The winners of the 2022 Breakthrough Prizes have been announced. There are eight recipients in mathematics: Takuro Mochizuki, Aaron Brown, Sebastian Hurtado Salazar, Jack Thorne, Jacob Tsimerman, Sarah Peluse, Hong Wang, and Yilin Wang.

The prizes and prize winners are listed below. The citations can be found on the Breakthrough Prize website. After the list, I have copied over some reviews of the work of the winners. Congratulations to the Breakthrough Prize Winners!

### 2022 Breakthrough Prize in Mathematics

**Takuro Mochizuki**, Kyoto University

### 2022 New Horizons in Mathematics Prize

**Aaron Brown**, Northwestern University

**Sebastian Hurtado Salazar**, University of Chicago**Jack Thorne**, University of Cambridge**Jacob Tsimerman**, University of Toronto

### 2022 Maryam Mirzakhani New Frontiers Prize

**Sarah Peluse**, Institute for Advanced Study and Princeton University (PhD Stanford University 2019)**Hong Wang**, University of California, Los Angeles (PhD MIT 2019)**Yilin Wang**, MIT (PhD ETH Zürich 2019)

### Some reviews from MathSciNet of the work of the Prize Winners

**MR2919903**

Mochizuki, Takuro (J-KYOT-R)

Wild harmonic bundles and wild pure twistor D-modules. (English, French summary)

Astérisque No. 340 (2011), x+607 pp. ISBN: 978-2-85629-332-4

14J60 (14F10 32L99)

This monograph provides a systematic analysis of the asymptotic behaviour of wild harmonic bundles on complex analytic manifolds. Important applications are then given to the study of the structure of (possibly irregular) flat meromorphic connections and to some open questions about $\scr{D}$-modules: among these problems is the complete proof of a stimulating conjecture of M. Kashiwara [in Topological field theory, primitive forms and related topics (Kyoto, 1996), 267–271, Progr. Math., 160, Birkhäuser Boston, Boston, MA, 1998; MR1653028] about an extension of the Hard Lefschetz Theorem and other nice properties from pure sheaves to semisimple $\scr{D}$-modules, which has drawn the attention also of notable researchers not necessarily working in the close vicinity of the subject [see, e.g., V. G. Drinfeld, Math. Res. Lett. 8 (2001), no. 5-6, 713–728; MR1879815]. Actually, Kashiwara’s conjecture appears even to be the original motivation under which the author started ten years ago a deep analysis of tame harmonic bundles, corresponding to (Fuchsian) regular differential equations or more generally to regular flat meromorphic connections “à la Deligne” [see T. Mochizuki, Geom. Topol. 13 (2009), no. 1, 359–455; MR2469521 and the references therein], and in this sense the present work is the ultimate step of the successful generalization of these results from the tame to the wild case.

As we said above, the consequences for the study of the structure of flat meromorphic connections (in particular the existence of resolutions of the bundle around the problematic “turning points”) are remarkable: for a more concrete and accessible presentation of these results we refer the interested reader also to the paper by the author in [J. Inst. Math. Jussieu 10 (2011), no. 3, 675–712; MR2806465]. It must be mentioned that similar conclusions were reached later by K. S. Kedlaya through different methods [see Duke Math. J. 154 (2010), no. 2, 343–418; MR2682186; J. Amer. Math. Soc. 24 (2011), no. 1, 183–229; MR2726603].

The importance of this extensive work is widely acknowledged in the current research literature, e.g. in the works concerning the recent substantial progresses in the irregular Riemann-Hilbert correspondence [see C. Sabbah, Introduction to Stokes structures, Lecture Notes in Math., 2060, Springer, Heidelberg, 2013; MR2978128; A. D’Agnolo and M. Kashiwara, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 10, 178–183; MR3004235; “Riemann-Hilbert correspondence for holonomic ${\scr D}$-modules”, preprint, arXiv:1311.2374].

