We give a new and conceptually straightforward proof of the well-known presentation for the Temperley–Lieb algebra, via an alternative new presentation. Our method involves twisted semigroup algebras, and we make use of two apparently new submonoids of the Temperley–Lieb monoid.
{"title":"Presentations for Temperley–Lieb Algebras","authors":"James East","doi":"10.1093/qmath/haab001","DOIUrl":"https://doi.org/10.1093/qmath/haab001","url":null,"abstract":"We give a new and conceptually straightforward proof of the well-known presentation for the Temperley–Lieb algebra, via an alternative new presentation. Our method involves twisted semigroup algebras, and we make use of two apparently new submonoids of the Temperley–Lieb monoid.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1253-1269"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49947646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum to: Tamagawa Products of Elliptic Curves over Q","authors":"","doi":"10.1093/qmath/haab052","DOIUrl":"https://doi.org/10.1093/qmath/haab052","url":null,"abstract":"","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1545-1545"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are concerned with an optimal regularity for ω-minimizers to double phase variational problems with variable exponents where the associated energy density is allowed to be discontinuous. We identify basic structure assumptions on the density for the absence of Lavrentiev phenomenon and higher integrability. Moreover, we establish a local Calderón–Zygmund theory for such generalized minimizers under minimal regularity requirements regarding such double phase functionals to the frame of Lebesgue spaces with variable exponents.
{"title":"Gradient Estimates of ω-Minimizers to Double Phase Variational Problems with Variable Exponents","authors":"Sun-Sig Byun;Ho-Sik Lee","doi":"10.1093/qmath/haaa067","DOIUrl":"https://doi.org/10.1093/qmath/haaa067","url":null,"abstract":"We are concerned with an optimal regularity for ω-minimizers to double phase variational problems with variable exponents where the associated energy density is allowed to be discontinuous. We identify basic structure assumptions on the density for the absence of Lavrentiev phenomenon and higher integrability. Moreover, we establish a local Calderón–Zygmund theory for such generalized minimizers under minimal regularity requirements regarding such double phase functionals to the frame of Lebesgue spaces with variable exponents.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1191-1221"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa067","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49947668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A linear equation with coefficients in $mathbb{F}_q$ is common if the number of monochromatic solutions in any two-coloring of $mathbb{F}_q^{,n}$ is asymptotically (as $n to infty$ ) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of $mathbb{F}_q^{,n}$ is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf.
{"title":"Common and Sidorenko Linear Equations","authors":"Jacob Fox;Huy Tuan Pham;Yufei Zhao","doi":"10.1093/qmath/haaa068","DOIUrl":"https://doi.org/10.1093/qmath/haaa068","url":null,"abstract":"A linear equation with coefficients in \u0000<tex>$mathbb{F}_q$</tex>\u0000 is common if the number of monochromatic solutions in any two-coloring of \u0000<tex>$mathbb{F}_q^{,n}$</tex>\u0000 is asymptotically (as \u0000<tex>$n to infty$</tex>\u0000) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of \u0000<tex>$mathbb{F}_q^{,n}$</tex>\u0000 is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1223-1234"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa068","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49947644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let f be a Maass form for $SL(3,{mathbb{Z}})$ with coefficients A f (m, n). The aim of this paper is to study the asymptotic behaviour of the sum $sum_{mleqslant x,nleqslant y}|A_f(m,n)|^2$ and some other related sums.
{"title":"On Sums Of Coefficients Of Maass Forms For SL(3, ℤ)","authors":"Wenguang Zhai","doi":"10.1093/qmath/haab012","DOIUrl":"https://doi.org/10.1093/qmath/haab012","url":null,"abstract":"Let f be a Maass form for \u0000<tex>$SL(3,{mathbb{Z}})$</tex>\u0000 with coefficients A\u0000<inf>f</inf>\u0000(m, n). The aim of this paper is to study the asymptotic behaviour of the sum \u0000<tex>$sum_{mleqslant x,nleqslant y}|A_f(m,n)|^2$</tex>\u0000 and some other related sums.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1435-1460"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haab012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explicitly construct the Dirichlet series $$begin{equation*}L_{mathrm{Tam}}(s):=sum_{m=1}^{infty}frac{P_{mathrm{Tam}}(m)}{m^s},end{equation*}$$ where $P_{mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/mathbb{Q}$ in short Weierstrass form with Tamagawa product m. Although there are no $E/mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{mathrm{Tam}}(1)={0.5053dots}$ . As a corollary, we find that $L_{mathrm{Tam}}(-1)={1.8193dots}$ is the average Tamagawa product for elliptic curves over $mathbb{Q}$ . We give an application of these results to canonical and Weil heights.
