{"id":3285,"date":"2021-09-13T19:53:06","date_gmt":"2021-09-13T23:53:06","guid":{"rendered":"https:\/\/blogs.ams.org\/beyondreviews\/?p=3285"},"modified":"2021-09-13T19:53:06","modified_gmt":"2021-09-13T23:53:06","slug":"winners-of-the-2022-breakthrough-prizes-in-mathematics","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/beyondreviews\/2021\/09\/13\/winners-of-the-2022-breakthrough-prizes-in-mathematics\/","title":{"rendered":"Winners of the 2022 Breakthrough Prizes in Mathematics"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-full wp-image-2252\" src=\"http:\/\/blogs.ams.org\/beyondreviews\/files\/2018\/10\/Breakthrough-Prize-Logo-bded.png\" alt=\"Breakthrough Prize Logo\" width=\"167\" height=\"142\" \/>The\u00a0<a href=\"https:\/\/breakthroughprize.org\/News\/65\">winners of the 2022 Breakthrough Prizes<\/a> have been announced.\u00a0 There are eight recipients in mathematics:\u00a0 Takuro Mochizuki, Aaron Brown, Sebastian Hurtado Salazar, Jack Thorne, Jacob Tsimerman, Sarah Peluse, Hong Wang, and Yilin Wang.<\/p>\n<p><!--more--><\/p>\n<p>The prizes and prize winners are listed below.\u00a0 The citations can be found on the <a href=\"https:\/\/breakthroughprize.org\/\">Breakthrough Prize website<\/a>.\u00a0 After the list, I have copied over some reviews of the work of the winners.\u00a0 \u00a0Congratulations to the Breakthrough Prize Winners!<\/p>\n<hr \/>\n<h3>2022 Breakthrough Prize in Mathematics<\/h3>\n<ul>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/677716\"><strong>Takuro Mochizuki<\/strong><\/a>, Kyoto University<\/li>\n<\/ul>\n<h3>2022 New Horizons in Mathematics Prize<\/h3>\n<ul>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/912945\"><strong>Aaron Brown<\/strong><\/a>, Northwestern University<br \/>\n<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/1120207\"><strong>Sebastian Hurtado Salazar<\/strong><\/a>, University of Chicago<\/li>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/904689\"><strong>Jack Thorne<\/strong><\/a>, University of Cambridge<\/li>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/896479\"><strong>Jacob Tsimerman<\/strong><\/a>, University of Toronto<\/li>\n<\/ul>\n<h3>2022 Maryam Mirzakhani New Frontiers Prize<\/h3>\n<ul>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/1035915\"><strong>Sarah Peluse<\/strong><\/a>, Institute for Advanced Study and Princeton University (PhD Stanford University 2019)<\/li>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/1250315\"><strong>Hong Wang<\/strong><\/a>, University of California, Los Angeles (PhD MIT 2019)<\/li>\n<li><a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/MRAuthorID\/1310851\"><strong>Yilin Wang<\/strong><\/a>, MIT (PhD ETH Z\u00fcrich 2019)<\/li>\n<\/ul>\n<hr \/>\n<h3>Some reviews from <a href=\"https:\/\/mathscinet.ams.org\/mathscinet\">MathSciNet<\/a> of the work of the Prize Winners<\/h3>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=2919903\"><strong>MR2919903<\/strong><\/a><br \/>\nMochizuki, Takuro (J-KYOT-R)<br \/>\nWild harmonic bundles and wild pure twistor D-modules. (English, French summary)<br \/>\nAst\u00e9risque No. 340 (2011), x+607 pp. ISBN: 978-2-85629-332-4<br \/>\n14J60 (14F10 32L99)<\/p>\n<p class=\"review\">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\u00a0<span class=\"MathTeX\">$\\scr{D}$<\/span>-modules: among these problems is the complete proof of a stimulating conjecture of M. Kashiwara [in\u00a0<span class=\"it\">Topological field theory, primitive forms and related topics (Kyoto, 1996)<\/span>, 267\u2013271, Progr. Math., 160, Birkh\u00e4user Boston, Boston, MA, 1998;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1653028&amp;loc=fromrevtext\">MR1653028<\/a>] about an extension of the Hard Lefschetz Theorem and other nice properties from pure sheaves to semisimple\u00a0<span class=\"MathTeX\">$\\scr{D}$<\/span>-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.\u00a0<span class=\"bf\">8<\/span>\u00a0(2001), no. 5-6, 713\u2013728;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1879815&amp;loc=fromrevtext\">MR1879815<\/a>]. Actually, Kashiwara&#8217;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 &#8220;\u00e0 la Deligne&#8221; [see T. Mochizuki, Geom. Topol.\u00a0<span class=\"bf\">13<\/span>\u00a0(2009), no. 1, 359\u2013455;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=2469521&amp;loc=fromrevtext\">MR2469521<\/a>\u00a0and 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.