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作者：Moreira, Carlos Gustavo
摘要：We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra. In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and we may assume any real value in the interval [0, 1].
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作者：Tsimerman, Jacob
摘要：We give a proof of the Andre-Oort conjecture for A(g) - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven averaged version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The Andre-Oort conjecture then follows from previous work of Pila and the author.
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作者：Dasgupta, Samit; Kakde, Mahesh; Ventullo, Kevin
摘要：In 1980, Gross conjectured a formula for the expected leading term at s = 0 of the Deligne-Ribet p-adic L-function associated to a totally even character psi of a totally real field F. The conjecture states that after scaling by L(psi omega(-1), 0), this value is equal to a p-adic regulator of units in the abelian extension of F cut out by psi omega(-1). In this paper, we prove Gross's conjecture.
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作者：Adiprasito, Karim; Huh, June; Katz, Eric
摘要：We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.
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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...