Reviewed by Corrado Marastoni

**MR3702679**

Brown, Aaron (1-CHI-NDM); Rodriguez Hertz, Federico (1-PAS-NDM); Wang, Zhiren (1-PAS-NDM)

Global smooth and topological rigidity of hyperbolic lattice actions. (English summary)

Ann. of Math. (2) 186 (2017), no. 3, 913–972.

37C85 (37D20)

Let $G$ be a connected semi-simple Lie group with finite centre, no compact factors, and all almost-simple factors having real rank at least 2 and let $\Gamma<G$ be a lattice. The superrigidity theorem of G. A. Margulis shows that any linear representation of $\Gamma$ into ${\rm{PSL}}_d(\Bbb{R})$ extends, up to a compact error, to a continuous representation of $G$. R. J. Zimmer subsequently put forward a series of conjectures and questions related to representations into the group ${\rm{Diff}}^{\infty}(M)$ for a compact manifold $M$, based on the analogy between linear groups and diffeomorphism groups. This initiated what is now called the Zimmer program for understanding and classifying smooth actions by lattices of higher rank. This paper is a significant contribution to one line of enquiry in this program, studying the global rigidity of actions of lattices of higher rank on nilmanifolds under the hypothesis of hyperbolic linear data. Under some mild hypotheses a rather complete picture emerges of global rigidity phenomena in this setting. Using these results, the authors establish $C^{\infty}$ global rigidity for Anosov actions by uniform lattices, for Anosov actions of ${\rm{SL}}_n(\Bbb{Z})$ on $\Bbb{T}^n$ for $n\geqslant5$, and for probability-preserving actions of lattices of higher rank on nilmanifolds.

Reviewed by Thomas Ward

**MR3375524**

Hurtado, Sebastian (1-CA)

Continuity of discrete homomorphisms of diffeomorphism groups. (English summary)

Geom. Topol. 19 (2015), no. 4, 2117–2154.

57S05

This work is about the continuity of certain (discrete) homomorphisms between groups of diffeomorphisms of smooth manifolds and the classification of such homomorphisms when the manifolds involved are of the same dimension.

Let $M$ be a $C^\infty$ manifold and denote by $\mathrm{Diff}_c(M)$ its group of $C^\infty$ compactly supported diffeomorphisms isotopic to the identity endowed with the (metrizable) weak topology [see M. W. Hirsch, Differential topology, Springer, New York, 1976; MR0448362]. Let $d_{C^\infty}$ be a metric compatible with the weak topology. For any compact set $K\subseteq M$, let $\mathrm{Diff}_K(M)$ denote the group of diffeomorphisms in $\mathrm{Diff}_c(M)$ supported in $K$ with the induced topology. Let $N$ be another smooth manifold. A group homomorphism $\Phi\:\mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ is weakly continuous if for every compact set $K\subseteq M$, the restriction $\Phi_{\mathrm{Diff}_K(M)}$ of $\Phi$ to $\mathrm{Diff}_K(M)$ is continuous.

Based on a theorem by E. Militon [“Éléments de distorsion du groupe des difféomorphismes isotopes à l’identité d’une variété compacte”, preprint, arXiv:1005.1765] the author proves the following lemma:

Let $\Phi\:\mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ be a (discrete) group homomorphism. Let $K\subseteq M$ be compact and suppose $\{h_n\}_n$ is a sequence in $\mathrm{Diff}_K(M)$ such that $\mathrm{\lim}_{n\rightarrow\infty}\ d_{C^\infty}(h_n,{\rm Id})=0$. Then $\{\Phi(h_n)\}_n$ contains a subsequence converging to a diffeomorphism $H$, which is an isometry for a $C^\infty$ Riemannian metric on $N$.

Using the above result, the main theorem of the present work is shown, which asserts that any discrete group homomorphism $\Phi\:\mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ is weakly continuous.

Generalizing previous work of K. Mann [Ergodic Theory Dynam. Systems 35 (2015), no. 1, 192–214; MR3294298], the author proves that when $N$ is a closed manifold, $\dim(M)\geq\dim(N)$ and $\Phi \: \mathrm{Diff}_c(M)\longrightarrow\mathrm{Diff}_c(N)$ is a nontrivial homomorphism, then $M$ and $N$ are of the same dimension and $\Phi $ is “extended topologically diagonal”.

