Pub Date : 2023-09-01DOI: 10.4007/annals.2023.198.2.2
D. Kang, T. Kelly, Daniela Kühn, Abhishek Methuku, D. Osthus
The Erdős-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this paper, we prove this conjecture for every large n. We also provide stability versions of this result, which confirm a prediction of Kahn.
{"title":"A proof of the Erdős--Faber--Lovász conjecture","authors":"D. Kang, T. Kelly, Daniela Kühn, Abhishek Methuku, D. Osthus","doi":"10.4007/annals.2023.198.2.2","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.2","url":null,"abstract":"The Erdős-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this paper, we prove this conjecture for every large n. We also provide stability versions of this result, which confirm a prediction of Kahn.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49217505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.4007/annals.2023.198.2.9
Z. Klagsbrun, B. Mazur, K. Rubin
{"title":"Erratum to \"Disparity in Selmer ranks of quadratic twists of elliptic curves\"","authors":"Z. Klagsbrun, B. Mazur, K. Rubin","doi":"10.4007/annals.2023.198.2.9","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.9","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46257052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.4007/annals.2023.198.2.3
Marco d’Addezio
{"title":"Parabolicity conjecture of $F$-isocrystals","authors":"Marco d’Addezio","doi":"10.4007/annals.2023.198.2.3","DOIUrl":"https://doi.org/10.4007/annals.2023.198.2.3","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45300981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-01DOI: 10.4007/annals.2023.198.1.1
Subhash Khot, Dor Minzer, Muli Safra
We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the $2$-to-$2$ Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. ${bf 162}$ (2005), 439--485], and new hardness gaps for Unique-Games. The Grassmann graph ${sf Gr}_{sf{global}}$ contains induced subgraphs ${sf Gr}_{sf{local}}$ that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is $o(1)$ inside all subgraphs ${sf Gr}_{sf{local}}$ whose order is $O(1)$ lower than that of ${sf Gr}_{sf{global}}$. We prove that pseudorandom sets have expansion $1-o(1)$, greatly extending the results and techniques of a previous work of the authors with Dinur and Kindler.
{"title":"Pseudorandom sets in Grassmann graph have near-perfect expansion","authors":"Subhash Khot, Dor Minzer, Muli Safra","doi":"10.4007/annals.2023.198.1.1","DOIUrl":"https://doi.org/10.4007/annals.2023.198.1.1","url":null,"abstract":"We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the $2$-to-$2$ Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. ${bf 162}$ (2005), 439--485], and new hardness gaps for Unique-Games. The Grassmann graph ${sf Gr}_{sf{global}}$ contains induced subgraphs ${sf Gr}_{sf{local}}$ that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is $o(1)$ inside all subgraphs ${sf Gr}_{sf{local}}$ whose order is $O(1)$ lower than that of ${sf Gr}_{sf{global}}$. We prove that pseudorandom sets have expansion $1-o(1)$, greatly extending the results and techniques of a previous work of the authors with Dinur and Kindler.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135111020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.4007/annals.2023.197.3.4
L. Simon
With respect to a C ∞ metric which is close to the standard Euclidean metric on R N + 1 + ℓ , where N ≥ 7 and ℓ ≥ 1 are given, we construct a class of embedded ( N + ℓ )-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset K ⊂ { 0 } × R ℓ . Thus the question is settled, with a strong affirmative, as to whether there can be “gaps” or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and in particular for the standard Euclidean metric. The construction used here involves the analysis of solutions u of the Symmetric Minimal Surface Equation on domains Ω ⊂ R n whose symmetric graphs (i.e. { ( x , ξ ) ∈ Ω × R m : | ξ | = u ( x ) } ) lie on one side of a cylindrical minimal cone, including in particular a Liouville type theorem for complete solutions (i.e. the case Ω = R n ).
{"title":"Stable minimal hypersurfaces in ℝ^N+1+ℓ with singular set an arbitrary closed K⊂{0}×ℝ^ℓ","authors":"L. Simon","doi":"10.4007/annals.2023.197.3.4","DOIUrl":"https://doi.org/10.4007/annals.2023.197.3.4","url":null,"abstract":"With respect to a C ∞ metric which is close to the standard Euclidean metric on R N + 1 + ℓ , where N ≥ 7 and ℓ ≥ 1 are given, we construct a class of embedded ( N + ℓ )-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset K ⊂ { 0 } × R ℓ . Thus the question is settled, with a strong affirmative, as to whether there can be “gaps” or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and in particular for the standard Euclidean metric. The construction used here involves the analysis of solutions u of the Symmetric Minimal Surface Equation on domains Ω ⊂ R n whose symmetric graphs (i.e. { ( x , ξ ) ∈ Ω × R m : | ξ | = u ( x ) } ) lie on one side of a cylindrical minimal cone, including in particular a Liouville type theorem for complete solutions (i.e. the case Ω = R n ).","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45845829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $lambda$ denote the Liouville function. We show that, as $X rightarrow infty$, $$ int^{2X}_X sup_{begin{smallmatrix} P(Y)inmathbb{R}[Y] mathrm{deg}{P} leq kend{smallmatrix}} left| sum_{xleq n leq x+H} lambda(n) e(-P(n))right| dx=o (XH) $$ for all fixed $k$ and $X^{theta} leq H leq X$ with $0 lt theta lt 1$ fixed but arbitrarily small. Previously this was only established for $k leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $overline{F}(g(n) Gamma )$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$ int_{X}^{2X} | lambda |_{U^{k+1}([x,x+H])} dx = o ( X ) $$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $lambda$ over short polynomial progressions $(n+P_1(m),ldots , n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $Hgeq mathrm{exp}((mathrm{log} X)^{5/8+varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.
