize for best research article (2018), Wilhelm T. Magnus Prize for Significant Contributions to Mathematical Sciences Courant Institute, New York University (2005)

Faculty of Natural Sciences Award for Excellence in Teaching, Imperial College London (2018)

Wolf Prize (2021), Wolf Prize in Mathematics (2020), Invited speaker - International Congress of Mathematics (2018), Breakthrough prize in Mathematics (2014), Royal Society Wolfson Research Merit Award (2012-2017), Queen's New Year's Honours List (2012), Fellow of the American Mathematical Society (2012), Shaw Prize Laureates in Mathematics (2009), Nemmers Prize (2008), King Faisal International Prize (2006), LMS Polya Prize (1999), Crafoord Prize (1994), Royal Medal (1992), Fields Medal (1986), Fellow of the Royal Society, Junior Whitehead Prize (1985)

Boris Dubrovin Medal (2024), Adams Prize (2024), LMS Whitehead Prize (2023)

Royal Society Wolfson Research Merit Award (2016-2020), America Mathematical Society Fellows (2014), Philip Leverhulme Prize (2012), London Mathematical Society Whitehead Prize (2012)

2022 foreign member at the Chinese Academy of Sciences , 2022 King Faisal Prize in Science, 2021 Breakthrough Prize in Mathematics, Invited speaker - European Congress of Mathematics (2020), Substantive Knight Commander of the Order of the British Empire (2019), Honorary Knight Commander of the Order of the British Empire (2016), Fellow of the American Mathematical Society (2015), Member of the Austrian Academy of Sciences (2015), Member of the Academy of Sciences Leopoldina (2015), Elected Fellow of the Royal Society (2014), Fröhlich Prize (2014), Fields Medal (2014), International Congress of Mathematics (2014), Fermat Prize (2013), Leverhulme Research Leadership Award (2012), Royal Society Wolfson Research Merit Award (2009-2014), Whitehead Prize (2008), Philip Leverhulme Prize (2008), Editors' Choice Award, Macworld (2007)

Fellow of the European Academy of Sciences (2020), Invited speaker - International Congress of Mathematical Physics (2018), Elected to the Royal Swedish Academy of Sciences (2012), Royal Society Wolfson Research Merit Award (2007-2012), President of the European Mathematical Society (2007 -2010)

LMS Whitehead Prize (2023)

London Mathematical Society Polya Prize (2020), Fellow of the American Mathematical Society (2020), President's award for Leadership in Culture & Community (2019), Simons Research Professor, MSRI (2018), Simons Research Rector's Medal for Outstanding Contribution to Teaching Excellence (2010), Johnson University Prize (1979), Senior Mathematical Prize (1979)

John Von Neumann Visiting Professorship, Technical University Munich (2020), The Ferran Sunyer I Balaguer Prize (2017)

The André Lichnerowicz Prize in Poisson Geometry (2016), Fellow of the American Institute of Mathematics (2008 - 2013)

LMS Frohlich Prize (2022), Member of the Academia Europaea (2019), Fellow of the American Mathematical Society (2018), Fellow of the Royal Society (2015), Royal Society Wolfson Research Merit Award (2010-2015), Invited Speaker - International Congress of Mathematicians in Hyderabad (2010), Philip Leverhulme Prize (2004), Whitehead Prize (2004)