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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：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：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：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.
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作者：Logunov, Alexander
摘要：Let M be a compact C-infinity-smooth Riemannian manifold of dimension n, n >= 3, and let phi lambda : Delta M phi lambda + lambda phi lambda = 0 denote the Laplace eigenfunction on M corresponding to the eigenvalue lambda. We show that Hn-1 ({phi lambda = 0}) <= C lambda(alpha), where alpha > 1/2 is a constant, which depends on n only, and C > 0 depends on M. This result is a consequence of our study of zero sets of harmonic functions on C-infinity-smooth Riemannian manifolds. We develop a tec...
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作者：Eskin, Alex; Filip, Simion; Wright, Alex
摘要：We compute the algebraic hull of the Kontsevich Zorich cocycle over any GL(2)(+) (R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.
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作者：Davison, Ben
摘要：Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjectu...
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作者：Brendle, Simon
摘要：We present a new curvature condition that is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, ...
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作者：Logunov, Alexander
摘要：Let u be a harmonic function in the unit ball B(0, 1) subset of R-n, n >= 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hn-1 ({u = 0} boolean AND B ) >= c. We prove Nadirashvili's conjecture as well as its counterpart on C-infinity-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact C-infinity-smooth Riemannian manifold M (without boundary...