The paper includes some illuminating examples and finishes with some relevant questions and remarks.

Reviewed by Ricardo Berlanga Zubiaga

**MR3327536**

Thorne, Jack A. (4-CAMB-NDM)

Automorphy lifting for residually reducible $l$-adic Galois representations.

J. Amer. Math. Soc. 28 (2015), no. 3, 785–870.

11F80 (13D10)

This paper proves important modularity theorems for $n$-dimensional $\ell$-adic Galois representations over totally real fields, in some residually reducible cases.

Let $F$ be a CM field, and denote by $F^+$ its maximally totally real subfield. Define $G_F=\mathrm{Gal}(\overline F/F)$. If an automorphic representation $\pi$ of $\mathrm{GL}_n(\Bbb A_F)$ is regular, algebraic, essentially conjugate self-dual and cuspidal, then there exists a continuous semisimple representation $$ \rho(\pi)\:G_F\longrightarrow \mathrm{GL}_n(\overline{\Bbb Q}_\ell) $$ attached to $\pi$, uniquely characterized (up to isomorphism) by the collection of local data coming essentially from the local Langlands correspondence [see G. Chenevier and M. H. Harris, Camb. J. Math. 1 (2013), no. 1, 53–73; MR3272052; A. Caraiani, Algebra Number Theory 8 (2014), no. 7, 1597–1646; MR3272276; Duke Math. J. 161 (2012), no. 12, 2311–2413; MR2972460].

One says that a Galois representation $\rho\:G_F\rightarrow \mathrm{GL}_n(\overline{\Bbb Q}_\ell)$ is modular if it is isomorphic to $\rho(\pi)$ for some $\pi$.

Fix a continuous irreducible Galois representation $$ \rho\:G_F\longrightarrow \mathrm{GL}_n(\overline{\Bbb Q}_\ell). $$ Suppose that it is conjugate self-dual: if $c$ denotes the non-trivial element in $\mathrm{Gal}(F/F^+)$, this condition means that $\rho^c\simeq \rho^\vee\epsilon^{1-n}$ where $\epsilon$ is the $\ell$-adic cyclotomic character. We also assume that $\rho$ is de Rham with distinct Hodge-Tate weights. The paper addresses the modularity of such a representation.

One may find a Galois stable lattice in $\Bbb Q_\ell^n$, and realize $\rho$ as a continuous representation $$ \rho\:G_F\longrightarrow \mathrm{GL}_n(\mathcal O_K) $$ where $K$ is a finite extension of $\Bbb Q_\ell$ and $\mathcal O_K$ is its ring of algebraic integers. In particular, it makes sense to consider the residual representation $$ \overline\rho\:G_F\longrightarrow \mathrm{GL}_n(k) $$ where $k$ is the residue field of $K$.

A previous result of the author [J. Inst. Math. Jussieu 11 (2012), no. 4, 855–920; MR2979825] proves modularity results under the condition that the residual representation $\overline\rho$ is absolutely irreducible and adequate in the terminology of [J. A. Thorne, op. cit.]. The paper under review addresses the modularity problem in some cases when $\overline\rho$ is not absolutely irreducible. The main problem with non-irreducible residual representations is that the universal deformation ring may not exist, and therefore there is no hope to address $R=T$ theorems.

