{"title":"Affine symmetries in quantum cohomology: corrections and new results","authors":"Pierre-Emmanuel Chaput, Nicolas Perrin","doi":"10.4310/mrl.2023.v30.n2.a3","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a3","url":null,"abstract":"In a previous paper Affine symmetries of the equivariant quantum cohomology of rational homogeneous spaces, a general formula was given for the multiplication by some special Schubert classes in the quantum cohomology of any homogeneous space. Although this formula is correct in the non equivariant setting, the stated equivariant version was wrong. We provide corrections for the equivariant formula, thus giving a correct argument for the non equivariant formula. We also give new formulas in the equivariant homology of the affine grassmannian that could lead to Pieri type formulas.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Derived categories of Quot schemes of locally free quotients via categorified Hall products","authors":"Yukinobu Toda","doi":"10.4310/mrl.2023.v30.n1.a10","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a10","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Refinements of strong multiplicity one for $mathrm{GL}(2)$","authors":"PENG-JIE Wong","doi":"10.4310/mrl.2022.v29.n2.a11","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n2.a11","url":null,"abstract":"For distinct unitary cuspidal automorphic representations π1 and π2 for GL(2) over a number field F and any α ∈ R, let Sα be the set of primes v of F for which λπ1(v) 6= eλπ2(v), where λπi(v) is the Fourier coefficient of πi at v. In this article, we show that the lower Dirichlet density of Sα is at least 1 16 . Moreover, if π1 and π2 are not twist-equivalent, we show that the lower Dirichlet densities of Sα and ∩α Sα are at least 2 13 and 1 11 , respectively. Furthermore, for non-twist-equivalent π1 and π2, if each πi corresponds to a non-CM newform of weight ki ≥ 2 and with trivial nebentypus, we obtain various upper bounds for the number of primes p ≤ x such that λπ1(p) 2 = λπ2(p) . These present refinements of the works of Murty-Pujahari, Murty-Rajan, Ramakrishnan, and Walji.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49562993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Surface Group Conjectures for groups with two generators","authors":"Giles Gardam, Dawid Kielak, Alan D. Logan","doi":"10.4310/mrl.2023.v30.n1.a5","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a5","url":null,"abstract":"The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46449068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Internal stabilization for KdV-BBM equation on a periodic domain","authors":"Melek Jellouli, M. Khenissi","doi":"10.4310/mrl.2022.v29.n6.a4","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n6.a4","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The local information of difference equations","authors":"Moisés Herradón Cueto","doi":"10.4310/mrl.2022.v29.n1.a5","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n1.a5","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Diagrams of $star$-trisections","authors":"Rom'an Aranda, Jesse Moeller","doi":"10.4310/mrl.2022.v29.n6.a1","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n6.a1","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Convex hull property for ancient harmonic map heat flows","authors":"C. Sung","doi":"10.4310/mrl.2022.v29.n5.a12","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n5.a12","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New $k$-th Yau algebras of isolated hypersurface singularities and weak Torelli-type theorem","authors":"Naveed Hussain, S. Yau, Huaiqing Zuo","doi":"10.4310/mrl.2022.v29.n2.a7","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n2.a7","url":null,"abstract":"Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : ( C n , 0) → ( C , 0). The Yau algebra L ( V ) is defined to be the Lie algebra of derivations of the moduli algebra A ( V ) := O n / ( f, ∂f∂x 1 , · · · , ∂f∂x n ), i.e., L ( V ) = Der( A ( V ) , A ( V )). It is known that L ( V ) is a finite dimensional Lie algebra and its dimension λ ( V ) is called Yau number. In this paper, we introduce a new series of Lie algebras, i.e., k -th Yau algebras L k ( V ), k ≥ 0, which are a generalization of Yau algebra. L k ( V ) is defined to be the Lie algebra of derivations of the k th moduli algebra A k ( V ) := O n / ( f, m k J ( f )) , k ≥ 0, i.e., L k ( V ) = Der( A k ( V ) , A k ( V )), where m is the maximal ideal of O n . The k -th Yau number is the dimension of L k ( V ) which we denote as λ k ( V ). In particular, L 0 ( V ) is exactly the Yau algebra, i.e., L 0 ( V ) = L ( V ) , λ 0 ( V ) = λ ( V ). These numbers λ k ( V ) are new numerical analytic invariants of singularities. In this paper we obtain the weak Torelli-type theorems of simple elliptic singularities using Lie algebras L 1 ( V ) and L 2 ( V ). We shall also characterize the simple singularities completely using L 1 ( V ).","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}