<\/p>\n<p class=\"review\">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 &#8220;turning points&#8221;) 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 <span class=\"bf\">10<\/span>\u00a0(2011), no. 3, 675\u2013712;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=2806465&amp;loc=fromrevtext\">MR2806465<\/a>]. It must be mentioned that similar conclusions were reached later by K. S. Kedlaya through different methods [see Duke Math. J.\u00a0<span class=\"bf\">154<\/span>\u00a0(2010), no. 2, 343\u2013418;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=2682186&amp;loc=fromrevtext\">MR2682186<\/a>; J. Amer. Math. Soc.\u00a0<span class=\"bf\">24<\/span>\u00a0(2011), no. 1, 183\u2013229;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=2726603&amp;loc=fromrevtext\">MR2726603<\/a>].<\/p>\n<p class=\"review\">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, <span class=\"it\">Introduction to Stokes structures<\/span>, Lecture Notes in Math., 2060, Springer, Heidelberg, 2013;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?pg1=MR&amp;s1=2978128&amp;loc=fromrevtext\">MR2978128<\/a>; A. D&#8217;Agnolo and M. Kashiwara, Proc. Japan Acad. Ser. A Math. Sci.\u00a0<span class=\"bf\">88<\/span>\u00a0(2012), no. 10, 178\u2013183;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?pg1=MR&amp;s1=3004235&amp;loc=fromrevtext\">MR3004235<\/a>; &#8220;Riemann-Hilbert correspondence for holonomic\u00a0<span class=\"MathTeX\">${\\scr D}$<\/span>-modules&#8221;, preprint,\u00a0<a href=\"https:\/\/mathscinet.ams.org\/leavingmsn?url=http:\/\/arxiv.org\/abs\/1311.2374&amp;from=url\" target=\"NEW\" rel=\"noopener\">arXiv:1311.2374<\/a>].<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=320739\">Corrado Marastoni<\/a><\/span><\/p>\n<hr \/>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3702679\"><strong>MR3702679<\/strong><\/a><br \/>\nBrown, Aaron (1-CHI-NDM); Rodriguez Hertz, Federico (1-PAS-NDM); Wang, Zhiren (1-PAS-NDM)<br \/>\nGlobal smooth and topological rigidity of hyperbolic lattice actions. (English summary)<br \/>\nAnn. of Math. (2) 186 (2017), no. 3, 913\u2013972.<br \/>\n37C85 (37D20)<\/p>\n<p class=\"review\">Let\u00a0<span class=\"MathTeX\">$G$<\/span>\u00a0be 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\u00a0<span class=\"MathTeX\">$\\Gamma&lt;G$<\/span>\u00a0be a lattice. The superrigidity theorem of G. A. Margulis shows that any linear representation of\u00a0<span class=\"MathTeX\">$\\Gamma$<\/span>\u00a0into\u00a0<span class=\"MathTeX\">${\\rm{PSL}}_d(\\Bbb{R})$<\/span>\u00a0extends, up to a compact error, to a continuous representation of\u00a0<span class=\"MathTeX\">$G$<\/span>. R. J. Zimmer subsequently put forward a series of conjectures and questions related to representations into the group\u00a0<span class=\"MathTeX\">${\\rm{Diff}}^{\\infty}(M)$<\/span>\u00a0for a compact manifold\u00a0<span class=\"MathTeX\">$M$<\/span>, 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\u00a0<span class=\"MathTeX\">$C^{\\infty}$<\/span>\u00a0global rigidity for Anosov actions by uniform lattices, for Anosov actions of\u00a0<span class=\"MathTeX\">${\\rm{SL}}_n(\\Bbb{Z})$<\/span>\u00a0on\u00a0<span class=\"MathTeX\">$\\Bbb{T}^n$<\/span>\u00a0for\u00a0<span class=\"MathTeX\">$n\\geqslant5$<\/span>, and for probability-preserving actions of lattices of higher rank on nilmanifolds.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=180610\">Thomas Ward<\/a><\/span><\/p>\n<hr \/>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3375524\"><strong>MR3375524<\/strong><\/a><br \/>\nHurtado, Sebastian (1-CA)<br \/>\nContinuity of discrete homomorphisms of diffeomorphism groups. (English summary)<br \/>\nGeom. Topol. 19 (2015), no. 4, 2117\u20132154.<br \/>\n57S05<\/p>\n<p class=\"review\">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.<\/p>\n<p class=\"review\">Let\u00a0<span class=\"MathTeX\">$M$<\/span>\u00a0be a\u00a0<span class=\"MathTeX\">$C^\\infty$<\/span>\u00a0manifold and denote by\u00a0<span class=\"MathTeX\">$\\mathrm{Diff}_c(M)$<\/span>\u00a0its group of\u00a0<span class=\"MathTeX\">$C^\\infty$<\/span>\u00a0compactly supported diffeomorphisms isotopic to the identity endowed with the (metrizable) weak topology [see M. W. Hirsch,\u00a0<span class=\"it\">Differential topology<\/span>, Springer, New York, 1976;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=448362&amp;loc=fromrevtext\">MR0448362<\/a>]. Let\u00a0<span class=\"MathTeX\">$d_{C^\\infty}$<\/span>\u00a0be a metric compatible with the weak topology. For any compact set\u00a0<span class=\"MathTeX\">$K\\subseteq