The cases treated in this paper are those when $\overline\rho$ is Schur. This technical condition is sufficient to guarantee the existence of universal deformation rings. This condition was first introduced in [L. Clozel, M. H. Harris and R. L. Taylor, Publ. Math. Inst. Hautes Études Sci. No. 108 (2008), 1–181; MR2470687]. The next problem is to apply the Taylor-Wiles method. Here a technical difficulty appears, since all relevant Galois cohomology computations require that the residual representation is absolutely irreducible. This problem is solved using an argument by C. M. Skinner and A. J. Wiles [Inst. Hautes Études Sci. Publ. Math. No. 89 (1999), 5–126 (2000); MR1793414]. The idea is to use Hida families and move from the residual representation $\overline\rho$ to an irreducible representation with coefficients in a one-dimensional quotient of the Iwasawa algebra, and then apply the usual arguments to a localization of the universal deformation ring $R$ at the dimension one prime corresponding to this representation. Note that to develop this method the author is forced to assume that $\rho$ is ordinary at primes dividing $\ell$. A third technical problem arises when one needs to show that the codimension of reducible Galois representations inside $\mathrm{Spec}(R)$ is large. In this case, there is no reason to expect such a property. For this, the author is led to assume another additional hypothesis to guarantee that the locus of reducible deformations is small. The condition he requires in the main result is that $\overline\rho$ admits a place $v$ at which the associated Weil-Deligne representation of the restriction of $\rho$ at $G_{F_v}$ corresponds under the local Langlands correspondence to a twist of the Steinberg representation.

Reviewed by Matteo Longo

**MR3744855**

Tsimerman, Jacob (3-TRNT-NDM)

The André-Oort conjecture for $\mathcal A_g$. (English summary)

Ann. of Math. (2) 187 (2018), no. 2, 379–390.

11G15 (11G18 14G35)

The author of this paper proves the following theorem: There exists $\delta_g > 0$ such that if $\Phi$ is a primitive CM type for a CM field $E$, and if $A$ is any $g$-dimensional abelian variety over $\overline{\Bbb{Q}}$ with endomorphism ring equal to the full ring of integers $\mathcal{O}_E$ and CM type $\Phi$, then the field of moduli $\Bbb{Q}(A)$ of $A$ satisfies $[\Bbb{Q}(A): \Bbb{Q}] > |{\rm Disc}(E)|^{\delta_g}$.

By a result of J. S. Pila and the author [Ann. of Math. (2) 179 (2014), no. 2, 659–681; MR3152943], this theorem implies the André-Oort conjecture for the coarse moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of fixed dimension $g \ge 1$. The André-Oort conjecture, as stated in this paper, asserts that an irreducible closed algebraic subvariety $V$ of a Shimura variety $S$ contains only finitely many maximal special subvarieties. Alternatively, every irreducible component of the Zariski closure of any set $\Sigma$ of special points in $S$ is a special subvariety of $S$ [see E. Ullmo and A. Yafaev, Ann. of Math. (2) 180 (2014), no. 3, 823–865; MR3245008]. In the latter cited paper and its sequel [B. Klingler and A. Yafaev, Ann. of Math. (2) 180 (2014), no. 3, 867–925; MR3245009], a conditional proof of the general conjecture was given which depends on the generalized Riemann hypothesis (GRH) for zeta functions of CM fields. The results of the paper under review are unconditional.

Reviewed by Patrick Morton

**MR4199235**

Peluse, Sarah (1-IASP-SM)

Bounds for sets with no polynomial progressions. (English summary)

Forum Math. Pi 8 (2020), e16, 55 pp.

11B30 (11B25)

This is a seriously impressive paper obtaining the first quantitative bounds for a large number of cases of the celebrated Bergelson-Leibman theorem [V. Bergelson and A. Leibman, J. Amer. Math. Soc. 9 (1996), no. 3, 725–753; MR1325795].

Recall that the Bergelson-Leibman theorem states the following: Let ${P_1,\dots,P_m\in\Bbb{Z}[X]}$ be polynomials with $P_i(0)=0$ for all $i$. Let $\alpha>0$. Then, provided that $N$ is sufficiently large in terms of $\alpha,P_1,\dots,P_m$, any set $A\subset\{1,\dots,N\}$ of cardinality at least $\alpha N$ contains a nontrivial configuration ${(x,x+P_1(d),\dots,x+P_m(d))}$. Note that the case $P_i(X)=iX$ is already Szemerédi’s theorem.

