Pub Date : 2021-02-07DOI: 10.1007/s11537-021-2116-3
Manuel Gonz'alez, Tomasz Kania
{"title":"Grothendieck spaces: the landscape and perspectives","authors":"Manuel Gonz'alez, Tomasz Kania","doi":"10.1007/s11537-021-2116-3","DOIUrl":"https://doi.org/10.1007/s11537-021-2116-3","url":null,"abstract":"","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"16 1","pages":"247 - 313"},"PeriodicalIF":1.5,"publicationDate":"2021-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48203409","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}
Pub Date : 2021-01-26DOI: 10.1007/s11537-021-2109-2
B. Bakalov, Alberto De Sole, Reimundo Heluani, V. Kac, V. Vignoli
{"title":"Classical and variational Poisson cohomology","authors":"B. Bakalov, Alberto De Sole, Reimundo Heluani, V. Kac, V. Vignoli","doi":"10.1007/s11537-021-2109-2","DOIUrl":"https://doi.org/10.1007/s11537-021-2109-2","url":null,"abstract":"","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"16 1","pages":"203 - 246"},"PeriodicalIF":1.5,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11537-021-2109-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45557578","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}
Pub Date : 2021-01-02DOI: 10.1007/s11537-020-1920-5
Shun-ichi Amari
Information geometry has emerged from the study of the invariant structure in families of probability distributions. This invariance uniquely determines a second-order symmetric tensor g and third-order symmetric tensor T in a manifold of probability distributions. A pair of these tensors (g, T) defines a Riemannian metric and a pair of affine connections which together preserve the metric. Information geometry involves studying a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. A dually flat Riemannian manifold is particularly useful for various applications, because a generalized Pythagorean theorem and projection theorem hold. The Wasserstein distance gives another important geometry on probability distributions, which is non-invariant but responsible for the metric properties of a sample space. I attempt to construct information geometry of the entropy-regularized Wasserstein distance.
{"title":"Information geometry","authors":"Shun-ichi Amari","doi":"10.1007/s11537-020-1920-5","DOIUrl":"https://doi.org/10.1007/s11537-020-1920-5","url":null,"abstract":"<p>Information geometry has emerged from the study of the invariant structure in families of probability distributions. This invariance uniquely determines a second-order symmetric tensor <i>g</i> and third-order symmetric tensor <i>T</i> in a manifold of probability distributions. A pair of these tensors (<i>g, T</i>) defines a Riemannian metric and a pair of affine connections which together preserve the metric. Information geometry involves studying a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. A dually flat Riemannian manifold is particularly useful for various applications, because a generalized Pythagorean theorem and projection theorem hold. The Wasserstein distance gives another important geometry on probability distributions, which is non-invariant but responsible for the metric properties of a sample space. I attempt to construct information geometry of the entropy-regularized Wasserstein distance.</p>","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537780","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}
Pub Date : 2021-01-01DOI: 10.1007/s11537-020-2036-7
M. Guest
{"title":"Topological-antitopological fusion and the quantum cohomology of Grassmannians","authors":"M. Guest","doi":"10.1007/s11537-020-2036-7","DOIUrl":"https://doi.org/10.1007/s11537-020-2036-7","url":null,"abstract":"","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":" 72","pages":"155 - 183"},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11537-020-2036-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41311134","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}
Pub Date : 2020-12-13DOI: 10.1007/s11537-021-2110-9
Adeel A. Khan
{"title":"K-theory and G-theory of derived algebraic stacks","authors":"Adeel A. Khan","doi":"10.1007/s11537-021-2110-9","DOIUrl":"https://doi.org/10.1007/s11537-021-2110-9","url":null,"abstract":"","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"17 1","pages":"1 - 61"},"PeriodicalIF":1.5,"publicationDate":"2020-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48623782","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}
Pub Date : 2020-11-07DOI: 10.1007/s11537-022-2073-5
S. V. Gusev, Edmond W. H. Lee, B. M. Vernikov
{"title":"The lattice of varieties of monoids","authors":"S. V. Gusev, Edmond W. H. Lee, B. M. Vernikov","doi":"10.1007/s11537-022-2073-5","DOIUrl":"https://doi.org/10.1007/s11537-022-2073-5","url":null,"abstract":"","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"17 1","pages":"117 - 183"},"PeriodicalIF":1.5,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48858224","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}
Pub Date : 2020-05-19DOI: 10.1007/s11537-020-1728-3
Shamgar Gurevich, Roger Howe
There is both theoretical and numerical evidence that the set of irreducible representations of a reductive group over local or finite fields is naturally partitioned into families according to analytic properties of representations. Examples of such properties are the rate of decay at infinity of “matrix coefficients” in the local field setting, and the order of magnitude of “character ratios” in the finite field situation.In these notes we describe known results, new results, and conjectures in the theory of “size” of representations of classical groups over finite fields (when correctly stated, most of them hold also in the local field setting), whose ultimate goal is to classify the above mentioned families of representations and accordingly to estimate the relevant analytic properties of each family.Specifically, we treat two main issues: the first is the introduction of a rigorous definition of a notion of size for representations of classical groups, and the second issue is a method to construct and obtain information on each family of representation of a given size.In particular, we propose several compatible notions of size that we call U-rank, tensor rank and asymptotic rank, and we develop a method called eta correspondence to construct the families of representation of each given rank.Rank suggests a new way to organize the representations of classical groups over finite and local fields—a way in which the building blocks are the “smallest” representations. This is in contrast to Harish-Chandra’s philosophy of cusp forms that is the main organizational principle since the 60s, and in it the building blocks are the cuspidal representations which are, in some sense, the “largest”. The philosophy of cusp forms is well adapted to establishing the Plancherel formula for reductive groups over local fields, and led to Lusztig’s classification of the irreducible representations of such groups over finite fields. However, the understanding of certain analytic properties, such as those mentioned above, seems to require a different approach.