{"title":"Higher uniformity of bounded multiplicative functions in short intervals on average","authors":"Kaisa Matomäki, Maksym Radziwiłł, Terence Tao, Joni Teräväinen, Tamar Ziegler","doi":"10.4007/annals.2023.197.2.3","DOIUrl":"https://doi.org/10.4007/annals.2023.197.2.3","url":null,"abstract":"Let $lambda$ denote the Liouville function. We show that, as $X rightarrow infty$, $$ int^{2X}_X sup_{begin{smallmatrix} P(Y)inmathbb{R}[Y] mathrm{deg}{P} leq kend{smallmatrix}} left| sum_{xleq n leq x+H} lambda(n) e(-P(n))right| dx=o (XH) $$ for all fixed $k$ and $X^{theta} leq H leq X$ with $0 lt theta lt 1$ fixed but arbitrarily small. Previously this was only established for $k leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $overline{F}(g(n) Gamma )$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$ int_{X}^{2X} | lambda |_{U^{k+1}([x,x+H])} dx = o ( X ) $$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $lambda$ over short polynomial progressions $(n+P_1(m),ldots , n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $Hgeq mathrm{exp}((mathrm{log} X)^{5/8+varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135693489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bonds between hydrogen and carbon atoms are the most frequent type of bonds in organic molecules. The ability to replace hydrogen atoms by making other types of bonds to carbon atoms can enable simpler access to complex organic molecules by substituting multistep synthetic sequences. The use of transition metal catalysts to activate C-H bonds is particularly attractive as it offers control over the reactivity and selectivity through catalyst design. However, such functionalization includes the difficult breaking of strong C-H bonds that are not activated by the presence of other groups. Additionally, the common presence of a number of C-H bonds in a molecule raises the issue of site-selectivity because differentiation of C-H bonds that are in sterically and electronically similar environments is a challenge. We discuss selected recent developments that are a part of the long-term research interest in mild and selective C-H activation reactions with a focus on the replacement of C-H bonds with C-aryl groups and an emphasis on the work of our group.
{"title":"Direct C-H Arylation.","authors":"Jyoti Dhankhar, Ilija Čorić","doi":"10.2533/chimia.2022.777","DOIUrl":"10.2533/chimia.2022.777","url":null,"abstract":"<p><p>Bonds between hydrogen and carbon atoms are the most frequent type of bonds in organic molecules. The ability to replace hydrogen atoms by making other types of bonds to carbon atoms can enable simpler access to complex organic molecules by substituting multistep synthetic sequences. The use of transition metal catalysts to activate C-H bonds is particularly attractive as it offers control over the reactivity and selectivity through catalyst design. However, such functionalization includes the difficult breaking of strong C-H bonds that are not activated by the presence of other groups. Additionally, the common presence of a number of C-H bonds in a molecule raises the issue of site-selectivity because differentiation of C-H bonds that are in sterically and electronically similar environments is a challenge. We discuss selected recent developments that are a part of the long-term research interest in mild and selective C-H activation reactions with a focus on the replacement of C-H bonds with C-aryl groups and an emphasis on the work of our group.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"14 1","pages":"777-783"},"PeriodicalIF":1.2,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90423479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-01DOI: 10.4007/annals.2022.196.2.4
F. Merle, P. Raphaël, I. Rodnianski, J. Szeftel
{"title":"On the implosion of a compressible fluid II: Singularity formation","authors":"F. Merle, P. Raphaël, I. Rodnianski, J. Szeftel","doi":"10.4007/annals.2022.196.2.4","DOIUrl":"https://doi.org/10.4007/annals.2022.196.2.4","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42422623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-01DOI: 10.4007/annals.2022.196.2.3
F. Merle, P. Raphaël, I. Rodnianski, J. Szeftel
{"title":"On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles","authors":"F. Merle, P. Raphaël, I. Rodnianski, J. Szeftel","doi":"10.4007/annals.2022.196.2.3","DOIUrl":"https://doi.org/10.4007/annals.2022.196.2.3","url":null,"abstract":"","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":" ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48369384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-27DOI: 10.4007/annals.2022.196.1.1
Jian Ding, Allan Sly, Nike Sun
We establish the satisfiability threshold for random k-SAT for all $kge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $alpha_{mathrm{SAT}}(k)$ such that a random k-SAT formula of clause density $alpha$ is with high probability satisfiable for $alphaltalpha_{mathrm{SAT}}$, and unsatisfiable for $alpha>alpha_{mathrm{SAT}}$. We show that the threshold $alpha_{mathrm{SAT}}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the 1-RSB universality class.
{"title":"Proof of the satisfiability conjecture for large $k$ | Annals of Mathematics","authors":"Jian Ding, Allan Sly, Nike Sun","doi":"10.4007/annals.2022.196.1.1","DOIUrl":"https://doi.org/10.4007/annals.2022.196.1.1","url":null,"abstract":"<p>We establish the satisfiability threshold for random k-SAT for all $kge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $alpha_{mathrm{SAT}}(k)$ such that a random k-SAT formula of clause density $alpha$ is with high probability satisfiable for $alphaltalpha_{mathrm{SAT}}$, and unsatisfiable for<br/>\u0000$alpha>alpha_{mathrm{SAT}}$. We show that the threshold $alpha_{mathrm{SAT}}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the 1-RSB universality class.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"282 ","pages":""},"PeriodicalIF":4.9,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138506499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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