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作者：Jitomirskaya, Svetlana; Liu, Wencai
摘要：We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regu...
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作者：Frantzikinakis, Nikos; Host, Bernard
摘要：The Mobius disjointness conjecture of Sarnak states that the Mobius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness ...
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作者：Irie, Kei; Marques, Fernando C.; Neves, Andre
摘要：For almost all Riemannian metrics (in the C-infinity Baire sense) on a closed manifold Mn+1, 3 <= (n + 1) <= 7, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces, thus proving a conjecture of Yau (1982) for generic metrics.
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作者：Naor, Assaf; Young, Robert
摘要：Given k epsilon N, the k'th discrete Heisenberg group, denoted H-z(2k+1), is the group generated by the elements a(1), b(1),..., a(k), b(k), c, subject to the commutator relations [a(1), b(1)] = ... = [a(k), b(k)] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j epsilon {1,..., k}, we have [a(i), a(j)] = [b(i), b(j)] = [a(i),b(j)] = [a(i), c] = [b(i), c] = 1. (In particular, this implies that c is in the center of H-z(2k...
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作者：Kaloshin, Vadim; Sorrentino, Alfonso
摘要：The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elli...
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作者：Bourgain, Jean; Dyatlov, Semyon
摘要：For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is, a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension delta of the limit set; in particular, we do not require the pressure condition delta <= 1/2. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov and Zahl. The main new ingredient is the...
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作者：Fujino, Osamu
摘要：We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Kollar. This completes Kollar's projectivity criterion for the moduli spaces of higher-dimensional stable varieties.
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作者：Frankel, Steven
摘要：We prove a conjecture of Calegari's, that every quasigeodesic flow on a closed hyperbolic 3-manifold contains a closed orbit.
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作者：Andreatta, Fabrizio; Goren, Eyal Z.; Howard, Benjamin; Pera, Keerthi Madapusi
摘要：Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over Q of signature (n, 2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions. As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.
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作者：Ionel, Eleny-Nicoleta; Parker, Thomas H.
摘要：The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.
