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作者：Thurston, Dylan P.
摘要：When is a topological branched self-cover of the sphere equivalent to a post-critically finite rational map on CP1? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive criterion: the branched self-cover is equivalent to a rational map if and only if there is an elastic graph spine for the complement of the post-critical set that gets looser under backwards iteration.
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作者：Smania, Daniel
摘要：We show that in a generic finite-dimensional real-analytic family of real-analytic multimodal maps, the subset of parameters on which the corresponding map has a solenoidal attractor with bounded combinatorics is a set with zero Lebesgue measure.
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作者：Judge, Chris; Mondal, Sugata
摘要：We show that a second Neumann eigenfunction u of a Euclidean triangle has at most one (non-vertex) critical point p, and if p exists, then it is a non-degenerate critical point of Morse index 1. Using this we deduce that (1) the extremal values of u are only achieved at a vertex of the triangle, and (2) a generic acute triangle has exactly one (non-vertex) critical point and that each obtuse triangle has no (non-vertex) critical points. This settles the hot spots conjecture for triangles in th...
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作者：Koukoulopoulos, Dimitris; Maynard, James
摘要：Let psi : N -> R->= 0 be an arbitrary function from the positive integers to the non-negative reals. Consider the set A of real numbers a for which there are infinitely many reduced fractions a/q such that vertical bar alpha-a/q vertical bar <= psi(q)/q. If Sigma(infinity)(q=1) psi(q)phi(q)/q = infinity, we show that A has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the i...
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作者：Chodosh, Otis; Mantoulidis, Christos
摘要：The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show that for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with m...
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作者：Kupiainen, Antti; Rhodes, Remi; Vargas, Vincent
摘要：Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called DOZZ formula, for the three point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given ...
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作者：Lie, Victor
摘要：We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the L-p-boundedness of the Polynomial Carleson operator for 1 < p < infinity. Our proof relies on two new ideas: (i) we develop a framework for higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a local analysis adapted to the concepts of mass and counting function, which yields a new tile discretization of the time-...
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作者：Mellit, Anton
摘要：We prove an explicit formula for the Poincare polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundle and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function c...
