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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：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：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：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...
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作者：Hahn, Jeremy; Wilson, Dylan
摘要：We equip BP(n) with an E3-BP-algebra structure for each prime p and height n. The algebraic K-theory of this ring is of chromatic height exactly n + 1, and the map K(BP(n))(p)-+ Lfn+1K(BP(n))(p) has bounded above fiber.
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作者：Brown, Aaron; Fisher, David; Hurtado, Sebastian
摘要：We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if 1-' is a cocompact lattice in SL(n, R), M is a compact manifold, and omega a volume form on M, we show that any homomorphism alpha: 1-'-+ Diff(M) has finite image if the dimension of M is less than n - 1 and that any homomor-phism alpha: 1-'-+ Diff(M, omega) has finite image if the dimension of M is less than n. The key step in the proof is to show that any...
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作者：Krause, Ben; Mirek, Mariusz; Tao, Terence
摘要：We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages AN(f,g)(x) := 1/N Sigma(N)(n=1) f(T(n)x)g(T(P(n))x) as N -> infinity, where T: X -> X is a measure-preserving transformation of a sigma-finite measure space (X, mu), P(n) is an element of Z[n] is a polynomial of degree d >= 2, and f is an element of L-P1 (X), g is an element of L-P2 (X) for some p1 ,p 2 > 1 with 1/p1 + 1/p2 <= 1. We also e...
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作者：Cohn, Henry; Kumar, Abhinav; Miller, Stephen D.; Radchenko, Danylo; Viazovska, Maryna
摘要：We prove that the E8 root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions eight and twenty-four, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently sh...
