Pub Date : 2021-01-01DOI: 10.17323/1609-4514-2021-21-4-737-766
Mario DeFranco DeFranco, P. Gunnells
The classical GUE matrix model of N×N Hermitian matrices equipped with the Gaussian measure can be used to count the orientable topological surfaces by genus obtained through gluing the edges of a polygon. We introduce a variation of the GUE matrix model that that enumerates certain edge-ramified CW complexes obtained from polygon gluings. We do this by replacing the Gaussian measure with a formal analogue related to generating functions that enumerate uniform hypergraphs. Our main results are three different ways to compute expectations of traces of powers. In particular, we show that our matrix model has a topological expansion.
{"title":"Hypergraph Matrix Models","authors":"Mario DeFranco DeFranco, P. Gunnells","doi":"10.17323/1609-4514-2021-21-4-737-766","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-4-737-766","url":null,"abstract":"The classical GUE matrix model of N×N Hermitian matrices equipped with the Gaussian measure can be used to count the orientable topological surfaces by genus obtained through gluing the edges of a polygon. We introduce a variation of the GUE matrix model that that enumerates certain edge-ramified CW complexes obtained from polygon gluings. We do this by replacing the Gaussian measure with a formal analogue related to generating functions that enumerate uniform hypergraphs. Our main results are three different ways to compute expectations of traces of powers. In particular, we show that our matrix model has a topological expansion.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67825750","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}
Pub Date : 2021-01-01DOI: 10.17323/1609-4514-2021-21-2-443-446
I. Arzhantsev, S. Gusein-Zade, Y. Il'yashenko, I. Losev, L. Rybnikov, O. Schwarzman, E. Smirnov, D. A. Timashev, M. Tsfasman
{"title":"Ernest Borisovich Vinberg (obituary)","authors":"I. Arzhantsev, S. Gusein-Zade, Y. Il'yashenko, I. Losev, L. Rybnikov, O. Schwarzman, E. Smirnov, D. A. Timashev, M. Tsfasman","doi":"10.17323/1609-4514-2021-21-2-443-446","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-2-443-446","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67826097","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}
Pub Date : 2021-01-01DOI: 10.17323/1609-4514-2021-21-2-365-382
Yury Kudryashov, Valente Ramírez
{"title":"Spectra of Quadratic Vector Fields on C 2 : the Missing Relation","authors":"Yury Kudryashov, Valente Ramírez","doi":"10.17323/1609-4514-2021-21-2-365-382","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-2-365-382","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67825986","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}
Pub Date : 2020-12-10DOI: 10.17323/1609-4514-2022-22-4-741-757
H. Murahara, Tomokazu Onozuka
In this paper, we present some results on the $a$-points of the symmetric sum of the Euler-Zagier multiple zeta function. Our first three results are for the $a$-points free region of the function. The fourth result is the Riemann-von Mangoldt type formula. In the last two results, we study the real parts of $a$-points of the function.
{"title":"On the a -Points of Symmetric Sum of Multiple Zeta Function","authors":"H. Murahara, Tomokazu Onozuka","doi":"10.17323/1609-4514-2022-22-4-741-757","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-4-741-757","url":null,"abstract":"In this paper, we present some results on the $a$-points of the symmetric sum of the Euler-Zagier multiple zeta function. Our first three results are for the $a$-points free region of the function. The fourth result is the Riemann-von Mangoldt type formula. In the last two results, we study the real parts of $a$-points of the function.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44573273","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}
Pub Date : 2020-11-20DOI: 10.17323/1609-4514-2023-23-1-47-58
Yurii Burman, V. Kulishov
For a finite-dimensional representation V of a group G we introduce and study the notion of a Lie element in the group algebra k[G]. The set L(V) subset k[G] of Lie elements is a Lie algebra and a G-module acting on the original representation V. Lie elements often exhibit nice combinatorial properties. Thus, for G = S_n and V, a permutation representation, we prove a formula for the characteristic polynomial of a Lie element similar to the classical matrix-tree theorem.
{"title":"Lie Elements and the Matrix-Tree Theorem","authors":"Yurii Burman, V. Kulishov","doi":"10.17323/1609-4514-2023-23-1-47-58","DOIUrl":"https://doi.org/10.17323/1609-4514-2023-23-1-47-58","url":null,"abstract":"For a finite-dimensional representation V of a group G we introduce and study the notion of a Lie element in the group algebra k[G]. The set L(V) subset k[G] of Lie elements is a Lie algebra and a G-module acting on the original representation V. \u0000Lie elements often exhibit nice combinatorial properties. Thus, for G = S_n and V, a permutation representation, we prove a formula for the characteristic polynomial of a Lie element similar to the classical matrix-tree theorem.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44218464","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}
Pub Date : 2020-11-09DOI: 10.17323/1609-4514-2022-22-3-377-392
D. Akhiezer, B. Kazarnovskii
It is shown how new integral-geometric formulae can be obtained from the existing formulae of Crofton type. In particular, for classical Crofton formulae in which the answer depends on the Riemannian volume, we obtain generalizations in terms of the mixed Riemannian volume defined in the paper. The method is based on the calculations in the ring of normal densities constructed in the previous work of the authors.
