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作者：Ghosh, Amit; Sarnak, Peter
作者单位：Oklahoma State University System; Oklahoma State University - Stillwater; Institute for Advanced Study - USA; Princeton University
摘要：For integers k, we consider the affine cubic surface V-k given by M(x) = x(1)(2) + x(2)(2) + x(3)(2) - x(1)x(2)x(3) = k. We show that for almost all k the Hasse Principle holds, namely that V-k(Z) is non-empty if V-k(Z(p)) is non-empty for all primes p, and that there are infinitely many k's for which it fails. The Markoff morphisms act on V-k(Z) with finitely many orbits and a numerical study points to some basic conjectures about these class numbers and Hasse failures. Some of the analysis m...
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作者：Angles, Bruno; Tuan Ngo Dac; Ribeiro, Floric Tavares
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Caen Normandie
摘要：In 2012 Taelman proved a class formula for Drinfeld F-q[t]-modules. For an arbitrary coefficient ring A, several deep but partial results in the direction of a class formula for Drinfeld A-modules have been obtained. In this paper, using a completely new approach based on the notion of Stark units, we establish the class formula for Drinfeld A-modules in full generality. Further we extend this class formula for A-finite or abelian Anderson modules and more generally for admissible Anderson mod...
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作者：Stokman, J., V; Reshetikhin, N.
作者单位：University of Amsterdam; Tsinghua University; Saint Petersburg State University
摘要：Let G be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on G in which the role of the quasi-simple admissible G-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular ...
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作者：Carrasco, Matias; Mackay, John M.
作者单位：Universidad de la Republica, Uruguay; University of Bristol
摘要：We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without 2-torsion) have confo...
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作者：Ghazouani, Selim
作者单位：Universite Paris Saclay
摘要：In this article we study local rigidity properties of generalised interval exchange maps using renormalisation methods. We study the dynamics of the renormalisation operator R acting on the space of C-3-generalised interval exchange transformations at fixed points (which are standard periodic type IETs). We show that R is hyperbolic and that the number of unstable direction is exactly that predicted by the ergodic theory of IETs and the work of Forni and Marmi-Moussa-Yoccoz. As a consequence w...
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作者：Meeks, William H., III; Tinaglia, Giuseppe
作者单位：University of Massachusetts System; University of Massachusetts Amherst; University of London; King's College London
摘要：We describe the lamination limits of sequences of compact disks M-n embedded in R-3 with constant mean curvature H-n, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi (Ann Math 160:573-615, 2004) for minimal disks. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in R-3 with constant mean curvature; this chord arc result generalizes Theorem...
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作者：Pigati, Alessandro; Stern, Daniel
作者单位：Swiss Federal Institutes of Technology Domain; ETH Zurich; Princeton University
摘要：Given a Hermitian line bundle L -> M over a closed, oriented Riemannian manifoldM, we study the asymptotic behavior, as epsilon -> 0, of couples(u(epsilon), del(epsilon)) critical for the rescalings E epsilon(u, del)= (M)integral(vertical bar del u vertical bar(2)+epsilon(2)vertical bar F-del vertical bar(2)+1/4 epsilon(2)(1-vertical bar u vertical bar(2))(2)) of the self-dual Yang-Mills-Higgs energy, whereuis a section of Land del is a Hermitian connection on L with curvature F-del. Under the...
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作者：Daskalopoulos, Georgios; Mese, Chikako
作者单位：Brown University; Johns Hopkins University
摘要：We prove the holomorphic rigidity conjecture of Teichmuller space which loosely speaking states that the action of the mapping class group uniquely determines theTeichmuller space as a complex manifold. The method of proof is through harmonicmaps. We prove that the singular set of a harmonic map from a smooth n-dimensional Riemannian domain to theWeil-Petersson completion T of Teichmuller space has Hausdorff dimension at most n - 2, and moreover, u has certain decay near the singular set. Comb...
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作者：Gwynne, Ewain; Miller, Jason
作者单位：University of Cambridge
摘要：We show that for each gamma is an element of (0, 2), there is a unique metric (i.e., distance function) associated with gamma-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric D-h associated with the Riemannian metric tensor e(gamma h)(dx(2) + dy(2)) on C which is characterized by a certain list of axioms: it is locally determined by h and it transforms appropriately when either adding a continuous fun...
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作者：Avramidi, Grigori; Okun, Boris; Schreve, Kevin
作者单位：Max Planck Society; University of Wisconsin System; University of Wisconsin Milwaukee; University of Chicago
摘要：We compute the mod p homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes p, we construct closed locally CAT(0) manifolds with nonzero mod p homology growth outside the middle dimension. These example...