{"title":"Rank and duality in representation theory","authors":"Shamgar Gurevich, Roger Howe","doi":"10.1007/s11537-020-1728-3","DOIUrl":"https://doi.org/10.1007/s11537-020-1728-3","url":null,"abstract":"There is both theoretical and numerical evidence that the set of irreducible representations of a reductive group over local or finite fields is naturally partitioned into families according to analytic properties of representations. Examples of such properties are the rate of decay at infinity of “matrix coefficients” in the local field setting, and the order of magnitude of “character ratios” in the finite field situation.In these notes we describe known results, new results, and conjectures in the theory of “size” of representations of classical groups over finite fields (when correctly stated, most of them hold also in the local field setting), whose ultimate goal is to classify the above mentioned families of representations and accordingly to estimate the relevant analytic properties of each family.Specifically, we treat two main issues: the first is the introduction of a rigorous definition of a notion of size for representations of classical groups, and the second issue is a method to construct and obtain information on each family of representation of a given size.In particular, we propose several compatible notions of size that we call <i>U-rank, tensor rank and asymptotic rank</i>, and we develop a method called <i>eta correspondence</i> to construct the families of representation of each given rank.Rank suggests a new way to organize the representations of classical groups over finite and local fields—a way in which the building blocks are the “smallest” representations. This is in contrast to Harish-Chandra’s philosophy of cusp forms that is the main organizational principle since the 60s, and in it the building blocks are the cuspidal representations which are, in some sense, the “largest”. The philosophy of cusp forms is well adapted to establishing the Plancherel formula for reductive groups over local fields, and led to Lusztig’s classification of the irreducible representations of such groups over finite fields. However, the understanding of certain analytic properties, such as those mentioned above, seems to require a different approach.","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"137 ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505936","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}
Pub Date : 2020-03-04DOI: 10.1007/s11537-019-1822-6
Nicolas Bergeron, Pierre Charollois, Luis E. Garcia
These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh.In this work-in-progress we give a new construction of some Eisenstein classes for GLN (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SLN (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GLN, GL1). This suggests looking to reductive dual pairs (GLN, GLk) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms.In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical.
{"title":"Transgressions of the Euler class and Eisenstein cohomology of GL N (Z)","authors":"Nicolas Bergeron, Pierre Charollois, Luis E. Garcia","doi":"10.1007/s11537-019-1822-6","DOIUrl":"https://doi.org/10.1007/s11537-019-1822-6","url":null,"abstract":"These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh.In this work-in-progress we give a new construction of some Eisenstein classes for GL<sub><i>N</i></sub> (<b>Z</b>) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SL<sub><i>N</i></sub> (<b>Z</b>) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a <i>regularized theta lift</i> for the reductive dual pair (GL<sub><i>N</i></sub>, GL<sub>1</sub>). This suggests looking to reductive dual pairs (GL<sub><i>N</i></sub>, GL<sub><i>k</i></sub>) with <i>k</i> ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms.In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical.","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"129 ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505941","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}
Pub Date : 2020-02-10DOI: 10.1007/s11537-020-2034-9
B. Bakalov, Alberto De Sole, V. Kac
{"title":"Computation of cohomology of vertex algebras","authors":"B. Bakalov, Alberto De Sole, V. Kac","doi":"10.1007/s11537-020-2034-9","DOIUrl":"https://doi.org/10.1007/s11537-020-2034-9","url":null,"abstract":"","PeriodicalId":54908,"journal":{"name":"Japanese Journal of Mathematics","volume":"16 1","pages":"81 - 154"},"PeriodicalIF":1.5,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s11537-020-2034-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47482716","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}
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