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Word Measures on Unitary Groups: Improved Bounds for Small Representations 单元群上的词量：小表征的改进界限
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-21 DOI: 10.1093/imrn/rnae100
Yaron Brodsky
{"title":"Word Measures on Unitary Groups: Improved Bounds for Small Representations","authors":"Yaron Brodsky","doi":"10.1093/imrn/rnae100","DOIUrl":"https://doi.org/10.1093/imrn/rnae100","url":null,"abstract":"Let $F$ be a free group of rank $r$ and fix some $win F$. For any compact group $G$ we can define a measure $mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $mu _{w,U(n)}$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"7 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Wasserstein Distance for Ricci Shrinkers 利玛窦收缩器的瓦瑟斯坦距离
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-15 DOI: 10.1093/imrn/rnae099
Franciele Conrado, Detang Zhou
{"title":"The Wasserstein Distance for Ricci Shrinkers","authors":"Franciele Conrado, Detang Zhou","doi":"10.1093/imrn/rnae099","DOIUrl":"https://doi.org/10.1093/imrn/rnae099","url":null,"abstract":"Let $(M^{n},g,f)$ be a Ricci shrinker such that $text{Ric}_{f}=frac{1}{2}g$ and the measure induced by the weighted volume element $(4pi )^{-frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $pin M$, we consider two probability measures defined in the tangent space $T_{p}M$, namely the Gaussian measure $gamma $ and the measure $overline{nu }$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_{0}$ between the measures $overline{nu }$ and $gamma $, and which also elucidates the rigidity implications resulting from this estimate.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"33 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Equivariant Cohomology and Conditional Oriented Matroids 等价同调与条件定向矩阵
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-14 DOI: 10.1093/imrn/rnad025
Galen Dorpalen-Barry, Nicholas Proudfoot, Jidong Wang
{"title":"Equivariant Cohomology and Conditional Oriented Matroids","authors":"Galen Dorpalen-Barry, Nicholas Proudfoot, Jidong Wang","doi":"10.1093/imrn/rnad025","DOIUrl":"https://doi.org/10.1093/imrn/rnad025","url":null,"abstract":"We give a cohomological interpretation of the Heaviside filtration on the Varchenko–Gelfand ring of a pair $({mathcal{A}},{mathcal{K}})$, where ${mathcal{A}}$ is a real hyperplane arrangement and ${mathcal{K}}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where ${mathcal{K}}$ is the ambient vector space. We also define the Gelfand–Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand–Rybnikov ring of an oriented matroid and the aforementioned Varchenko–Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"12 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Annihilating Branching Brownian Motion 湮没分支布朗运动
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-09 DOI: 10.1093/imrn/rnae068
Daniel Ahlberg, Omer Angel, Brett Kolesnik
{"title":"Annihilating Branching Brownian Motion","authors":"Daniel Ahlberg, Omer Angel, Brett Kolesnik","doi":"10.1093/imrn/rnae068","DOIUrl":"https://doi.org/10.1093/imrn/rnae068","url":null,"abstract":"We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"20 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Induced Subgraph Density. I. A loglog Step Towards Erd̋s–Hajnal 诱导子图密度。I. 迈向 Erd̋s-Hajnal 的日志步骤
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-07 DOI: 10.1093/imrn/rnae065
Matija Bucić, Tung Nguyen, Alex Scott, Paul Seymour
{"title":"Induced Subgraph Density. I. A loglog Step Towards Erd̋s–Hajnal","authors":"Matija Bucić, Tung Nguyen, Alex Scott, Paul Seymour","doi":"10.1093/imrn/rnae065","DOIUrl":"https://doi.org/10.1093/imrn/rnae065","url":null,"abstract":"In 1977, Erd̋s and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^{c}$, and they proved that this is true with $ |G|^{c}$ replaced by $2^{csqrt{log |G|}}$. Until now, there has been no improvement on this result (for general $H$). We prove a strengthening: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|ge 2$ has a clique or stable set of size at least $$ begin{align*} &2^{csqrt{log |G|loglog|G|}}.end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"42 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Arthur Packets for Quasisplit GSp(2n) and GO(2n) Over a p-Adic Field p 阿基米德域上的 Quasisplit GSp(2n) 和 GO(2n) 的亚瑟数据包
