{"title":"Relative perversity","authors":"David Hansen, P. Scholze","doi":"10.1090/cams/21","DOIUrl":"https://doi.org/10.1090/cams/21","url":null,"abstract":"<p>We define and study a relative perverse <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-structure associated with any finitely presented morphism of schemes <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper X right-arrow upper S\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi>S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">f: Xto S</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, with relative perversity equivalent to perversity of the restrictions to all geometric fibres of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The existence of this <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-structure is closely related to perverse <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-exactness properties of nearby cycles. This <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal e normal r normal v Superscript normal upper U normal upper L normal upper A Baseline left-parenthesis upper X slash upper S right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">P</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">e</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">r</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">v</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">U</mml:mi>\u0000 <mml:mi ","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"47 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120992600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules","authors":"Cain EDIE-MICHELL","doi":"10.1090/cams/19","DOIUrl":"https://doi.org/10.1090/cams/19","url":null,"abstract":"<p>This paper is the first of a pair that aims to classify a large number of the type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I upper I\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mi>I</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">II</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> quantum subgroups of the categories <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript r plus 1 Baseline comma k right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">s</mml:mi>\u0000 <mml:mi mathvariant=\"fraktur\">l</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {C}(mathfrak {sl}_{r+1}, k)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this work we classify the braided auto-equivalences of the categories of local modules for all known type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\u0000 <mml:semantics>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> quantum subgroups of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper C left-parenthesis German s German l Subscript r plus 1 Baseline comma k right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">s</mml:mi>\u0000 <mml:mi mathvariant=\"fraktur\">l</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>r</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {C}(mathfrak {sl}_{r+1}, k)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We find that the symmetries are all non-exceptional e","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127256285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Self-Bäcklund curves in centroaffine geometry and Lamé’s equation","authors":"M. Bialy, Gil Bor, S. Tabachnikov","doi":"10.1090/cams/9","DOIUrl":"https://doi.org/10.1090/cams/9","url":null,"abstract":"Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centroaffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centroaffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves.\u0000\u0000Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam’s problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics.\u0000\u0000We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116757699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tumor growth with nutrients: Regularity and stability","authors":"M. Jacobs, Inwon C. Kim, Jiajun Tong","doi":"10.1090/cams/20","DOIUrl":"https://doi.org/10.1090/cams/20","url":null,"abstract":"In this paper, we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the incompressibility of the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics thanks to an unexpected comparison principle. Using the comparison principle, we provide quantitative \u0000\u0000 \u0000 \u0000 L\u0000 1\u0000 \u0000 L^1\u0000 \u0000\u0000-contraction estimates and establish the \u0000\u0000 \u0000 \u0000 C\u0000 \u0000 1\u0000 ,\u0000 α\u0000 \u0000 \u0000 C^{1,alpha }\u0000 \u0000\u0000-boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient \u0000\u0000 \u0000 \u0000 n\u0000 0\u0000 \u0000 n_0\u0000 \u0000\u0000 either lies entirely above or entirely below the critical value \u0000\u0000 \u0000 \u0000 \u0000 n\u0000 0\u0000 \u0000 =\u0000 1\u0000 \u0000 n_0=1\u0000 \u0000\u0000, we are able to give a complete characterization of the long-time behavior of the system. When \u0000\u0000 \u0000 \u0000 n\u0000 0\u0000 \u0000 n_0\u0000 \u0000\u0000 is constant, we can even describe the dynamics of the full system in terms of some simpler nutrient-free and parameter-free model problems. These results are in sharp contrast to the observed behavior of the models either with nutrient diffusion or with death rate in tumor cells.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126947183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ribbon concordance of knots is a partial ordering","authors":"I. Agol","doi":"10.1090/cams/15","DOIUrl":"https://doi.org/10.1090/cams/15","url":null,"abstract":"In this note we show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon [Math. Ann. 257 (1981), pp. 157–170, Conjecture 1.1]. The proof makes use of representation varieties of the knot groups to \u0000\u0000 \u0000 \u0000 S\u0000 O\u0000 (\u0000 N\u0000 )\u0000 \u0000 SO(N)\u0000 \u0000\u0000 and the subquotient relation between them induced by a ribbon concordance.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"64 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120919741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Smooth local rigidity for hyperbolic toral automorphisms","authors":"B. Kalinin, V. Sadovskaya, Zhenqi Wang","doi":"10.1090/cams/22","DOIUrl":"https://doi.org/10.1090/cams/22","url":null,"abstract":"<p>We study the regularity of a conjugacy <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> between a hyperbolic toral automorphism <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and its smooth perturbation <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We show that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is weakly differentiable then it is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript 1 plus upper H reverse-solidus quotation-mark older\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mtext>H\"older</mml:mtext>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^{1+text {H\"older}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and, if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is also weakly irreducible, then <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H\">\u0000 <mml:semantics>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">H</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. As a part of the proof, we establish results of independent interest on Hölder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, w","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129741365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotic vanishing of syzygies of algebraic varieties","authors":"Jinhyung Park","doi":"10.1090/cams/7","DOIUrl":"https://doi.org/10.1090/cams/7","url":null,"abstract":"The purpose of this paper is to prove Ein–Lazarsfeld’s conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein–Lazarsfeld’s asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114231278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere","authors":"Shachar Carmeli, Allen Yuan","doi":"10.1090/cams/17","DOIUrl":"https://doi.org/10.1090/cams/17","url":null,"abstract":"<p>We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K left-parenthesis 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K(1)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-local sphere <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {S}_{K(1)}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> at the prime <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, in particular realizing the non-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-adic rational element <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 plus epsilon element-of pi 0 double-struck upper S Subscript upper K left-parenthesis 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">1+varepsilon in pi _0mathbb {S}_{K(1)}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> as a “semiadditive cardinality.” As a fur","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134144285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jane Breen, Alexander W. N. Riasanovsky, Michael Tait, John C. Urschel
{"title":"Maximum spread of graphs and bipartite graphs","authors":"Jane Breen, Alexander W. N. Riasanovsky, Michael Tait, John C. Urschel","doi":"10.1090/cams/14","DOIUrl":"https://doi.org/10.1090/cams/14","url":null,"abstract":"<p>Given any graph <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the spread of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the maximum difference between any two eigenvalues of the adjacency matrix of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-vertex graph <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that maximizes spread is the join of a clique and an independent set, with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left floor 2 n slash 3 right floor\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⌊<!-- ⌊ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⌋<!-- ⌋ --></mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">lfloor 2n/3 rfloor</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left ceiling n slash 3 right ceiling\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⌈<!-- ⌈ --></mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125845519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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