Communications of the American Mathematical Society最新文献

{"title":"Eigenvalue problems in 𝐿^{∞}: optimality conditions, duality, and relations with optimal transport","authors":"Leon Bungert, Yury Korolev","doi":"10.1090/cams/11","DOIUrl":"https://doi.org/10.1090/cams/11","url":null,"abstract":"<p>In this article we characterize the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper L Superscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathrm {L}^infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> eigenvalue problem associated to the Rayleigh quotient <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-vertical-bar nabla u double-vertical-bar Subscript normal upper L Sub Superscript normal infinity Baseline slash double-vertical-bar u double-vertical-bar Subscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" />\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:msub>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-CLOSE\">\u0000\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"true\">/</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-OPEN\">\u0000\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:msub>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" />\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">left .{|nabla u|_{mathrm {L}^infty }}middle /{|u|_infty }right .</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and relate it to a divergence-form PDE, similarly to what is known for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper L Superscript p\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathrm {L}^p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> eigenvalue problems and the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/199","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116874004","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":"Friendly bisections of random graphs","authors":"Asaf Ferber, Matthew Kwan, Bhargav P. Narayanan, A. Sah, Mehtaab Sawhney","doi":"10.1090/cams/13","DOIUrl":"https://doi.org/10.1090/cams/13","url":null,"abstract":"Resolving a conjecture of Füredi from 1988, we prove that with high probability, the random graph \u0000\u0000 \u0000 \u0000 \u0000 G\u0000 \u0000 (\u0000 n\u0000 ,\u0000 1\u0000 \u0000 /\u0000 \u0000 2\u0000 )\u0000 \u0000 mathbb {G}(n,1/2)\u0000 \u0000\u0000 admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which \u0000\u0000 \u0000 \u0000 n\u0000 −\u0000 o\u0000 (\u0000 n\u0000 )\u0000 \u0000 n-o(n)\u0000 \u0000\u0000 vertices have more neighbours in their own part as across. Our proof is constructive, and in the process, we develop a new method to study stochastic processes driven by degree information in random graphs; this involves combining enumeration techniques with an abstract second moment argument.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126799527","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":"On the fast spreading scenario","authors":"Si-ming He, E. Tadmor, Andrej Zlatovs","doi":"10.1090/cams/6","DOIUrl":"https://doi.org/10.1090/cams/6","url":null,"abstract":"We study two types of divergence-free fluid flows on unbounded domains in two and three dimensions—hyperbolic and shear flows—and their influence on chemotaxis and combustion. We show that fast spreading by these flows, when they are strong enough, can suppress growth of solutions to PDE modeling these phenomena. This includes prevention of singularity formation and global regularity of solutions to advective Patlak-Keller-Segel equations on \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 2\u0000 \u0000 mathbb {R}^2\u0000 \u0000\u0000 and \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 3\u0000 \u0000 mathbb {R}^3\u0000 \u0000\u0000, confirming numerical observations by Khan, Johnson, Cartee, and Yao [Involve 9 (2016), pp. 119–131], as well as quenching in advection-reaction-diffusion equations.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122728233","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":"The structure group for quasi-linear equations via universal enveloping algebras","authors":"P. Linares, F. Otto, Markus Tempelmayr","doi":"10.1090/cams/16","DOIUrl":"https://doi.org/10.1090/cams/16","url":null,"abstract":"We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule.\u0000\u0000Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants).\u0000\u0000We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the stochastic heat equation.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115454982","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":"Geodesic length and shifted weights in first-passage percolation","authors":"Arjun Krishnan, F. Rassoul-Agha, T. Seppalainen","doi":"10.1090/cams/18","DOIUrl":"https://doi.org/10.1090/cams/18","url":null,"abstract":"We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the \u0000\u0000 \u0000 \u0000 ℓ\u0000 1\u0000 \u0000 ell ^1\u0000 \u0000\u0000 distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produces singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"129 8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124245487","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":"Colored fermionic vertex models and symmetric functions","authors":"A. Aggarwal, A. Borodin, M. Wheeler","doi":"10.1090/cams/24","DOIUrl":"https://doi.org/10.1090/cams/24","url":null,"abstract":"In this text we introduce and analyze families of symmetric functions arising as partition functions for colored fermionic vertex models associated with the quantized affine Lie superalgebra $U_q big( widehat{mathfrak{sl}} (1 | n) big)$. We establish various combinatorial results for these vertex models and symmetric functions, which include the following. (1) We apply the fusion procedure to the fundamental $R$-matrix for $U_q big( widehat{mathfrak{sl}} (1 | n) big)$ to obtain an explicit family of vertex weights satisfying the Yang-Baxter equation. (2) We define families of symmetric functions as partition functions for colored, fermionic vertex models under these fused weights. We further establish several combinatorial properties for these symmetric functions, such as branching rules and Cauchy identities. (3) We show that the Lascoux-Leclerc-Thibon (LLT) polynomials arise as special cases of these symmetric functions. This enables us to show both old and new properties about the LLT polynomials, including Cauchy identities, contour integral formulas, stability properties, and branching rules under a certain family of plethystic transformations. (4) A different special case of our symmetric functions gives rise to a new family of polynomials called factorial LLT polynomials. We show they generalize the LLT polynomials, while also satisfying a vanishing condition reminiscent of that satisfied by the factorial Schur functions. (5) By considering our vertex model on a cylinder, we obtain fermionic partition function formulas for both the symmetric and nonsymmetric Macdonald polynomials. (6) We prove combinatorial formulas for the coefficients of the LLT polynomials when expanded in the modified Hall-Littlewood basis, as partition functions for a $U_q big( widehat{mathfrak{sl}} (2 | n) big)$ vertex model.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126299042","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":"Universal selection of pulled fronts","authors":"M. Avery, A. Scheel","doi":"10.1090/cams/8","DOIUrl":"https://doi.org/10.1090/cams/8","url":null,"abstract":"We establish selection of critical pulled fronts in invasion processes as predicted by the marginal stability conjecture. Our result shows convergence to a pulled front with a logarithmic shift for open sets of steep initial data, including one-sided compactly supported initial conditions. We rely on robust, conceptual assumptions, namely existence and marginal spectral stability of a front traveling at the linear spreading speed and demonstrate that the assumptions hold for open classes of spatially extended systems. Previous results relied on comparison principles or probabilistic tools with implied nonopen conditions on initial data and structure of the equation. Technically, we describe the invasion process through the interaction of a Gaussian leading edge with the pulled front in the wake. Key ingredients are sharp linear decay estimates to control errors in the nonlinear matching and corrections from initial data.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132980578","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":"Expressive curves","authors":"S. Fomin, E. Shustin","doi":"10.1090/cams/12","DOIUrl":"https://doi.org/10.1090/cams/12","url":null,"abstract":"We initiate the study of a class of real plane algebraic curves which we call expressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity.\u0000\u0000We prove that a plane curve \u0000\u0000 \u0000 C\u0000 C\u0000 \u0000\u0000 is expressive if (a) each irreducible component of \u0000\u0000 \u0000 C\u0000 C\u0000 \u0000\u0000 can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of \u0000\u0000 \u0000 C\u0000 C\u0000 \u0000\u0000 in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of \u0000\u0000 \u0000 C\u0000 C\u0000 \u0000\u0000 in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity.\u0000\u0000We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115220542","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":"Gap sets for the spectra of cubic graphs","authors":"Alicia J. Koll'ar, P. Sarnak","doi":"10.1090/cams/3","DOIUrl":"https://doi.org/10.1090/cams/3","url":null,"abstract":"<p>We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 2 StartRoot 2 EndRoot comma 3 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:msqrt>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msqrt>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(2 sqrt {2},3)</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-bracket negative 3 comma negative 2 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">[-3,-2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket negative 3 comma 3 right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">[-3,3]</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket negative 3 comma 3 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">[-3,3)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.</p>","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124846536","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":"Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant","authors":"M. Hopkins, Jianfeng Lin, Xiaolin Shi, Zhouli Xu","doi":"10.1090/cams/4","DOIUrl":"https://doi.org/10.1090/cams/4","url":null,"abstract":"In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of \u0000\u0000 \u0000 \u0000 Pin\u0000 ⁡\u0000 (\u0000 2\u0000 )\u0000 \u0000 operatorname {Pin}(2)\u0000 \u0000\u0000-equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the \u0000\u0000 \u0000 \u0000 Pin\u0000 ⁡\u0000 (\u0000 2\u0000 )\u0000 \u0000 operatorname {Pin}(2)\u0000 \u0000\u0000-equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem.\u0000\u0000We prove our theorem by analyzing maps between certain finite spectra arising from \u0000\u0000 \u0000 \u0000 B\u0000 Pin\u0000 ⁡\u0000 (\u0000 2\u0000 )\u0000 \u0000 Boperatorname {Pin}(2)\u0000 \u0000\u0000 and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the \u0000\u0000 \u0000 j\u0000 j\u0000 \u0000\u0000-based Atiyah–Hirzebruch spectral sequence.","PeriodicalId":285678,"journal":{"name":"Communications of the American Mathematical Society","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128820601","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}