{"title":"Tamagawa Products of Elliptic Curves Over ℚ","authors":"Michael Griffin;Ken Onowei-Lun Tsai;Wei-Lun Tsai","doi":"10.1093/qmath/haab042","DOIUrl":"https://doi.org/10.1093/qmath/haab042","url":null,"abstract":"We explicitly construct the Dirichlet series \u0000<tex>$$begin{equation*}L_{mathrm{Tam}}(s):=sum_{m=1}^{infty}frac{P_{mathrm{Tam}}(m)}{m^s},end{equation*}$$</tex>\u0000 where \u0000<tex>$P_{mathrm{Tam}}(m)$</tex>\u0000 is the proportion of elliptic curves \u0000<tex>$E/mathbb{Q}$</tex>\u0000 in short Weierstrass form with Tamagawa product m. Although there are no \u0000<tex>$E/mathbb{Q}$</tex>\u0000 with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is \u0000<tex>$P_{mathrm{Tam}}(1)={0.5053dots}$</tex>\u0000. As a corollary, we find that \u0000<tex>$L_{mathrm{Tam}}(-1)={1.8193dots}$</tex>\u0000 is the average Tamagawa product for elliptic curves over \u0000<tex>$mathbb{Q}$</tex>\u0000. We give an application of these results to canonical and Weil heights.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 4","pages":"1517-1543"},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49978799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe a general procedure, based on Gerstenhaber–Schack complexes, for extending to quantized twistor spaces the Donaldson–Friedman gluing of twistor spaces via deformation theory of singular spaces. We consider in particular various possible quantizations of twistor spaces that leave the underlying spacetime manifold classical, including the geometric quantization of twistor spaces originally constructed by the second author, as well as some variants based on non-commutative geometry. We discuss specific aspects of the gluing construction for these different quantization procedures.
{"title":"Gluing Non-commutative Twistor Spaces","authors":"Matilde Marcolli;Roger Penrose","doi":"10.1093/qmath/haab024","DOIUrl":"10.1093/qmath/haab024","url":null,"abstract":"We describe a general procedure, based on Gerstenhaber–Schack complexes, for extending to quantized twistor spaces the Donaldson–Friedman gluing of twistor spaces via deformation theory of singular spaces. We consider in particular various possible quantizations of twistor spaces that leave the underlying spacetime manifold classical, including the geometric quantization of twistor spaces originally constructed by the second author, as well as some variants based on non-commutative geometry. We discuss specific aspects of the gluing construction for these different quantization procedures.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"417-454"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42262147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multivector fields and differential forms at the continuum level have respectively two commutative associative products, a third composition product between them and various operators like $partial$ , d and ‘∗’ which are used to describe many nonlinear problems. The point of this paper is to construct consistent direct and inverse systems of finite dimensional approximations to these structures and to calculate combinatorially how these finite dimensional models differ from their continuum idealizations. In a Euclidean background, there is an explicit answer which is natural statistically.
{"title":"Quantitative towers in finite difference calculus approximating the continuum","authors":"R Lawrence;N Ranade;D Sullivan","doi":"10.1093/qmath/haaa060","DOIUrl":"https://doi.org/10.1093/qmath/haaa060","url":null,"abstract":"Multivector fields and differential forms at the continuum level have respectively two commutative associative products, a third composition product between them and various operators like \u0000<tex>$partial$</tex>\u0000, d and ‘∗’ which are used to describe many nonlinear problems. The point of this paper is to construct consistent direct and inverse systems of finite dimensional approximations to these structures and to calculate combinatorially how these finite dimensional models differ from their continuum idealizations. In a Euclidean background, there is an explicit answer which is natural statistically.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"515-545"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qmath/haaa060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).
{"title":"Moduli Spaces of Generalized Hyperpolygons","authors":"Steven Rayan;Laura P Schaposnik","doi":"10.1093/qmath/haaa036","DOIUrl":"https://doi.org/10.1093/qmath/haaa036","url":null,"abstract":"We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"137-161"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Atiyah–Drinfeld–Hitchin–Manin transform and its various generalizations are examples of nonlinear integral transforms between finite-dimensional moduli spaces. This note describes a natural infinite-dimensional generalization, where the transform becomes a map from boundary data to a family of solutions of the self-duality equations in a domain.
{"title":"Infinite-Parameter ADHM Transform","authors":"R. S. Ward","doi":"10.1093/qmath/haaa054","DOIUrl":"https://doi.org/10.1093/qmath/haaa054","url":null,"abstract":"The Atiyah–Drinfeld–Hitchin–Manin transform and its various generalizations are examples of nonlinear integral transforms between finite-dimensional moduli spaces. This note describes a natural infinite-dimensional generalization, where the transform becomes a map from boundary data to a family of solutions of the self-duality equations in a domain.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"72 1-2","pages":"407-415"},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49950711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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