M$<\/span>, let\u00a0<span class=\"MathTeX\">$\\mathrm{Diff}_K(M)$<\/span>\u00a0denote the group of diffeomorphisms in\u00a0<span class=\"MathTeX\">$\\mathrm{Diff}_c(M)$<\/span>\u00a0supported in\u00a0<span class=\"MathTeX\">$K$<\/span>\u00a0with the induced topology. Let\u00a0<span class=\"MathTeX\">$N$<\/span>\u00a0be another smooth manifold. A group homomorphism\u00a0<span class=\"MathTeX\">$\\Phi\\:\\mathrm{Diff}_c(M)\\longrightarrow\\mathrm{Diff}_c(N)$<\/span>\u00a0is weakly continuous if for every compact set\u00a0<span class=\"MathTeX\">$K\\subseteq M$<\/span>, the restriction\u00a0<span class=\"MathTeX\">$\\Phi_{\\mathrm{Diff}_K(M)}$<\/span>\u00a0of\u00a0<span class=\"MathTeX\">$\\Phi$<\/span>\u00a0to\u00a0<span class=\"MathTeX\">$\\mathrm{Diff}_K(M)$<\/span> is continuous.<\/p>\n<p class=\"review\">Based on a theorem by E. Militon [&#8220;\u00c9l\u00e9ments de distorsion du groupe des diff\u00e9omorphismes isotopes \u00e0 l&#8217;identit\u00e9 d&#8217;une vari\u00e9t\u00e9 compacte&#8221;, preprint,\u00a0<a href=\"https:\/\/mathscinet.ams.org\/leavingmsn?url=http:\/\/arxiv.org\/abs\/1005.1765&amp;from=url\" target=\"NEW\" rel=\"noopener\">arXiv:1005.1765<\/a>] the author proves the following lemma:<\/p>\n<p class=\"review\">Let\u00a0<span class=\"MathTeX\">$\\Phi\\:\\mathrm{Diff}_c(M)\\longrightarrow\\mathrm{Diff}_c(N)$<\/span>\u00a0be a (discrete) group homomorphism. Let\u00a0<span class=\"MathTeX\">$K\\subseteq M$<\/span>\u00a0be compact and suppose\u00a0<span class=\"MathTeX\">$\\{h_n\\}_n$<\/span>\u00a0is a sequence in\u00a0<span class=\"MathTeX\">$\\mathrm{Diff}_K(M)$<\/span>\u00a0such that\u00a0<span class=\"MathTeX\">$\\mathrm{\\lim}_{n\\rightarrow\\infty}\\ d_{C^\\infty}(h_n,{\\rm Id})=0$<\/span>. Then\u00a0<span class=\"MathTeX\">$\\{\\Phi(h_n)\\}_n$<\/span>\u00a0contains a subsequence converging to a diffeomorphism\u00a0<span class=\"MathTeX\">$H$<\/span>, which is an isometry for a\u00a0<span class=\"MathTeX\">$C^\\infty$<\/span>\u00a0Riemannian metric on\u00a0<span class=\"MathTeX\">$N$<\/span>.<\/p>\n<p class=\"review\">Using the above result, the main theorem of the present work is shown, which asserts that any discrete group homomorphism\u00a0<span class=\"MathTeX\">$\\Phi\\:\\mathrm{Diff}_c(M)\\longrightarrow\\mathrm{Diff}_c(N)$<\/span> is weakly continuous.<\/p>\n<p class=\"review\">Generalizing previous work of K. Mann [Ergodic Theory Dynam. Systems\u00a0<span class=\"bf\">35<\/span>\u00a0(2015), no. 1, 192\u2013214;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=3294298&amp;loc=fromrevtext\">MR3294298<\/a>], the author proves that when\u00a0<span class=\"MathTeX\">$N$<\/span>\u00a0is a closed manifold,\u00a0<span class=\"MathTeX\">$\\dim(M)\\geq\\dim(N)$<\/span>\u00a0and\u00a0<span class=\"MathTeX\">$\\Phi \\: \\mathrm{Diff}_c(M)\\longrightarrow\\mathrm{Diff}_c(N)$<\/span>\u00a0is a nontrivial homomorphism, then\u00a0<span class=\"MathTeX\">$M$<\/span>\u00a0and\u00a0<span class=\"MathTeX\">$N$<\/span>\u00a0are of the same dimension and\u00a0<span class=\"MathTeX\">$\\Phi $<\/span> is &#8220;extended topologically diagonal&#8221;.<\/p>\n<p class=\"review\">The paper includes some illuminating examples and finishes with some relevant questions and remarks.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=644546\">Ricardo Berlanga Zubiaga<\/a><\/span><\/p>\n<p>&nbsp;<\/p>\n<hr \/>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3327536\"><strong>MR3327536<\/strong><\/a><br \/>\nThorne, Jack A. (4-CAMB-NDM)<br \/>\nAutomorphy lifting for residually reducible $l$-adic Galois representations.<br \/>\nJ. Amer. Math. Soc. 28 (2015), no. 3, 785\u2013870.<br \/>\n11F80 (13D10)<\/p>\n<p class=\"review\">This paper proves important modularity theorems for\u00a0<span class=\"MathTeX\">$n$<\/span>-dimensional\u00a0<span class=\"MathTeX\">$\\ell$<\/span>-adic Galois representations over totally real fields, in some residually reducible cases.<\/p>\n<p class=\"review\">Let <span class=\"MathTeX\">$F$<\/span>\u00a0be a CM field, and denote by\u00a0<span class=\"MathTeX\">$F^+$<\/span>\u00a0its maximally totally real subfield. Define\u00a0<span class=\"MathTeX\">$G_F=\\mathrm{Gal}(\\overline F\/F)$<\/span>. If an automorphic representation\u00a0<span class=\"MathTeX\">$\\pi$<\/span>\u00a0of\u00a0<span class=\"MathTeX\">$\\mathrm{GL}_n(\\Bbb A_F)$<\/span>\u00a0is regular, algebraic, essentially conjugate self-dual and cuspidal, then there exists a continuous semisimple representation\u00a0<span class=\"MathTeX\">$$ \\rho(\\pi)\\:G_F\\longrightarrow \\mathrm{GL}_n(\\overline{\\Bbb Q}_\\ell) $$<\/span>\u00a0attached to\u00a0<span class=\"MathTeX\">$\\pi$<\/span>, 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.