We provide a short and self-contained overview of the techniques based on motivic integration as they are applied in harmonic analysis on p-adic groups; our target audience is mainly representation theorists with no back- ground in model theory (and model theorists with an interest in recent ap- plications of motivic integration in representation theory, though we do not provide any representation theory background). We aim to give a fairly com- prehensive survey of the results in harmonic analysis that were proved by such techniques in the last ten years, with emphasis on the most recent techniques and applications from (13), (8), and (32, Appendix B).
{"title":"Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups","authors":"R. Cluckers, J. Gordon, Immanuel Halupczok","doi":"10.3934/ERA.2014.21.137","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.137","url":null,"abstract":"We provide a short and self-contained overview of the techniques based on motivic integration as they are applied in harmonic analysis on p-adic groups; our target audience is mainly representation theorists with no back- ground in model theory (and model theorists with an interest in recent ap- plications of motivic integration in representation theory, though we do not provide any representation theory background). We aim to give a fairly com- prehensive survey of the results in harmonic analysis that were proved by such techniques in the last ten years, with emphasis on the most recent techniques and applications from (13), (8), and (32, Appendix B).","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"137-152"},"PeriodicalIF":0.0,"publicationDate":"2014-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An example is given of a compact quotient of the unit ball in $mathbb{C}^2$ by an arithmetic group acting freely such that the Albanese variety is not of CM type. Such examples do not exist for congruence subgroups.
{"title":"An arithmetic ball quotient surface whose Albanese variety is not of CM type","authors":"Chad Schoen","doi":"10.3934/ERA.2014.21.132","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.132","url":null,"abstract":"An example is given of a compact quotient of the unit ball in $mathbb{C}^2$ by an arithmetic group acting freely such that the Albanese variety is not of CM type. \u0000Such examples do not exist for congruence subgroups.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"132-136"},"PeriodicalIF":0.0,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3934/ERA.2014.21.132","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We calculate the the sharp constant and characterize the extremal initial data in $dot{H}^{frac{3}{4}} times dot{H}^{-frac{1}{4}}$ for the $L^4$ Sobolev--Strichartz estimate for the wave equation in four spatial dimensions.
我们计算了四个空间维度波动方程的$L^4$ Sobolev—Strichartz估计的尖锐常数，并在$dot{H}^{frac{3}{4}} times dot{H}^{-frac{1}{4}}$中描述了极值初始数据。
{"title":"A sharp Sobolev-Strichartz estimate for the wave equation","authors":"N. Bez, Chris Jeavons","doi":"10.3934/ERA.2015.22.46","DOIUrl":"https://doi.org/10.3934/ERA.2015.22.46","url":null,"abstract":"We calculate the the sharp constant and characterize the extremal initial data in $dot{H}^{frac{3}{4}} times dot{H}^{-frac{1}{4}}$ for the $L^4$ Sobolev--Strichartz estimate for the wave equation in four spatial dimensions.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"22 1","pages":"46-54"},"PeriodicalIF":0.0,"publicationDate":"2014-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70234044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we announce the following result. We show that any $n$-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of $mathbb{R}^n$ by an action of a crystallographic group. We describe all such actions. We start with a history, results and open questions concerning estimates on the number of extremal subsets in Alexandrov spaces. Then we give the plan of the proof of our result; the complete proof will published elsewhere.
{"title":"Number of extremal subsets in Alexandrov spaces and rigidity","authors":"N. Lebedeva","doi":"10.3934/ERA.2014.21.120","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.120","url":null,"abstract":"In this paper we announce the following result. We show that any $n$-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of $mathbb{R}^n$ by an action of a crystallographic group. We describe all such actions. We start with a history, results and open questions concerning estimates on the number of extremal subsets in Alexandrov spaces. Then we give the plan of the proof of our result; the complete proof will published elsewhere.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"44 1","pages":"120-125"},"PeriodicalIF":0.0,"publicationDate":"2014-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Hladký, Diana Piguet, M. Simonovits, M. Stein, E. Szemerédi
Loebl, Komlos and Sos conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. For our proof, we use a structural decomposition which can be seen as an analogue of Szemeredi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].
{"title":"THE APPROXIMATE LOEBL-KOMLOS-SOS CONJECTURE AND EMBEDDING TREES IN SPARSE GRAPHS","authors":"J. Hladký, Diana Piguet, M. Simonovits, M. Stein, E. Szemerédi","doi":"10.3934/era.2015.22.1","DOIUrl":"https://doi.org/10.3934/era.2015.22.1","url":null,"abstract":"Loebl, Komlos and Sos conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices \u0000of degree at least $k$ contains each tree $T$ of order $k+1$ as a \u0000subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. \u0000 \u0000For our proof, we use a structural decomposition which can be seen as an analogue of Szemeredi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$. \u0000 \u0000The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"8 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2014-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.
