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作者：Greenfeld, Rachel; Tao, Terence
摘要：The periodic tiling conjecture asserts that any finite subset of a lattice Zd that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large d, which also implies a disproof of the corresponding conjecture for Euclidean spaces Rd. In fact, we also obtain a counterexample in a group of the form Z2 x G0 for some finite abelian 2-group G0. Our methods rely on encoding a Sudoku puzzle whose rows and other non-horizontal lines ar...
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作者：Bergstrom, Jonas; Faber, Carel; Payne, Sam
摘要：We compute the number of F-q-points on (M) over bar (4,n) for n <= 3 and show that it is a polynomial in q , using a sieve based on Hasse-Weil zeta functions. As an application, we prove that the rational singular cohomology group H-k((M) over bar (g,n)) vanishes for all odd k <= 9. Both results confirm predictions of the Langlands program, via the conjectural correspondence with polarized algebraic cuspidal automorphic representations of conductor 1, which are classified in low weight. Our va...
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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.
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作者：De Bobadilla, Javier Fernandez; Pelka, Tomasz
摘要：We show that every family of isolated hypersurface singularities with constant Milnor number has constant multiplicity. To achieve this, we endow the A'Campo model of radius zero monodromy with a symplectic structure. This new approach allows us to generalize a spectral sequence of McLean converging to fixed point Floer homology of iterates of the monodromy to a more general setting that is well suited to study mu -constant families.
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作者：Park, Jinyoung; Pham, Huy tuan
摘要：For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-p setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.
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作者：Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S.
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作者：Kusakabe, Yuta
摘要：Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in Cn (n > 1) is Oka. Furthermore, we obtain new examples of non -elliptic Oka manifolds which negatively answer Gromov's question. The relative version of the main theorem is also proved. As an applicatio...
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作者：Aizenman, Michael; Duminil-Copin, Hugo