\u00a0<span class=\"bf\">1<\/span>\u00a0(2013), no. 1, 53\u201373;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?pg1=MR&amp;s1=3272052&amp;loc=fromrevtext\">MR3272052<\/a>; A. Caraiani, Algebra Number Theory\u00a0<span class=\"bf\">8<\/span>\u00a0(2014), no. 7, 1597\u20131646;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?pg1=MR&amp;s1=3272276&amp;loc=fromrevtext\">MR3272276<\/a>; Duke Math. J.\u00a0<span class=\"bf\">161<\/span>\u00a0(2012), no. 12, 2311\u20132413;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?pg1=MR&amp;s1=2972460&amp;loc=fromrevtext\">MR2972460<\/a>].<\/p>\n<p class=\"review\">One says that a Galois representation\u00a0<span class=\"MathTeX\">$\\rho\\:G_F\\rightarrow \\mathrm{GL}_n(\\overline{\\Bbb Q}_\\ell)$<\/span>\u00a0is\u00a0<span class=\"it\">modular<\/span>\u00a0if it is isomorphic to\u00a0<span class=\"MathTeX\">$\\rho(\\pi)$<\/span>\u00a0for some\u00a0<span class=\"MathTeX\">$\\pi$<\/span>.<\/p>\n<p class=\"review\">Fix a continuous irreducible Galois representation\u00a0<span class=\"MathTeX\">$$ \\rho\\:G_F\\longrightarrow \\mathrm{GL}_n(\\overline{\\Bbb Q}_\\ell). $$<\/span>\u00a0Suppose that it is conjugate self-dual: if\u00a0<span class=\"MathTeX\">$c$<\/span>\u00a0denotes the non-trivial element in\u00a0<span class=\"MathTeX\">$\\mathrm{Gal}(F\/F^+)$<\/span>, this condition means that\u00a0<span class=\"MathTeX\">$\\rho^c\\simeq \\rho^\\vee\\epsilon^{1-n}$<\/span>\u00a0where\u00a0<span class=\"MathTeX\">$\\epsilon$<\/span>\u00a0is the\u00a0<span class=\"MathTeX\">$\\ell$<\/span>-adic cyclotomic character. We also assume that\u00a0<span class=\"MathTeX\">$\\rho$<\/span> is de Rham with distinct Hodge-Tate weights. The paper addresses the modularity of such a representation.<\/p>\n<p class=\"review\">One may find a Galois stable lattice in\u00a0<span class=\"MathTeX\">$\\Bbb Q_\\ell^n$<\/span>, and realize\u00a0<span class=\"MathTeX\">$\\rho$<\/span>\u00a0as a continuous representation\u00a0<span class=\"MathTeX\">$$ \\rho\\:G_F\\longrightarrow \\mathrm{GL}_n(\\mathcal O_K) $$<\/span>\u00a0where\u00a0<span class=\"MathTeX\">$K$<\/span>\u00a0is a finite extension of\u00a0<span class=\"MathTeX\">$\\Bbb Q_\\ell$<\/span>\u00a0and\u00a0<span class=\"MathTeX\">$\\mathcal O_K$<\/span>\u00a0is its ring of algebraic integers. In particular, it makes sense to consider the residual representation\u00a0<span class=\"MathTeX\">$$ \\overline\\rho\\:G_F\\longrightarrow \\mathrm{GL}_n(k) $$<\/span>\u00a0where\u00a0<span class=\"MathTeX\">$k$<\/span>\u00a0is the residue field of\u00a0<span class=\"MathTeX\">$K$<\/span>.<\/p>\n<p class=\"review\">A previous result of the author [J. Inst. Math. Jussieu\u00a0<span class=\"bf\">11<\/span>\u00a0(2012), no. 4, 855\u2013920;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?pg1=MR&amp;s1=2979825&amp;loc=fromrevtext\">MR2979825<\/a>] proves modularity results under the condition that the residual representation\u00a0<span class=\"MathTeX\">$\\overline\\rho$<\/span>\u00a0is absolutely irreducible and\u00a0<span class=\"it\">adequate<\/span>\u00a0in the terminology of [J. A. Thorne, op. cit.]. The paper under review addresses the modularity problem in some cases when\u00a0<span class=\"MathTeX\">$\\overline\\rho$<\/span>\u00a0is\u00a0<span class=\"it\">not absolutely irreducible<\/span>. 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\u00a0<span class=\"MathTeX\">$R=T$<\/span> theorems.<\/p>\n<p class=\"review\">The cases treated in this paper are those when\u00a0<span class=\"MathTeX\">$\\overline\\rho$<\/span>\u00a0is\u00a0<span class=\"it\">Schur<\/span>. 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 \u00c9tudes Sci. No. 108 (2008), 1\u2013181;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=2470687&amp;loc=fromrevtext\">MR2470687<\/a>]. 