{"title":"Crofton Formulae for Products","authors":"D. Akhiezer, B. Kazarnovskii","doi":"10.17323/1609-4514-2022-22-3-377-392","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-3-377-392","url":null,"abstract":"It is shown how new integral-geometric formulae can be obtained from the existing formulae of Crofton type. In particular, for classical Crofton formulae in which the answer depends on the Riemannian volume, we obtain generalizations in terms of the mixed Riemannian volume defined in the paper. The method is based on the calculations in the ring of normal densities constructed in the previous work of the authors.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47997792","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}
Pub Date : 2020-09-19DOI: 10.17323/1609-4514-2022-22-4-705-739
V. Lunts
This preprint supersedes the previous version, which was only about Kontsevich's conjecture on the relation between the monodromy of a family of (weakly) CY varieties and the action on cohomology of the group of autoequivalences of the derived category of varieties in the mirror dual family. Here we add another conjecture about the relation of the group of autoequivalence of the derived category of a CY variety and its Neron-Severi Lie algebra.
{"title":"Néron–Severi Lie Algebra, Autoequivalences of the Derived Category, and Monodromy","authors":"V. Lunts","doi":"10.17323/1609-4514-2022-22-4-705-739","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-4-705-739","url":null,"abstract":"This preprint supersedes the previous version, which was only about Kontsevich's conjecture on the relation between the monodromy of a family of (weakly) CY varieties and the action on cohomology of the group of autoequivalences of the derived category of varieties in the mirror dual family. Here we add another conjecture about the relation of the group of autoequivalence of the derived category of a CY variety and its Neron-Severi Lie algebra.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41987982","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}
Pub Date : 2020-08-17DOI: 10.17323/1609-4514-2021-21-3-639-652
Hendrik Süß
{"title":"Toric Topology of the Grassmannian of Planes in C 5 and the Del Pezzo Surface of Degree 5","authors":"Hendrik Süß","doi":"10.17323/1609-4514-2021-21-3-639-652","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-3-639-652","url":null,"abstract":"","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46084513","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}
Pub Date : 2020-07-03DOI: 10.17323/1609-4514-2023-23-1-113-120
R. Hidalgo
If $Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $Gamma'$ is co-compact and torsion free, then $S={mathbb H}^{2}/Gamma'$ is a closed Riemann surface of genus $g geq 2$ admitting the abelian group $A=Gamma/Gamma'$ as a group of conformal automorphisms. We say that $A$ is a homology group of $S$. A natural question is if $S$ admits unique homology groups or not, in other words, is there are different Fuchsian groups $Gamma_{1}$ and $Gamma_{2}$ with $Gamma_{1}'=Gamma'_{2}$? It is known that if $Gamma_{1}$ and $Gamma_{2}$ are both of the same signature $(0;k,ldots,k)$, for some $k geq 2$, then the equality $Gamma_{1}'=Gamma_{2}'$ ensures that $Gamma_{1}=Gamma_{2}$. Generalizing this, we observe that if $Gamma_{j}$ has signature $(0;k_{j},ldots,k_{j})$ and $Gamma_{1}'=Gamma'_{2}$, then $Gamma_{1}=Gamma_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${rm Aut}(S)$ of each homology group $A$ is also obtained.
{"title":"Homology Group Automorphisms of Riemann Surfaces","authors":"R. Hidalgo","doi":"10.17323/1609-4514-2023-23-1-113-120","DOIUrl":"https://doi.org/10.17323/1609-4514-2023-23-1-113-120","url":null,"abstract":"If $Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $Gamma'$ is co-compact and torsion free, then $S={mathbb H}^{2}/Gamma'$ is a closed Riemann surface of genus $g geq 2$ admitting the abelian group $A=Gamma/Gamma'$ as a group of conformal automorphisms. We say that $A$ is a homology group of $S$. A natural question is if $S$ admits unique homology groups or not, in other words, is there are different Fuchsian groups $Gamma_{1}$ and $Gamma_{2}$ with $Gamma_{1}'=Gamma'_{2}$? It is known that if $Gamma_{1}$ and $Gamma_{2}$ are both of the same signature $(0;k,ldots,k)$, for some $k geq 2$, then the equality $Gamma_{1}'=Gamma_{2}'$ ensures that $Gamma_{1}=Gamma_{2}$. Generalizing this, we observe that if $Gamma_{j}$ has signature $(0;k_{j},ldots,k_{j})$ and $Gamma_{1}'=Gamma'_{2}$, then $Gamma_{1}=Gamma_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${rm Aut}(S)$ of each homology group $A$ is also obtained.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47771051","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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