Bergelson and Leibman’s proof uses ergodic theory and does not lead to effective bounds (of any kind, even in principle). Finding such bounds is a major open problem in additive combinatorics. In the paper under review, the author finds the first “reasonable” bounds in the case that the $P_i$ have distinct degrees. She shows that in this case $A$ contains a configuration of the stated type provided that $|A|\ll N/(\log\log N)^c$, where $c=c_{P_1,\dots,P_m}$. For comparison, we remark that this is a bound of the same strength as that obtained by W. T. Gowers [Geom. Funct. Anal. 11 (2001), no. 3, 465–588; MR1844079] in his famous work on Szemerédi’s theorem, although it should be noted that the work under review does not directly extend that work since, in the Szemerédi case, the degrees of the $P_i$ are not distinct.

Previously, bounds of the strength the author obtains were known only when $m=1$ (where, in fact, better bounds are known, the state of the art being work of T. F. Bloom and J. Maynard [“A new upper bound for sets with no square difference”, preprint, arXiv:2011.13266]) and the case $P_1(X)=X$, $P_2(X)=X^2$, which was handled by the author and S. Prendiville [“Quantitative bounds in the nonlinear Roth theorem”, preprint, arXiv:1903.02592]. The present work, while it builds from that case, contains some substantial new innovations and the overall scheme of argument is vastly more complicated, though at its heart it remains a density-increment argument as with almost all quantitative bounds for problems of this kind over the integers. A key feature, critical in obtaining good bounds, is the use of comparatively “soft” arguments to avoid any need to invoke the inverse theory of Gowers norms or to introduce any discussion of nilsequences. For this to be possible the distinct degree condition is essential.

There seems little point in trying to sketch the argument here, not least because Section 3 of the paper does just that, providing in addition a stylish diagram illustrating the dependencies between various arguments. From Section 4 onwards, the technical details are at times quite formidable.

A number of intermediate results in the paper could be of independent interest. Foremost in this category is probably Theorem 3.5, a “quantitative concatenation” result for Gowers norms, which roughly speaking asserts that certain averages of box norms are controlled by Gowers norms. The author also highlights Lemma 5.1, a more technical result of a similar flavour, as being potentially portable elsewhere.

Whilst the bounds obtained in this paper are impressive compared to what went before, it is plausibly true that the true bound in all of the cases considered is polynomial! I am not aware of any counterexamples to such a possibility.

Reviewed by Ben Joseph Green

**MR4055179**

Guth, Larry (1-MIT); Iosevich, Alex (1-RCT); Ou, Yumeng (1-CUNY2); Wang, Hong (1-MIT)

On Falconer’s distance set problem in the plane. (English summary)

Invent. Math. 219 (2020), no. 3, 779–830.

42B20 (28A80)

Falconer’s distance problem is a famous and difficult problem in geometric measure theory. Roughly speaking it asks for the relationships between the dimensions of a Borel set $F \subseteq \Bbb{R}^d$ and the distance set of $F$, defined by $$ D(F) = \{ |x-y| : x,y \in F\}. $$ From now on all sets $F$ are Borel. There are various ways to formulate the problem precisely. One conjecture is that if $\dim_H F >d/2$, then $D(F)$ should have positive Lebesgue measure. Here $\dim_H$ denotes the Hausdorff dimension.

In his original paper [Mathematika 32 (1985), no. 2, 206–212 (1986); MR0834490], K. J. Falconer proved that $\dim_H F >d/2+1/2$ ensures that $D(F)$ has positive Lebesgue measure. T. H. Wolff [Internat. Math. Res. Notices 1999, no. 10, 547–567; MR1692851; addendum, J. Anal. Math. 88 (2002), 35–39; MR1979770] improved this in the plane, proving that $\dim_H F >1+1/3$ ensures that $D(F)$ has positive Lebesgue measure. The main result of the paper under review is a further improvement on this. Specifically, if $F$ is a planar Borel set with $\dim_H F >1+1/4$, then $D(F)$ has positive Lebesgue measure. The proof is long and technical with many new insights and techniques coming from harmonic analysis and geometric measure theory. The results also hold for pinned distance sets and distance sets where the distances are taken with respect to a norm which has a unit ball with a smooth boundary of non-vanishing Gaussian curvature.