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-07 DOI: 10.1093/imrn/rnae086
Bin Xu
{"title":"Arthur Packets for Quasisplit GSp(2n) and GO(2n) Over a p-Adic Field","authors":"Bin Xu","doi":"10.1093/imrn/rnae086","DOIUrl":"https://doi.org/10.1093/imrn/rnae086","url":null,"abstract":"We construct the Arthur packets for symplectic and even orthogonal similitude groups over a $p$-adic field and show that they are stable and satisfy the twisted endoscopic character relations.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"20 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Elekes–Rónyai Theorem for Sets With Few Products 少乘积集合的 Elekes-Rónyai 定理
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-07 DOI: 10.1093/imrn/rnae087
Akshat Mudgal
{"title":"An Elekes–Rónyai Theorem for Sets With Few Products","authors":"Akshat Mudgal","doi":"10.1093/imrn/rnae087","DOIUrl":"https://doi.org/10.1093/imrn/rnae087","url":null,"abstract":"Given $n in mathbb{N}$, we call a polynomial $F in mathbb{C}[x_{1},dots ,x_{n}]$ degenerate if there exist $Pin mathbb{C}[y_{1}, dots , y_{n-1}]$ and monomials $m_{1}, dots , m_{n-1}$ with fractional exponents, such that $F = P(m_{1}, dots , m_{n-1})$. Our main result shows that whenever a polynomial $F$, with degree $d geq 1$, is non-degenerate, then for every finite, non-empty set $Asubset mathbb{C}$ such that $|Acdot A| leq K|A|$, one has $$ begin{align*} & |F(A, dots, A)| gg |A|^{n} 2^{-O_{d,n}((log 2K)^{3 + o(1)})}. end{align*} $$This is sharp since for every degenerate $F$ and finite set $A subset mathbb{C}$ with $|Acdot A| leq K|A|$, one has $$ begin{align*} & |F(A,dots,A)| ll K^{O_{F}(1)}|A|^{n-1}.end{align*} $$Our techniques rely on Freiman type inverse theorems and Schmidt’s subspace theorem.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"45 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Regularity Theory for Nonlocal Equations with General Growth in the Heisenberg Group 海森堡群中具有一般增长的非局部方程的正则理论
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-02 DOI: 10.1093/imrn/rnae072
Yuzhou Fang, Chao Zhang
{"title":"Regularity Theory for Nonlocal Equations with General Growth in the Heisenberg Group","authors":"Yuzhou Fang, Chao Zhang","doi":"10.1093/imrn/rnae072","DOIUrl":"https://doi.org/10.1093/imrn/rnae072","url":null,"abstract":"We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate in the spirit of De Giorgi-Nash-Moser theory, some local properties of weak solutions to such kind of problems, involving boundedness, Hölder continuity and Harnack inequality. To this end, an improved nonlocal Caccioppoli-type estimate as the main auxiliary ingredient is exploited several times.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"30 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Parabolic Analog of a Theorem of Beilinson and Schechtman 贝林森和谢赫特曼定理的抛物线类比
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-02 DOI: 10.1093/imrn/rnae085
Indranil Biswas, Swarnava Mukhopadhyay, Richard Wentworth
{"title":"A Parabolic Analog of a Theorem of Beilinson and Schechtman","authors":"Indranil Biswas, Swarnava Mukhopadhyay, Richard Wentworth","doi":"10.1093/imrn/rnae085","DOIUrl":"https://doi.org/10.1093/imrn/rnae085","url":null,"abstract":"For a simple, simply connected, complex group $G$, we prove an explicit formula to compute the Atiyah class of parabolic determinant of cohomology line bundle on the moduli space of parabolic $G$-bundles. This generalizes an earlier result of Beilinson-Schechtman.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"24 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Irreducibility of the Spaces of Rational Curves on del Pezzo Manifolds 德尔佩佐积分上有理曲线空间的不可还原性
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-29 DOI: 10.1093/imrn/rnae080
Fumiya Okamura
{"title":"The Irreducibility of the Spaces of Rational Curves on del Pezzo Manifolds","authors":"Fumiya Okamura","doi":"10.1093/imrn/rnae080","DOIUrl":"https://doi.org/10.1093/imrn/rnae080","url":null,"abstract":"We prove the irreducibility of the spaces of rational curves on del Pezzo manifolds of Picard rank $1$ and dimension $n ge 4$ by analyzing the fibers of evaluation maps. As a corollary, we prove Geometric Manin’s Conjecture in these cases.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"59 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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