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作者：Prasanna, Kartik
作者单位：University System of Maryland; University of Maryland College Park
摘要：We prove that the theta correspondence for the dual pair ((SL2) over tilde, PBx), for B an indefinite quaternion algebra over Q, acting on modular forms of odd square-free level, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to give a direct construction of isogenies b...
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作者：Fukaya, Kenji; Seidel, Paul; Smith, Ivan
作者单位：Kyoto University; Massachusetts Institute of Technology (MIT); University of Cambridge
摘要：We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different...
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作者：Cautis, Sabin; Kamnitzer, Joel
作者单位：Rice University; University of California System; University of California Berkeley
摘要：Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.
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作者：Kebekus, Stefan; Kovacs, Sandor J.
作者单位：University of Cologne; University of Washington; University of Washington Seattle
摘要：Shafarevich's hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this paper, we relate the variation of a family to the logarithmic Kodaira dimension of the base and give an affirmative ...
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作者：Bartels, Arthur; Lueck, Wolfgang; Reich, Holger
作者单位：University of Munster; Heinrich Heine University Dusseldorf
摘要：We prove the K-theoretic Farrell-Jones conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.
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作者：Hausel, Tamas; Rodriguez-Villegas, Fernando
作者单位：University of Oxford
摘要：We calculate the E-polynomials of certain twisted GL( n, C)character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie- type GL( n, Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL( n, C)- character variety. The calculation also leads to several conjectures about the cohomolo...
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作者：Caffarelli, Luis A.; Salsa, Sandro; Silvestre, Luis
作者单位：University of Texas System; University of Texas Austin; Polytechnic University of Milan; New York University
摘要：We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.
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作者：Ciocan-Fontanine, Ionut; Kim, Bumsig; Sabbah, Claude
作者单位：University of Minnesota System; University of Minnesota Twin Cities; Korea Institute for Advanced Study (KIAS); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Institut Polytechnique de Paris; Ecole Polytechnique
摘要：We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov-Witten invariants of nonabelian quotients X//G and those of the corresponding abelianized quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.
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作者：Krieger, J.; Schlag, W.; Tataru, D.
作者单位：Harvard University; University of Chicago; University of California System; University of California Berkeley
摘要：We prove the existence of equivariant finite time blow-up solutions for the wave map problem from R2+ 1 -> S-2 of the form u( t, r) = Q(lambda( t) r) + R(t, r) where u is the polar angle on the sphere, Q(r) = 2 arctan r is the ground state harmonic map, lambda(t) = t(-1-nu), and R(t, r) is a radiative error with local energy going to zero as t -> 0. The number nu > 1/2 can be prescribed arbitrarily. This is accomplished by first renormalizing the blow-up profile, followed by a perturbative ana...
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作者：Burq, Nicolas; Tzvetkov, Nikolay
作者单位：Universite Paris Saclay; Institut Universitaire de France; Universite de Lille
摘要：We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H-s(M), s < 1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in theHadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H-s(M), where s >= 1/4 in the case of a boundary less manifold and s >= 8/21 in the case of a manifold...