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 \u00c9tudes Sci. Publ. Math. No. 89 (1999), 5\u2013126 (2000);\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1793414&amp;loc=fromrevtext\">MR1793414<\/a>]. The idea is to use Hida families and move from the residual representation\u00a0<span class=\"MathTeX\">$\\overline\\rho$<\/span>\u00a0to 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\u00a0<span class=\"MathTeX\">$R$<\/span>\u00a0at the dimension one prime corresponding to this representation. Note that to develop this method the author is forced to assume that\u00a0<span class=\"MathTeX\">$\\rho$<\/span>\u00a0is\u00a0<span class=\"it\">ordinary<\/span>\u00a0at primes dividing\u00a0<span class=\"MathTeX\">$\\ell$<\/span>. A third technical problem arises when one needs to show that the codimension of reducible Galois representations inside\u00a0<span class=\"MathTeX\">$\\mathrm{Spec}(R)$<\/span>\u00a0is 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\u00a0<span class=\"it\">the locus of reducible deformations is small<\/span>. The condition he requires in the main result is that\u00a0<span class=\"MathTeX\">$\\overline\\rho$<\/span>\u00a0admits a place\u00a0<span class=\"MathTeX\">$v$<\/span>\u00a0at which the associated Weil-Deligne representation of the restriction of\u00a0<span class=\"MathTeX\">$\\rho$<\/span>\u00a0at\u00a0<span class=\"MathTeX\">$G_{F_v}$<\/span>\u00a0corresponds under the local Langlands correspondence to a twist of the Steinberg representation.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=790759\">Matteo Longo<\/a><\/span><\/p>\n<hr \/>\n<p class=\"reflist \"><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3744855\"><strong>MR3744855<\/strong><\/a><br \/>\nTsimerman, Jacob (3-TRNT-NDM)<br \/>\nThe Andr\u00e9-Oort conjecture for $\\mathcal A_g$. (English summary)<br \/>\nAnn. of Math. (2) 187 (2018), no. 2, 379\u2013390.<br \/>\n11G15 (11G18 14G35)<\/p>\n<p class=\"review\">The author of this paper proves the following theorem: There exists\u00a0<span class=\"MathTeX\">$\\delta_g &gt; 0$<\/span>\u00a0such that if\u00a0<span class=\"MathTeX\">$\\Phi$<\/span>\u00a0is a primitive\u00a0<span class=\"rm\">CM<\/span>\u00a0type for a\u00a0<span class=\"rm\">CM<\/span>\u00a0field\u00a0<span class=\"MathTeX\">$E$<\/span>, and if\u00a0<span class=\"MathTeX\">$A$<\/span>\u00a0is any\u00a0<span class=\"MathTeX\">$g$<\/span>-dimensional abelian variety over\u00a0<span class=\"MathTeX\">$\\overline{\\Bbb{Q}}$<\/span>\u00a0with endomorphism ring equal to the full ring of integers\u00a0<span class=\"MathTeX\">$\\mathcal{O}_E$<\/span>\u00a0and\u00a0<span class=\"rm\">CM<\/span>\u00a0type\u00a0<span class=\"MathTeX\">$\\Phi$<\/span>, then the field of moduli\u00a0<span class=\"MathTeX\">$\\Bbb{Q}(A)$<\/span>\u00a0of\u00a0<span class=\"MathTeX\">$A$<\/span>\u00a0satisfies\u00a0<span class=\"MathTeX\">$[\\Bbb{Q}(A): \\Bbb{Q}] &gt; |{\\rm Disc}(E)|^{\\delta_g}$<\/span>.<\/p>\n<p class=\"review\">By a result of J. S. Pila and the author [Ann. of Math. (2) <span class=\"bf\">179<\/span>\u00a0(2014), no. 2, 659\u2013681;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=3152943&amp;loc=fromrevtext\">MR3152943<\/a>], this theorem implies the Andr\u00e9-Oort conjecture for the coarse moduli space\u00a0<span class=\"MathTeX\">$\\mathcal{A}_g$<\/span>\u00a0of principally polarized abelian varieties of fixed dimension\u00a0<span class=\"MathTeX\">$g \\ge 1$<\/span>. The Andr\u00e9-Oort conjecture, as stated in this paper, asserts that an irreducible closed algebraic subvariety\u00a0<span class=\"MathTeX\">$V$<\/span>\u00a0of a Shimura variety\u00a0<span class=\"MathTeX\">$S$<\/span>\u00a0contains only finitely many maximal special subvarieties. Alternatively, every irreducible component of the Zariski closure of any set\u00a0<span class=\"MathTeX\">$\\Sigma$<\/span>\u00a0of special points in\u00a0<span class=\"MathTeX\">$S$<\/span>\u00a0is a special subvariety of\u00a0<span class=\"MathTeX\">$S$<\/span>\u00a0[see E. Ullmo and A. Yafaev, Ann. of Math. (2)\u00a0<span class=\"bf\">180<\/span>\u00a0(2014), no. 3, 823\u2013865;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=3245008&amp;loc=fromrevtext\">MR3245008<\/a>]. In the latter cited paper and its sequel [B. Klingler and A. Yafaev, Ann. of Math. (2)\u00a0<span class=\"bf\">180<\/span>\u00a0(2014), no. 3, 867\u2013925;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=3245009&amp;loc=fromrevtext\">MR3245009<\/a>], 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.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=191163\">Patrick Morton<\/a><\/span><\/p>\n<hr \/>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=4199235\"><strong>MR4199235<\/strong><\/a><br \/>\nPeluse, Sarah (1-IASP-SM)<br \/>\nBounds for sets with no polynomial progressions. (English summary)<br \/>\nForum Math. Pi 8 (2020), e16, 55 pp.<br \/>\n11B30 (11B25)<\/p>\n<p class=\"review\">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.\u00a0<span class=\"bf\">9<\/span>\u00a0(1996), no. 3, 725\u2013753;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1325795&amp;loc=fromrevtext\">MR1325795<\/a>].