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作者：Das, Tushar; Simmons, David
作者单位：University of Wisconsin System; University of Wisconsin La Crosse; University of York - UK
摘要：We construct a self-affine sponge in R-3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem tha...
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作者：Marques, Fernando C.; Neves, Andre
作者单位：Princeton University; Imperial College London
摘要：In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open pro...
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作者：Leble, Thomas; Serfaty, Sylvia
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; Institut Universitaire de France; New York University
摘要：We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which ar...
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作者：Chodosh, Otis; Ketover, Daniel; Maximo, Davi
作者单位：University of Cambridge; Princeton University; Stanford University; University of Pennsylvania
摘要：We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold (M-n, g), 3 <= n <= 7 can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems...
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作者：Chhaibi, Reda; Najnudel, Joseph; Nikeghbali, Ashkan
摘要：We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called microscopic level, that is we consider the characteristic polynomial at points whose distance to 1 has order 1 / n. We prove that the rescaled characteristic polynomial does not even have a moment of order one, henc...
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作者：Lomonaco, Luna; Petersen, Carsten Lunde
作者单位：Universidade de Sao Paulo; Roskilde University
摘要：Douady and Hubbard (Ann Sci Ec Norm Suppl 4 18(2):287-343, 1985) introduced the notion of polynomial-like maps. They used it to identify homeomorphic copies of the Mandelbrot set inside the Mandelbrot set . These copies can be primitive (with a root cusp) or satellite (without a root cusp). They conjectured that the primitive copies are quasiconformally homeomorphic to , and that the satellite ones are quasiconformally homeomorphic to outside any small neighbourhood of the root. These conjectu...
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作者：Wang, Zhiren
作者单位：Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park
摘要：We show that the Mobius function is disjoint to every analytic skew product dynamical system on over a rotation of the circle.
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作者：Boutonnet, Remi; Ioana, Adrian; Golsefidy, Alireza Salehi
作者单位：Centre National de la Recherche Scientifique (CNRS); Inria; Universite de Bordeaux; University of California System; University of California San Diego
摘要：We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action , whenever is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83-121, 2008; J Eur Math Soc (JEMS) 14:1455-1511, 2012), and Benoist and de Saxc, (Invent Math 205:337-361, 2016). We present se...
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作者：Caffarelli, Luis; Ros-Oton, Xavier; Serra, Joaquim
作者单位：University of Texas System; University of Texas Austin; Universitat Politecnica de Catalunya
摘要：We study the obstacle problem for integro-differential operators of order 2s, with s is an element of(0,1). Our main results establish the optimal C<^>{1,s} regularity of solutions u, and the C<^>{1,\alpha} regularity of the free boundary near regular points. Namely, we prove the following dichotomy at all free boundary points x_0\in\partial\{u=\varphi\}: (i) either u(x)-\varphi(x)=c\,d<^>{1+s}(x)+o(|x-x_0|<^>{1+s+\alpha}) for some c>0, (ii) or u(x)-\varphi(x)=o(|x-x_0|<^>{1+s+\alpha}), where ...
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作者：Sam, Steven V.
作者单位：University of Wisconsin System; University of Wisconsin Madison
摘要：Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C(r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for p...
