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作者：Judge, Chris; Mondal, Sugata
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作者：Raskin, Sam
摘要：We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of GL(2) over a non-Archimedean field.
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作者：Merle, Frank; Raphael, Pierre; Rodnianski, Igor; Szeftel, Jeremie
摘要：In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension d >= 2 with decaying density at spatial infinity. The p...
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作者：Hahn, Jeremy; Wilson, Dylan
摘要：We equip BP(n) with an E3-BP-algebra structure for each prime p and height n. The algebraic K-theory of this ring is of chromatic height exactly n + 1, and the map K(BP(n))(p)-+ Lfn+1K(BP(n))(p) has bounded above fiber.
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作者：Wang, Qian
摘要：We prove the local-in-time well-posedness for the solution of the compressible Euler equations in 3-D for the Cauchy data of the velocity, density and vorticity (upsilon, rho, w) is an element of H-s x H-s x H-s('), 2 < s' < s. The result extends the sharp results of Smith-Tataru and of Wang, established in the irrotational case, i.e., w = 0, which is known to be optimal for s > 2. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold f...
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作者：Liu, Yuchen; Xu, Chenyang; Zhuang, Ziquan
摘要：We prove that on any log Fano pair of dimension n whose stability threshold is less than n+1/n, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of K...
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作者：Sawin, Will; Shusterman, Mark
摘要：Using geometric methods, we improve on the function field version of the Burgess bound and show that, when restricted to certain special subspaces, the Mobius function over F-q[T] can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to 1 for the Mobius function in arithmetic progressions and resolve Chowla's k-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential ...
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作者：Bialy, Misha; Mironov, Andrey E.
摘要：In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric C-2-smooth convex planar billiards. We assume that the domain A between the invariant curve of 4-periodic orbits and the boundary of the phase cylinder is foliated by C-0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C-1-smooth foliation by convex caustics ...
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作者：Witaszek, Jakub
摘要：We develop techniques of mimicking the Frobenius action in the study of universal homeomorphisms in mixed characteristic. As a consequence, we show a mixed characteristic Keel's base point free theorem obtaining applications towards the mixed characteristic Minimal Model Program, we generalise Kollar's theorem on the existence of quotients by finite equivalence relations to mixed characteristic, and we provide a new proof of the existence of quotients by affine group schemes.
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作者：Avila, Artur; Lyubich, Mikhail
摘要：We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family Pc : z -> z2 + c the corresponding set of parameters c is shown to have positive Hausdorff dimension. Our examples include renormalization fixed points, and the corresponding quadratic polynomials in their stable manifold are the first known rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set.