<\/p>\n<p class=\"review\">Recall that the Bergelson-Leibman theorem states the following: Let <span class=\"MathTeX\">${P_1,\\dots,P_m\\in\\Bbb{Z}[X]}$<\/span>\u00a0be polynomials with\u00a0<span class=\"MathTeX\">$P_i(0)=0$<\/span>\u00a0for all\u00a0<span class=\"MathTeX\">$i$<\/span>. Let\u00a0<span class=\"MathTeX\">$\\alpha&gt;0$<\/span>. Then, provided that\u00a0<span class=\"MathTeX\">$N$<\/span>\u00a0is sufficiently large in terms of\u00a0<span class=\"MathTeX\">$\\alpha,P_1,\\dots,P_m$<\/span>, any set\u00a0<span class=\"MathTeX\">$A\\subset\\{1,\\dots,N\\}$<\/span>\u00a0of cardinality at least\u00a0<span class=\"MathTeX\">$\\alpha N$<\/span>\u00a0contains a nontrivial configuration\u00a0<span class=\"MathTeX\">${(x,x+P_1(d),\\dots,x+P_m(d))}$<\/span>. Note that the case\u00a0<span class=\"MathTeX\">$P_i(X)=iX$<\/span> is already Szemer\u00e9di&#8217;s theorem.<\/p>\n<p class=\"review\">Bergelson and Leibman&#8217;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 &#8220;reasonable&#8221; bounds in the case that the <span class=\"MathTeX\">$P_i$<\/span>\u00a0have distinct degrees. She shows that in this case\u00a0<span class=\"MathTeX\">$A$<\/span>\u00a0contains a configuration of the stated type provided that\u00a0<span class=\"MathTeX\">$|A|\\ll N\/(\\log\\log N)^c$<\/span>, where\u00a0<span class=\"MathTeX\">$c=c_{P_1,\\dots,P_m}$<\/span>. For comparison, we remark that this is a bound of the same strength as that obtained by W. T. Gowers [Geom. Funct. Anal.\u00a0<span class=\"bf\">11<\/span>\u00a0(2001), no. 3, 465\u2013588;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1844079&amp;loc=fromrevtext\">MR1844079<\/a>] in his famous work on Szemer\u00e9di&#8217;s theorem, although it should be noted that the work under review does not directly extend that work since, in the Szemer\u00e9di case, the degrees of the\u00a0<span class=\"MathTeX\">$P_i$<\/span> are not distinct.<\/p>\n<p class=\"review\">Previously, bounds of the strength the author obtains were known only when\u00a0<span class=\"MathTeX\">$m=1$<\/span>\u00a0(where, in fact, better bounds are known, the state of the art being work of T. F. Bloom and J. Maynard [&#8220;A new upper bound for sets with no square difference&#8221;, preprint,\u00a0<a href=\"https:\/\/mathscinet.ams.org\/leavingmsn?url=http:\/\/arxiv.org\/abs\/2011.13266&amp;from=url\" target=\"NEW\" rel=\"noopener\">arXiv:2011.13266<\/a>]) and the case\u00a0<span class=\"MathTeX\">$P_1(X)=X$<\/span>,\u00a0<span class=\"MathTeX\">$P_2(X)=X^2$<\/span>, which was handled by the author and S. Prendiville [&#8220;Quantitative bounds in the nonlinear Roth theorem&#8221;, preprint,\u00a0<a href=\"https:\/\/mathscinet.ams.org\/leavingmsn?url=http:\/\/arxiv.org\/abs\/1903.02592&amp;from=url\" target=\"NEW\" rel=\"noopener\">arXiv:1903.02592<\/a>]. 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 &#8220;soft&#8221; 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.<\/p>\n<p class=\"review\">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.<\/p>\n<p class=\"review\">A number of intermediate results in the paper could be of independent interest. Foremost in this category is probably Theorem 3.5, a &#8220;quantitative concatenation&#8221; 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.<\/p>\n<p class=\"review\">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.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=685855\">Ben Joseph Green<\/a><\/span><\/p>\n<hr \/>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=4055179\"><strong>MR4055179<\/strong><\/a><br \/>\nGuth, Larry (1-MIT); Iosevich, Alex (1-RCT); Ou, Yumeng (1-CUNY2); Wang, Hong (1-MIT)<br \/>\nOn Falconer&#8217;s distance set problem in the plane. (English summary)<br \/>\nInvent. Math. 219 (2020), no. 3, 779\u2013830.<br \/>\n42B20 (28A80)<\/p>\n<p class=\"review\">Falconer&#8217;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\u00a0<span class=\"MathTeX\">$F \\subseteq \\Bbb{R}^d$<\/span>\u00a0and the\u00a0<span class=\"it\">distance set<\/span>\u00a0of\u00a0<span class=\"MathTeX\">$F$<\/span>, defined by\u00a0<span class=\"MathTeX\">$$ D(F) = \\{ |x-y| : x,y \\in F\\}. $$<\/span>\u00a0From now on all sets\u00a0<span class=\"MathTeX\">$F$<\/span>\u00a0are Borel. There are various ways to formulate the problem precisely. One conjecture is that if\u00a0<span class=\"MathTeX\">$\\dim_H F &gt;d\/2$<\/span>, then\u00a0<span class=\"MathTeX\">$D(F)$<\/span>\u00a0should have positive Lebesgue measure. Here\u00a0<span class=\"MathTeX\">$\\dim_H$<\/span> denotes the Hausdorff dimension.