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作者：Matomaeki, Kaisa; Radziwill, Maksym; Tao, Terence; Teraevaeinen, Joni; Ziegler, Tamar
摘要：Let lambda denote the Liouville function. We show that, as X-+ oo,2X Zsup X P(Y)ER[Y] degP 0 < theta < 1 fiixed but arbitrarily small. Previously this was only established for k < 1. We obtain this result as a special case of the corresponding statement for (non-pretentious) 1 -bounded multiplicative functions that we prove.In fact, we are able to replace the polynomial phases e(-P (n)) by degree k nilsequences F(g(n)Gamma). By the inverse theory for the Gowers norms this implies the higher or...
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作者：Yuan, Xinyi; Zhang, Shou-Wu
摘要：We will fix an error in our paper on the average Colmez conjecture by proving a slightly weaker statement than Theorem 2.7, which is sufficient for application to the main results.
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作者：Browning, Tim; Le Boudec, Pierre; Sawin, Will
摘要：It is known that the Brauer-Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least 3 over any number field. Moreover, for such varieties it follows from a general con-jecture of Colliot-The ' le`ne that the Brauer-Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hy-persurfaces of fixed degree and dimension by height, we p...
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作者：Song, Antoine
摘要：Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.
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作者：Dasgupta, Samit; Kakde, Mahesh
摘要：Let H/F be a finite abelian extension of number fields with F totally real and H a CM field. Let S and T be disjoint finite sets of places of F satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element Theta H/F S,T annihilates the T-smoothed class group ClT (H). We prove this conjecture away from p = 2, that is, after tensoring with Z[1/2]. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting id...
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作者：Simon, Leon
摘要：With respect to a C infinity metric which is close to the standard Euclidean metric on RN+1+e, where N > 7 and $ > 1 are given, we construct a class of embedded (N + )-dimensional pound hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset K C {0} x Re. Thus the question is settled, with a strong affirmative, as to whether there can be gaps or even fractional dimensional parts in the singular set. Suc...
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作者：Belkale, Prakash
摘要：We give a construction that produces irreducible complex rigid local systems on P-C(1) - {p(1), ..., P-s} via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU(n) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals...
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作者：Shelukhin, Egor
摘要：We prove that if a Hamiltonian diffeomorphism of a closed monotone symplectic manifold with semisimple quantum homology has more contractible fixed points, counted homologically, than the total dimension of the homology of the manifold, then it must have an infinite number of contractible periodic points. This constitutes a higher-dimensional homological generalization of a celebrated result of Franks from 1992, as conjectured by Hofer and Zehnder in 1994.
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作者：Ding, Jian; Sly, Allan; Sun, Nike
摘要：We establish the satisfiability threshold for random k-SAT for all k >= k(0), with k(0) an absolute constant. That is, there exists a limiting density alpha(sat) (k) such that a random k-SAT formula of clause density alpha is with high probability satisfiable for alpha < alpha(sat), and unsatisfiable for alpha > alpha(sat). We show that the threshold alpha(sat)(k) is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analyti...
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作者：Albritton, Dallas; Brue, Elia; Colombo, Maria
摘要：In a seminal work, Leray (1934) demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a ???background??? solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional ...
