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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 ...
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作者：Ryabogin, Dmitry
摘要：We give a negative answer to Ulam's Problem 19 from the Scottish Book asking is a solid of uniform density which will float in water in every position a sphere? Assuming that the density of water is 1, we show that there exists a strictly convex body of revolution K subset of R-3 of uniform density 1/2, which is not a Euclidean ball, yet floats in equilibrium in every orientation. We prove an analogous result in all dimensions d >= 3.
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作者：Bhattacharya, Prasit; Kitchloo, Nitu
摘要：In this paper, we provide a new proof of the stable Adams conjecture. Our proof constructs a canonical null-homotopy of the stable J-homomorphism composed with a virtual Adams operation, by applying the K-theory functor to a multinatural transformation. We also point out that the original proof of the stable Adams conjecture is incorrect and present a correction. This correction is crucial to our main application. We settle the question on the height of higher associative structures on the mod...
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作者：Bhatt, Bhargav; Scholze, Peter
摘要：We introduce the notion of a prism, which may be regarded as a deperfection of the notion of a perfectoid ring. Using prisms, we attach a ringed site - the prismatic site - to a p-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral p-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free de...
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作者：Hezari, Hamid; Zelditch, Steve
摘要：We show that if the eccentricity of an ellipse is sufficiently small, then up to isometries it is spectrally unique among all smooth domains. We do not assume any symmetry, convexity, or closeness to the ellipse, on the class of domains.In the course of the proof we also show that for nearly circular domains, the lengths of periodic orbits that are shorter than the perimeter of the domain must belong to the singular support of the wave trace. As a result we also obtain a Laplace spectral rigid...
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作者：Buzzi, Jerome; Crovisier, Sylvain; Sarig, Omri
摘要：We show that C-infinity-surface diffeomorphisms with positive topological entropy have finitely many ergodic measures of maximal entropy in general, and exactly one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irred...
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作者：Brown, Aaron; Hertz, Federico; Wang, Zhiren
摘要：We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers r(G) and m(G) associated with the roots system of the Lie algebra of a Lie group G. If the dimension of the manifold is smaller than r(G), then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most m(G), we show there is a quasi-invariant measure on the manifold such that the action is measurably isomorphic to a relatively measure-prese...
