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作者：Haag, Summer; Kertzer, Clyde; Rickards, James; Stange, Katherine E.
摘要：In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes appears as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski clo...
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作者：Kollar, Janos; Voisin, Claire
摘要：We prove in this paper the smoothability of cycles modulo rational equivalence below the middle dimension, that is, when the dimension is strictly smaller than the co dimension. We introduce and study the class of cycles obtained as flat pushforwards of Chern classes (or equivalently, flat pushforwards of products of divisors) and prove that they are smooth- able below the middle dimension. Our main result is that all cycles (of any dimension) on a smooth projective variety are flat pushforwar...
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作者：Poltoratski, A.
摘要：We prove pointwise convergence for the scattering data of a Dirac system of differential equations. Equivalently, we prove an analog of Carleson's theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. Our proofs are based on the study of resonances of Dirac systems using families of meromorphic inner functions, generated by a Riccati equation corresponding to the system.
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作者：Landesman, Aaron; Litt, Daniel
摘要：Let Sigma(g,n) be an orientable surface of genus g with n punctures. We study actions of the mapping class group Mod(g,n) of Sigma(g,n) via Hodge -theoretic and arithmetic techniques. We show that if rho : pi(1)(Sigma(g,n)) -> GL(r)(C) is a representation whose conjugacy class has finite orbit under Mod(g,n), and r < root g + 1, then rho has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman -Wieland conjecture, the Fonta...
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作者：Maulik, Davesh; Shen, Junliang
摘要：We prove the P = W conjecture for GL(n) for all ranks n and curves of arbitrary genus g >= 2. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the perversity statement, we apply the vanishing cycles constructions in our earlier work to global Springer theory in the sense of Yun, and prove a parabolic support theorem.
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作者：Binyamini, Gal; Novikov, Dmitry; Zak, Benny
摘要：1.1. Sharply o-minimal structures. Our general results in this paper are stated in the context of sharply o-minimal structures admitting sharp derivatives. For a complete definition of these notions, see Section 2. Briefly, a sharply o-minimal structure (abbreviated #o-minimal) is a pair (S, S2), where S is an o-minimal expansion of the real field and we identify S with the collection of all definable sets in S; and S2 = {S2 ,D c S} ,DEN is filtration on S depending on the format and degree D....
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作者：Li, Penghui; Nadler, David; Yun, Zhiwei
摘要：For a complex reductive group, we construct a semi-orthogonal decomposition of the cocenter of the universal variant of its affine Hecke category. We use this to calculate the endomorphisms of a Whittaker object in the cocenter via a diagram organizing parabolic induction of character sheaves. Assuming a universal variant of Bezrukavnikov's spectral description of the affine Hecke category, we deduce a formula for the dg algebra of global functions on commuting stacks of complex reductive grou...
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作者：Kanigowski, Adam; Lemanczyk, Mariusz; Radziwill, Maksym
摘要：We establish a prime number theorem for all uniquely ergodic, analytic skew products on the 2 -torus T 2 . More precisely, for every irrational alpha and every 1 -periodic real analytic g : R -+ R of zero mean, let T alpha,g : T 2 -+ T 2 be defined by ( x, y ) 7-+ ( x + alpha, y + g ( x )). We prove that if T alpha,g is uniquely ergodic then, for every ( x, y ) is an element of T 2 , the sequence { T p alpha,g ( x, y )} is equidistributed on T 2 as p traverses prime numbers. This is the first ...
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作者：Deng, Yu; Nahmod, Andrea R.; Yue, Haitian
摘要：We consider the defocusing nonlinear Schrodinger equation on T-2 with Wick ordered power nonlinearity, and prove almost sure global well-posedness with respect to the associated Gibbs measure. The heart of the matter is the uniqueness of the solution as limit of solutions to canonically truncated systems, and the invariance of the Gibbs measure under the global dynamics follows as a consequence. The proof relies on the novel idea of random averaging operators.
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作者：Raskin, Sam; Yang, David
摘要：We prove the Frenkel-Gaitsgory localization conjecture describing regular Kac-Moody representations at critical level via eigensheaves on the affine Grassmannian using categorical Moy-Prasad theory. This extends previous work of the authors.
