We determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is exponential in n. This is surprising in that the essential dimension of the orthogonal groups is smaller in characteristic 2. We also find the essential dimension of Spin(n) in characteristic 2 for n at most 10.
{"title":"Essential dimension of the spin groups in characteristic 2","authors":"B. Totaro","doi":"10.4171/CMH/452","DOIUrl":"https://doi.org/10.4171/CMH/452","url":null,"abstract":"We determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is exponential in n. This is surprising in that the essential dimension of the orthogonal groups is smaller in characteristic 2. \u0000We also find the essential dimension of Spin(n) in characteristic 2 for n at most 10.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/452","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46223302","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}
Let $G$ be a semisimple Lie group and $Gamma$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $Gammabackslash G$.
{"title":"On the rate of equidistribution of expanding translates of horospheres in $Gammabackslash G$","authors":"Samuel C. Edwards","doi":"10.4171/cmh/513","DOIUrl":"https://doi.org/10.4171/cmh/513","url":null,"abstract":"Let $G$ be a semisimple Lie group and $Gamma$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $Gammabackslash G$.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44232039","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}
For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincare-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or the boundary of $U$ is contained in the basin of a finite family of topological "rotational" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids.
{"title":"A Poincaré–Bendixson theorem for translation lines and applications to prime ends","authors":"Andres Koropecki, A. Passeggi","doi":"10.4171/CMH/457","DOIUrl":"https://doi.org/10.4171/CMH/457","url":null,"abstract":"For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincare-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or the boundary of $U$ is contained in the basin of a finite family of topological \"rotational\" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/457","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42648664","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}
The moduli space QMg of non-zero genus g quadratic differentials has a natural action of G D GL 2 .R/= ̋ ̇ 1 0 0 1 ̨ . The Veech group PSL.X; q/ is the stabilizer of .X; q/ 2 QMg in G. We describe a new algorithm for finding elements of PSL.X; q/ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of H=PSL.X; q/ is greater than zero. Mathematics Subject Classification (2010). 32G15, 30F30.
{"title":"Fundamental domains and generators for lattice Veech groups","authors":"R. E. Mukamel","doi":"10.4171/CMH/406","DOIUrl":"https://doi.org/10.4171/CMH/406","url":null,"abstract":"The moduli space QMg of non-zero genus g quadratic differentials has a natural action of G D GL 2 .R/= ̋ ̇ 1 0 0 1 ̨ . The Veech group PSL.X; q/ is the stabilizer of .X; q/ 2 QMg in G. We describe a new algorithm for finding elements of PSL.X; q/ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of H=PSL.X; q/ is greater than zero. Mathematics Subject Classification (2010). 32G15, 30F30.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/406","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70840626","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}
P. F. D. Santos, R. Hardt, James Lewis, P. Lima-filho
{"title":"An explicit cycle map for the motivic cohomology of real varieties","authors":"P. F. D. Santos, R. Hardt, James Lewis, P. Lima-filho","doi":"10.4171/CMH/416","DOIUrl":"https://doi.org/10.4171/CMH/416","url":null,"abstract":"","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/416","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70840764","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}
We construct connections on $S^1$-equivariant Hamiltonian Floer cohomology, which differentiate with respect to certain formal parameters.
我们构造了$S^1$-等变哈密顿花上同调上的连接，这些连接对某些形式参数微分。
{"title":"Connections on equivariant Hamiltonian Floer cohomology","authors":"P. Seidel","doi":"10.4171/CMH/445","DOIUrl":"https://doi.org/10.4171/CMH/445","url":null,"abstract":"We construct connections on $S^1$-equivariant Hamiltonian Floer cohomology, which differentiate with respect to certain formal parameters.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2016-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/445","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841563","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}
Camillo De Lellis, Andrea Marchese, E. Spadaro, Daniele Valtorta
In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.
{"title":"Rectifiability and upper Minkowski bounds for singularities of harmonic $Q$-valued maps","authors":"Camillo De Lellis, Andrea Marchese, E. Spadaro, Daniele Valtorta","doi":"10.4171/CMH/449","DOIUrl":"https://doi.org/10.4171/CMH/449","url":null,"abstract":"In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2016-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/449","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841972","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}
Let 0 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t∈[0,T] is (c(e)+o(1))T, for all but at most countably many e>0. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of T. Our method is flexible, and can be generalized to other L-functions in the t-aspect, as well as to families of L-functions in the conductor aspect.
{"title":"An effective universality theorem for the Riemann zeta function","authors":"Youness Lamzouri, S. Lester, Maksym Radziwill","doi":"10.4171/CMH/448","DOIUrl":"https://doi.org/10.4171/CMH/448","url":null,"abstract":"Let 0 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t∈[0,T] is (c(e)+o(1))T, for all but at most countably many e>0. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of T. Our method is flexible, and can be generalized to other L-functions in the t-aspect, as well as to families of L-functions in the conductor aspect.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2016-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/448","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841795","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}
We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.
{"title":"Non-arithmetic ball quotients from a configuration of elliptic curves in an Abelian surface","authors":"M. Deraux","doi":"10.4171/CMH/443","DOIUrl":"https://doi.org/10.4171/CMH/443","url":null,"abstract":"We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with Parker and Paupert.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2016-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/443","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841834","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}
We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy. The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation. We partially compute these invariants for certain Lagrangian tori.
{"title":"Lagrangian isotopies and symplectic function theory","authors":"Michael Entov, Y. Ganor, Cedric Membrez","doi":"10.4171/CMH/451","DOIUrl":"https://doi.org/10.4171/CMH/451","url":null,"abstract":"We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy. The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation. We partially compute these invariants for certain Lagrangian tori.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2016-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/CMH/451","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70841763","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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