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 $mu _{w,U(n)}$.
{"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}
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.
{"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}
Galen Dorpalen-Barry, Nicholas Proudfoot, Jidong Wang
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.
{"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}
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.
{"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}
Matija Bucić, Tung Nguyen, Alex Scott, Paul Seymour
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.
{"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}
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.
{"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}
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.
给定 $n in mathbb{N}$，如果存在 $Pin mathbb{C}[y_{1}、dots , y_{n-1}]$ 中存在 $P 和小数指数的单项式 $m_{1}, dots , m_{n-1}$，使得 $F = P(m_{1}, dots , m_{n-1})$ 退化。我们的主要结果表明，每当阶数为 $d geq 1$ 的多项式 $F$ 是非退化的，那么对于每一个有限非空集 $Asubset mathbb{C}$ ，使得 $|Acdot A| leq K|A|$，都有 $$ begin{align*} &；|F(A, dots, A)| gg |A|^{n} 2^{-O_{d,n}((log 2K)^{3 + o(1)})}.end{align*}$$This is sharp since for every degenerate $F$ and finite set $A subset mathbb{C}$ with $|Acdot A| leq K|A|$, one has $$ begin{align*} & |F(A,dots,A)| ll K^{O_{F}(1)}|A|^{n-1}.end{align*}.$$我们的技术依赖于 Freiman 型逆定理和施密特子空间定理。
{"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}
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.
{"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}
Indranil Biswas, Swarnava Mukhopadhyay, Richard Wentworth
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.
{"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}
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.
{"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}
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