The distance set problem has seen a lot of activity in the last few years—for example [T. Orponen, Adv. Math. 307 (2017), 1029–1045; MR3590535; T. Keleti and P. S. Shmerkin, Geom. Funct. Anal. 29 (2019), no. 6, 1886–1948; MR4034924] and various other works.

Reviewed by Jonathan MacDonald Fraser

**MR3959854**

Wang, Yilin (CH-ETHZ)

The energy of a deterministic Loewner chain: reversibility and interpretation via ${\rm SLE}_{0+}$. (English summary)

J. Eur. Math. Soc. (JEMS) 21 (2019), no. 7, 1915–1941.

30C55 (30C62 60J67)

The author studies some features of the energy of a deterministic Loewner chain. According to the chordal Loewner description, a simple curve $\gamma$ from 0 to infinity in the upper half-plane $\Bbb H=\{z\in\Bbb C\:{\rm Im}\,z>0\}$ is parameterized so that the conformal map $g_t$ from $\Bbb H\setminus\gamma[0,t]$ onto $\Bbb H$ with $g_t(z)=z+o(1)$ as $z\to\infty$ satisfies in fact $g_t(z)=z+2t/z+o(1/z)$ as $z\to\infty$. Extend $g_t$ continuously to the boundary point $\gamma_t$. The real-valued driving function $W_t := g_t(\gamma_t)$ of $\gamma$ is continuous. The Loewner energy is the Dirichlet energy of $W_t$ given by $$ I(\gamma):= \frac{1}{2}\int_0^{\infty}\left(\frac{dW_t}{dt}\right)^2 dt=\frac{1}{2}\int_0^{\infty}\left(\frac{d(g_t(\gamma_t))}{dt}\right)^2dt. $$ The scale invariance $I(u\gamma)=I(\gamma)$ for $u>0$ makes it possible to define the energy $I_{D,a,b}(\eta)$ of any simple curve $\eta$ from a boundary point $a$ of a simply connected domain $D$ to another boundary point $b$ to be the energy of the conformal image of $\eta$ via any uniformizing map $\Psi$ from $(D,a,b)$ to $\Bbb (H,0,\infty)$. For such $\eta$, define its time-reversal $\widehat\eta$ that has the same trace as $\eta$, but which is viewed as going from $b$ to $a$. The first main contribution of the paper is presented in the following theorem.

Main Theorem 1.1. The Loewner energy of the time-reversal $\widehat\eta$ of a simple curve $\eta$ from $a$ to $b$ in $D$ is equal to the Loewner energy of $\eta$: $I_{D,a,b}(\eta)=I_{D,b,a}(\widehat\eta)$.

When $\lambda\in C([0,\infty))$, consider the Loewner differential equation $$ \partial_tg_t(z)=\frac{2}{g_t(z)-\lambda_t}$$ with the initial condition $$ g_0(z)=z. $$ The chordal Loewner chain in $\Bbb H$ driven by $\lambda$ (or the Loewner transform of $\lambda$) is the increasing family $(K_t)_{t>0}$ defined by $K_t=\{z\in\Bbb H\:\tau(z)\leq t\}$, where $\tau(z)$ is the maximum survival time of the solution $g_t(z)$. Let $H\subset C[0,\infty)$ be the set of finite $I$ energy functions. The author proves the following statement.

Proposition 2.1. For every $\lambda\in H$, there exists $K=K(I(\lambda))$, depending only on $I(\lambda)$, such that the trace of the Loewner transform $\gamma$ of $\lambda$ is a $K$-quasiconformal curve.

The deterministic results in the paper are closely linked with the Schramm-Loewner Evolutions (SLE) theory. In the final section, the author establishes some connections with ideas from SLE restriction properties and SLE commutation relations.

Reviewed by Dmitri Valentinović Prokhorov