The next International Congress of Mathematicians (ICM) will be next year in Rio de Janeiro, Brazil. The present question is the 2018 version of similar questions from [2014](https://mathoverflow.net/questions/145065/work-of-plenary-speakers-at-icm-2014) and [2010](https://mathoverflow.net/questions/29485/work-of-icm-2010-plenary-speakers-and-other-humans/). Can you, please, for the benefit of others give a short description of the work of one of the plenary speakers?


List of [plenary speakers at ICM 2018](http://www.icm2018.org/portal/en/icm-speakers):


1. [Alex Lubotzky](http://www.ma.huji.ac.il/~alexlub/) (Israel)
2. [Andrei Okounkov](http://www.math.columbia.edu/~okounkov/) (Russia/USA)
3. [Assaf Naor](https://web.math.princeton.edu/~naor/) (USA)
4. [Carlos Gustavo Moreira](http://w3.impa.br/~gugu/) (Brazil)
5. [Catherine Goldstein](https://webusers.imj-prg.fr/~catherine.goldstein/) (France)
6. [Christian Lubich](https://na.uni-tuebingen.de/~lubich/) (Germany)
7. [Geordie Williamson](http://www.maths.usyd.edu.au/u/geordie/) (Australia/Germany)
8. [Gil Kalai](http://www.ma.huji.ac.il/~kalai/) (Israel)
9. [Greg Lawler](http://math.uchicago.edu/~lawler/) (USA)
10. [Lai-Sang Young](http://www.cims.nyu.edu/~lsy/) (USA)
11. [Luigi Ambrosio](http://cvgmt.sns.it/person/3/) (Italy)
12. [Michael Jordan](https://www.stat.berkeley.edu/~jordan/) (USA)
13. [Nalini Anantharaman](http://www-irma.u-strasbg.fr/~anantharaman/) (France)
14. [Peter Kronheimer](http://www.math.harvard.edu/~kronheim/) (USA) and [Tom Mrowka](http://www-math.mit.edu/~mrowka/) (USA)
15. [Peter Scholze](http://www.math.uni-bonn.de/people/scholze/) (Germany)
16. [Rahul Pandharipande](https://people.math.ethz.ch/~rahul/) (Switzerland)
17. [Ronald Coifman](http://cpsc.yale.edu/people/ronald-coifman) (USA)
18. [Sanjeev Arora](https://www.cs.princeton.edu/~arora/) (USA)
19. [Simon Donaldson](http://wwwf.imperial.ac.uk/~skdona/) (UK/USA)
20. [Sylvia Serfaty](https://www.math.nyu.edu/~serfaty/) (France/USA)
21. [Vincent Lafforgue](http://vlafforg.perso.math.cnrs.fr/) (France)



Work of plenary speakers at ICM 2018
Kronheimer and Mrowka have both spoken at the ICM before. Most likely, the current invitation is based on their proof that [Khovanov homology](https://en.wikipedia.org/wiki/Khovanov_homology) detects the unknot (although they have other spectacular work since their previous ICM talks, such as the [proof of Property (P)](http://www.ams.org/mathscinet-getitem?mr=2023280)). The corresponding question for the Jones polynomial is a well-known open problem. 


*Kronheimer, P.B.; Mrowka, T.S.*, [**Khovanov homology is an unknot-detector**](http://dx.doi.org/10.1007/s10240-010-0030-y), Publ. Math., Inst. Hautes Étud. Sci. 113, 97-208 (2011). [ZBL1241.57017](https://zbmath.org/?q=an:1241.57017). [MR2805599](http://www.ams.org/mathscinet-getitem?mr=2805599) 


The strategy of the proof is to consider (a modification of) the [instanton](http://www.ams.org/mathscinet-getitem?mr=2860345) [Floer homology](https://en.wikipedia.org/wiki/Floer_homology) invariant of knots which (roughly) counts representations of the knot group to $SU(2)$ in which the meridian has trace $=0$. They show that this invariant is always non-trivial for non-trivial knots. This mimics a similar proof of non-triviality for knot Floer homology (defined by Rasmussen and Oszvath-Zsabo) by Juhasz, who [showed that](http://www.ams.org/mathscinet-getitem?mr=2390347) the highest grading of the knot Floer homology is sutured Floer homology of the complement of a minimal genus Seifert surface in the knot complement (using an adjunction inequality). Kronheimer and Mrowka had previously formulated an [instanton version of sutured Floer homology](http://www.ams.org/mathscinet-getitem?mr=2652464) so that they could mimic Juhasz's proof. 


Then they show that there is a spectral sequence going from Khovanov homology to their knot instanton homology, and hence the Khovanov homology of non-trivial knots has rank at least 2. This part of the proof was modeled on a [spectral sequence that Oszvath-Zsabo](http://www.ams.org/mathscinet-getitem?mr=2141852) found from Khovanov homology to the Heegaard-Floer homology of the double branched cover. The proof of the existence of this spectral sequence is based on the TQFT-like properties of the instanton knot invariant for cobordisms between knots by surfaces in 4-manifolds, which they develop further in this paper. 