<\/p>\n<p class=\"review\">In his original paper [Mathematika\u00a0<span class=\"bf\">32<\/span>\u00a0(1985), no. 2, 206\u2013212 (1986);\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=834490&amp;loc=fromrevtext\">MR0834490<\/a>], K. J. Falconer proved that\u00a0<span class=\"MathTeX\">$\\dim_H F &gt;d\/2+1\/2$<\/span>\u00a0ensures that\u00a0<span class=\"MathTeX\">$D(F)$<\/span>\u00a0has positive Lebesgue measure. T. H. Wolff [Internat. Math. Res. Notices\u00a0<span class=\"bf\">1999<\/span>, no. 10, 547\u2013567;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1692851&amp;loc=fromrevtext\">MR1692851<\/a>; addendum, J. Anal. Math.\u00a0<span class=\"bf\">88<\/span>\u00a0(2002), 35\u201339;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=1979770&amp;loc=fromrevtext\">MR1979770<\/a>] improved this in the plane, proving that\u00a0<span class=\"MathTeX\">$\\dim_H F &gt;1+1\/3$<\/span>\u00a0ensures that\u00a0<span class=\"MathTeX\">$D(F)$<\/span>\u00a0has positive Lebesgue measure. The main result of the paper under review is a further improvement on this. Specifically, if\u00a0<span class=\"MathTeX\">$F$<\/span>\u00a0is a planar Borel set with\u00a0<span class=\"MathTeX\">$\\dim_H F &gt;1+1\/4$<\/span>, then\u00a0<span class=\"MathTeX\">$D(F)$<\/span>\u00a0has 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\u00a0<span class=\"it\">pinned<\/span> 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.<\/p>\n<p class=\"review\">The distance set problem has seen a lot of activity in the last few years\u2014for example [T. Orponen, Adv. Math.\u00a0<span class=\"bf\">307<\/span>\u00a0(2017), 1029\u20131045;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=3590535&amp;loc=fromrevtext\">MR3590535<\/a>; T. Keleti and P. S. Shmerkin, Geom. Funct. Anal.\u00a0<span class=\"bf\">29<\/span>\u00a0(2019), no. 6, 1886\u20131948;\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/publdoc.html?r=1&amp;pg1=MR&amp;s1=4034924&amp;loc=fromrevtext\">MR4034924<\/a>] and various other works.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=946983\">Jonathan MacDonald Fraser<\/a><\/span><\/p>\n<hr \/>\n<p><a href=\"https:\/\/mathscinet.ams.org\/mathscinet-getitem?mr=3959854\"><strong>MR3959854<\/strong><\/a><br \/>\nWang, Yilin (CH-ETHZ)<br \/>\nThe energy of a deterministic Loewner chain: reversibility and interpretation via ${\\rm SLE}_{0+}$. (English summary)<br \/>\nJ. Eur. Math. Soc. (JEMS) 21 (2019), no. 7, 1915\u20131941.<br \/>\n30C55 (30C62 60J67)<\/p>\n<p class=\"review\">The author studies some features of the energy of a deterministic Loewner chain. According to the chordal Loewner description, a simple curve\u00a0<span class=\"MathTeX\">$\\gamma$<\/span>\u00a0from 0 to infinity in the upper half-plane\u00a0<span class=\"MathTeX\">$\\Bbb H=\\{z\\in\\Bbb C\\:{\\rm Im}\\,z&gt;0\\}$<\/span>\u00a0is parameterized so that the conformal map\u00a0<span class=\"MathTeX\">$g_t$<\/span>\u00a0from\u00a0<span class=\"MathTeX\">$\\Bbb H\\setminus\\gamma[0,t]$<\/span>\u00a0onto\u00a0<span class=\"MathTeX\">$\\Bbb H$<\/span>\u00a0with\u00a0<span class=\"MathTeX\">$g_t(z)=z+o(1)$<\/span>\u00a0as\u00a0<span class=\"MathTeX\">$z\\to\\infty$<\/span>\u00a0satisfies in fact\u00a0<span class=\"MathTeX\">$g_t(z)=z+2t\/z+o(1\/z)$<\/span>\u00a0as\u00a0<span class=\"MathTeX\">$z\\to\\infty$<\/span>. Extend\u00a0<span class=\"MathTeX\">$g_t$<\/span>\u00a0continuously to the boundary point\u00a0<span class=\"MathTeX\">$\\gamma_t$<\/span>. The real-valued driving function\u00a0<span class=\"MathTeX\">$W_t := g_t(\\gamma_t)$<\/span>\u00a0of\u00a0<span class=\"MathTeX\">$\\gamma$<\/span>\u00a0is continuous. The Loewner energy is the Dirichlet energy of\u00a0<span class=\"MathTeX\">$W_t$<\/span>\u00a0given by\u00a0<span class=\"MathTeX\">$$ I(\\gamma):= \\frac{1}{2}\\int_0^{\\infty}\\left(\\frac{dW_t}{dt}\\right)^2 dt=\\frac{1}{2}\\int_0^{\\infty}\\left(\\frac{d(g_t(\\gamma_t))}{dt}\\right)^2dt. $$<\/span>\u00a0The scale invariance\u00a0<span class=\"MathTeX\">$I(u\\gamma)=I(\\gamma)$<\/span>\u00a0for\u00a0<span class=\"MathTeX\">$u&gt;0$<\/span>\u00a0makes it possible to define the energy\u00a0<span class=\"MathTeX\">$I_{D,a,b}(\\eta)$<\/span>\u00a0of any simple curve\u00a0<span class=\"MathTeX\">$\\eta$<\/span>\u00a0from a boundary point\u00a0<span class=\"MathTeX\">$a$<\/span>\u00a0of a simply connected domain\u00a0<span class=\"MathTeX\">$D$<\/span>\u00a0to another boundary point\u00a0<span