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 detects the unknot (although they have other spectacular work since their previous ICM talks, such as the proof of Property (P)). 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, Publ. Math., Inst. Hautes Étud. Sci. 113, 97-208 (2011). ZBL1241.57017. MR2805599

The strategy of the proof is to consider (a modification of) the instanton 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 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 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 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.

Vincent Lafforgue's work span many topics and contain many striking results but the most probable recent work to be described in the spirit of the question is Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale. In slogan form, this work proves the Langlands correspondence in the "automorphic to Galois" direction for reductive group over global function field of positive characteristic.

More precisely, let $F$ be the function field of a smooth, projective, geometrically irreducible curve $X$ over $\mathbb F_{p}$, let $\mathbb A$ be its adele ring and let $G$ be a connected, reductive group over $F$ assumed split for simplicity. For $N$ a finite sub-scheme of $X$, denote by $K_N$ the compact, open subgroup of $G(\mathbb A)$ equal to the kernel of $G(\mathbb O)\rightarrow G(\mathcal O_N)$ where $\mathbb O$ is the product of the unit balls of the local fields $F_v$ and $\mathcal O_N$ is the ring of functions on $N$. Denote by $Z$ the center of $G$ and fix $\Xi$ a lattice inside $Z(F)\backslash Z(\mathbb A)$. Finally fix $E$ a finite extension of $\mathbb Q_\ell$ where $\ell\nmid p$.

Then the finite-dimensional $E$ vector space $$C^{\operatorname{cusp}}_c(G(F)\backslash G(\mathbb A)/K_N\Xi,E)$$ of cuspidal functions admits a direct-sum decomposition indexed by global Langlands parameter $$\sigma:\operatorname{Gal}(\bar{F}/F)\longrightarrow \widehat{G}(E)$$ with values in the Langlands dual group. This decomposition is compatible with the Satake isomorphism.

In addition to the result itself, the method he introduced (the so-called excursion operators) looks very promising even in the characteristic zero case.

Let's say something about what this method entails. The standard strategy, going back at least to Eichler-Shimura, for proving statements of this type is to study the cohomology groups of some space (in the classical case, a modular curve or Shimura variety, and in the function field case, a moduli space of Shtukas) and show that it admits a Galois action and an action of Hecke operators, and then show that the Hecke eigenspace is a Galois representation arising from a Langlands parameter $\sigma:\operatorname{Gal}(\bar{F}/F)\longrightarrow \widehat{G}(E)$ composed with some fixed representation of $\widehat{G}(E)$.

This works quite well when $G = GL_n$, and was used by Vincent's brother Laurent Lafforgue to great effect in that case. However, there are some difficulties for other groups. If we pick out the subspace of the cohomology corresponding to a particular automorphic form, it may be difficult to show that the Galois action factors through the group $\hat{G}$, or it might factor through the group $\hat{G}$ in multiple indistinguishable ways.

V. Lafforgue solves this by working simultaneously with the cohomology of a huge array of spaces, in which Langlands parameters are expected to appear via different representations. He defines several different maps between the cohomology of these different spaces. Composing these maps appropriately, he defines excursion operators on the original space $C^{cusp}_c$ of automorphic forms. These operators include the Hecke operators, but are not limited to them. They satisfy some relations, forming a ring, whose characters he checks by abstract group theory correspond to Langlands parameters. The compatibility with Satake can be established by comparing these operators to the classical Hecke operators.

Nalini Anantharaman is a french mathematician working in the fields of dynamical systems, partial differential equations and mathematical physics.

Her early works deal with the counting of closed geodesics on hyperbolic surfaces in a given homology class. She gave a full asymptotic expansion for the counting function following dynamical methods introduced by D. Dolgopyat, when the homology class lies in the interior of the set of winding cycles of the invariant measures of the geodesic flow. She also gave estimates when the homology class is in the boundary of this set. In that case, the problem is connected to the zero temperature limit in the theory of Markov processes and maximizing measures in the Aubry-Mather theory of Lagrangian systems.

She then got interested in semi-classical analysis and used entropy methods to study the weak limits of the sequence of probability measures $$ |\psi_k|^2 d\hbox{vol}$$ where $\psi_k$ are the eigenfunctions of the Laplacian defined on a negatively curved compact manifold. The quantum unique ergodicity conjecture asserts that the sequence should converge to the Liouville measure after a suitable lift of the measures to the unit tangent bundle of the manifold. She showed that any cluster points of the sequence must have positive entropy, thus ruling out a convergence to the Dirac mass on some closed orbit. So this is a remarkable application of ergodic theory to the study of the linear wave equation and Schrodinger equation. See a survey of P. Sarnak for additional details.

More recently, she studied related problems on billiards and regular graphs. Interesting pictures of cardioid billiards can be found in her joint work with Arnd Backer. Her webpage contains a few of her lectures in video format.