{"title":"Fixed frequency eigenfunction immersions and supremum norms of random waves","authors":"Y. Canzani, B. Hanin","doi":"10.3934/era.2015.22.76","DOIUrl":"https://doi.org/10.3934/era.2015.22.76","url":null,"abstract":"A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"22 1","pages":"76-86"},"PeriodicalIF":0.0,"publicationDate":"2014-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70234085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a family of examples of non-associative algebras ${R_alpha ,vert, 1
我们构造了一组非结合代数${R_alpha ,vert, 1
{"title":"ON EXISTENCE OF PI-EXPONENTS OF CODIMENSION GROWTH","authors":"M. Zaicev","doi":"10.3934/ERA.2014.21.113","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.113","url":null,"abstract":"We construct a family of examples of non-associative algebras ${R_alpha ,vert, 1<alphainmathbb R}$ such that $underline{exp}(R_alpha)=1$, $overline{exp}(R_alpha)=alpha$. In particular, it follows that for any $R_alpha$, an ordinary PI-exponent of codimension growth does not exist.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"128 1","pages":"113-119"},"PeriodicalIF":0.0,"publicationDate":"2014-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study dynamical properties of direction foliations on the complex plane pulled back from direction foliations on a half-translation torus $T$, i.e., a torus equipped with a strict and integrable quadratic differential. If the torus $T$ admits a pseudo-Anosov map we give a homological criterion for the appearance of dense leaves and leaves with bounded deviation on the universal covering of $T$, called Panov plane. Our result generalizes Dmitri Panov's explicit construction of dense leaves for certain arithmetic half-translation tori [33]. Certain Panov planes are related to the polygonal table of the periodic wind-tree model. In fact, we show that the dynamics on periodic wind-tree billiards can be converted to the dynamics on a pair of singular planes. Possible strategies to generalize our main dynamical result to larger sets of directions are discussed. Particularly we include recent results of Frączek and Ulcigrai [17, 18] and Delecroix [6] for the wind-tree model. Implicitly Panov planes appear in Frączek and Schmoll [15], where the authors consider Eaton Lens distributions.
{"title":"Pseudo-Anosov eigenfoliations on Panov planes","authors":"Christy Johnson, Martin Schmoll","doi":"10.3934/ERA.2014.21.89","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.89","url":null,"abstract":"We study dynamical properties of direction foliations on the complex plane pulled back from \u0000direction foliations on a half-translation torus $T$, i.e., a torus equipped with a strict \u0000and integrable quadratic differential. \u0000If the torus $T$ admits a pseudo-Anosov map we give a homological criterion for the appearance of dense leaves and leaves with bounded deviation on the universal covering of $T$, called Panov plane. \u0000Our result generalizes Dmitri Panov's explicit construction of dense leaves for certain \u0000arithmetic half-translation tori [33]. Certain Panov planes are related to \u0000the polygonal table of the periodic wind-tree model. In fact, we show that the dynamics \u0000on periodic wind-tree billiards can be converted to the dynamics on a pair of singular planes. \u0000 \u0000Possible strategies to generalize our main dynamical result to larger sets \u0000of directions are discussed. Particularly we include recent results \u0000of Frączek and Ulcigrai [17, 18] and Delecroix [6] \u0000for the wind-tree model. Implicitly Panov planes appear in Frączek and Schmoll [15], \u0000where the authors consider Eaton Lens distributions.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"89-108"},"PeriodicalIF":0.0,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a method to construct equilibrium states via inducing. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Holder continuous potentials. It allows us to prove the existence of phase transition.
{"title":"FROM LOCAL TO GLOBAL EQUILIBRIUM STATES: THERMODYNAMIC FORMALISM VIA AN INDUCING SCHEME","authors":"R. Leplaideur","doi":"10.3934/ERA.2014.21.72","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.72","url":null,"abstract":"We present a method to construct equilibrium states via inducing. This method can be used for some non-uniformly hyperbolic dynamical systems and for non-Holder continuous potentials. It allows us to prove the existence of phase transition.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"72-79"},"PeriodicalIF":0.0,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this note is to study the number of elements in Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone.
本文的目的是研究具有有理多面体伪有效锥的代数曲面上Minkowski基的元数。
{"title":"Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone","authors":"Piotr Pokora, T. Szemberg","doi":"10.3934/era.2014.21.126","DOIUrl":"https://doi.org/10.3934/era.2014.21.126","url":null,"abstract":"The purpose of this note is to study the number of elements in Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"126-131"},"PeriodicalIF":0.0,"publicationDate":"2014-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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