class=\"MathTeX\">$b$<\/span>\u00a0to be the energy of the conformal image of\u00a0<span class=\"MathTeX\">$\\eta$<\/span>\u00a0via any uniformizing map\u00a0<span class=\"MathTeX\">$\\Psi$<\/span>\u00a0from\u00a0<span class=\"MathTeX\">$(D,a,b)$<\/span>\u00a0to\u00a0<span class=\"MathTeX\">$\\Bbb (H,0,\\infty)$<\/span>. For such\u00a0<span class=\"MathTeX\">$\\eta$<\/span>, define its time-reversal\u00a0<span class=\"MathTeX\">$\\widehat\\eta$<\/span>\u00a0that has the same trace as\u00a0<span class=\"MathTeX\">$\\eta$<\/span>, but which is viewed as going from\u00a0<span class=\"MathTeX\">$b$<\/span>\u00a0to\u00a0<span class=\"MathTeX\">$a$<\/span>. The first main contribution of the paper is presented in the following theorem.<\/p>\n<p class=\"review\">Main Theorem 1.1. The Loewner energy of the time-reversal\u00a0<span class=\"MathTeX\">$\\widehat\\eta$<\/span>\u00a0of a simple curve\u00a0<span class=\"MathTeX\">$\\eta$<\/span>\u00a0from\u00a0<span class=\"MathTeX\">$a$<\/span>\u00a0to\u00a0<span class=\"MathTeX\">$b$<\/span>\u00a0in\u00a0<span class=\"MathTeX\">$D$<\/span>\u00a0is equal to the Loewner energy of\u00a0<span class=\"MathTeX\">$\\eta$<\/span>:\u00a0<span class=\"MathTeX\">$I_{D,a,b}(\\eta)=I_{D,b,a}(\\widehat\\eta)$<\/span>.<\/p>\n<p class=\"review\">When\u00a0<span class=\"MathTeX\">$\\lambda\\in C([0,\\infty))$<\/span>, consider the Loewner differential equation\u00a0<span class=\"MathTeX\">$$ \\partial_tg_t(z)=\\frac{2}{g_t(z)-\\lambda_t}$$<\/span>\u00a0with the initial condition\u00a0<span class=\"MathTeX\">$$ g_0(z)=z. $$<\/span>\u00a0The chordal Loewner chain in\u00a0<span class=\"MathTeX\">$\\Bbb H$<\/span>\u00a0driven by\u00a0<span class=\"MathTeX\">$\\lambda$<\/span>\u00a0(or the Loewner transform of\u00a0<span class=\"MathTeX\">$\\lambda$<\/span>) is the increasing family\u00a0<span class=\"MathTeX\">$(K_t)_{t&gt;0}$<\/span>\u00a0defined by\u00a0<span class=\"MathTeX\">$K_t=\\{z\\in\\Bbb H\\:\\tau(z)\\leq t\\}$<\/span>, where\u00a0<span class=\"MathTeX\">$\\tau(z)$<\/span>\u00a0is the maximum survival time of the solution\u00a0<span class=\"MathTeX\">$g_t(z)$<\/span>. Let\u00a0<span class=\"MathTeX\">$H\\subset C[0,\\infty)$<\/span>\u00a0be the set of finite\u00a0<span class=\"MathTeX\">$I$<\/span> energy functions. The author proves the following statement.<\/p>\n<p class=\"review\">Proposition 2.1. For every\u00a0<span class=\"MathTeX\">$\\lambda\\in H$<\/span>, there exists\u00a0<span class=\"MathTeX\">$K=K(I(\\lambda))$<\/span>, depending only on\u00a0<span class=\"MathTeX\">$I(\\lambda)$<\/span>, such that the trace of the Loewner transform\u00a0<span class=\"MathTeX\">$\\gamma$<\/span>\u00a0of\u00a0<span class=\"MathTeX\">$\\lambda$<\/span>\u00a0is a\u00a0<span class=\"MathTeX\">$K$<\/span>-quasiconformal curve.<\/p>\n<p class=\"review\">\u00a0The 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.<\/p>\n<p><span class=\"ReviewedBy\">Reviewed by\u00a0<a href=\"https:\/\/mathscinet.ams.org\/mathscinet\/search\/author.html?mrauthid=201559\">Dmitri Valentinovi\u0107 Prokhorov<\/a><\/span><\/p>\n<p>&nbsp;<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>The\u00a0winners of the 2022 Breakthrough Prizes have been announced.\u00a0 There are eight recipients in mathematics:\u00a0 Takuro Mochizuki, Aaron Brown, Sebastian Hurtado Salazar, Jack Thorne, Jacob Tsimerman, Sarah Peluse, Hong Wang, and Yilin Wang.<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/beyondreviews\/2021\/09\/13\/winners-of-the-2022-breakthrough-prizes-in-mathematics\/><\/div>\n","protected":false},"author":86,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[44],"tags":[],"class_list":["post-3285","post","type-post","status-publish","format-standard","hentry","category-prizes-and-awards"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p6C2KK-QZ","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/3285","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/users\/86"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/comments?post=3285"}],"version-history":[{"count":21,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/3285\/revisions"}],"predecessor-version":[{"id":3306,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/posts\/3285\/revisions\/3306"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/media?parent=3285"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/categories?post=3285"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/beyondreviews\/wp-json\/wp\/v2\/tags?post=3285"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}