| 1. |
- Alper, Jarod, et al.
(författare)
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A Luna etale slice theorem for algebraic stacks
- 2020
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Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 191:3, s. 675-738
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Tidskriftsartikel (refereegranskat)abstract
- We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is (tale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.
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| 2. |
- Andersson, Lars, et al.
(författare)
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Hidden symmetries and decay for the wave equation on the Kerr spacetime
- 2015
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Ingår i: Annals of Mathematics. - : Princeton University Press. - 0003-486X .- 1939-8980. ; 182:3, s. 787-853
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Tidskriftsartikel (refereegranskat)abstract
- Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.
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| 3. |
- Bergström, Jonas, et al.
(författare)
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Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves
- 2024
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Ingår i: Annals of Mathematics. - 0003-486X .- 1939-8980. ; 199:3, s. 1323-1365
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Tidskriftsartikel (refereegranskat)abstract
- We compute the number of F-q-points on (M) over bar (4,n) for n <= 3 and show that it is a polynomial in q , using a sieve based on Hasse-Weil zeta functions. As an application, we prove that the rational singular cohomology group H-k((M) over bar (g,n)) vanishes for all odd k <= 9. Both results confirm predictions of the Langlands program, via the conjectural correspondence with polarized algebraic cuspidal automorphic representations of conductor 1, which are classified in low weight. Our vanishing result for odd cohomology resolves a problem posed by Arbarello and Cornalba in the 1990s.
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| 4. |
- Björner, Anders, et al.
(författare)
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On the shape of Bruhat intervals
- 2009
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Ingår i: Annals of Mathematics. - Princeton : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 170:2, s. 799-817
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Tidskriftsartikel (refereegranskat)abstract
- Let (W, S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J subset of S. Let W-J denote the set of minimal coset representatives modulo the parabolic subgroup W-J. For w is an element of W-J, let f(i)(w,J) denote the number of elements of length i below w in Bruhat order on W-J (with notation simplified to f(i)(w) in the case when W-J = W). We show that 0 <= i < j <= l(w)-i implies f(i)(w,J) <= f(j)(w,J). Also, the case of equalities f(i)(w) = f(l(w)-i)(w) for i = 1,..., k is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial P-e,P-w (q). We show that if W is finite then the number sequence f(0)(w), f(1)(w),... f(l(w))(w) cannot grow too rapidly. Further, in the finite case, for any given k >= 1 and any w is an element of W of sufficiently great length (with respect to k), we show f(l(w)-k)(w) >= f(l(w)-k+1)(w) >= ... >= f(l(w))(w). The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if (X) over bar (w) is a Schubert variety of dimension d = l(w), and lambda = c(1) (L) is an element of H-2 ((X) over bar (w)) is the restriction to (X) over bar (w) of the Chem class of an ample line bundle, then (lambda(k)) . : Hd-k((X) over bar (w)) -> Hd+k((X) over bar (w)) is injective for all k >= 0.
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| 5. |
- Borcea, Julius, et al.
(författare)
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Polya-Schur master theorems for circular domains and their boundaries
- 2009
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Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 170:1, s. 465-492
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Tidskriftsartikel (refereegranskat)abstract
- We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Omega subset of C for arbitrary closed circular domains Omega (i.e., images of the closed unit disk under a Mobius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for Omega = R our results settle open questions that go back to Laguerre and Polya-Schur.
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| 6. |
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| 7. |
- Bränden, Petter, 1976-, et al.
(författare)
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Lorentzian polynomials
- 2020
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Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 192:3, s. 821-891
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Tidskriftsartikel (refereegranskat)abstract
- We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge-Riemann relations for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence properties. We show that matroids, and more generally M-convex sets, are characterized by the Lorentzian property, and develop a theory around Lorentzian polynomials. In particular, we provide a large class of linear operators that preserve the Lorentzian property and prove that Lorentzian measures enjoy several negative dependence properties. We also prove that the class of tropicalized Lorentzian polynomials coincides with the class of M-convex functions in the sense of discrete convex analysis. The tropical connection is used to produce Lorentzian polynomials from M-convex functions. We give two applications of the general theory. First, we prove that the homogenized multivariate Tutte polynomial of a matroid is Lorentzian whenever the parameter q satisfies 0 < q <= 1. Consequences are proofs of the strongest Mason's conjecture from 1972 and negative dependence properties of the random cluster model in statistical physics. Second, we prove that the multivariate characteristic polynomial of an M-matrix is Lorentzian. This refines a result of Holtz who proved that the coefficients of the characteristic polynomial of an M-matrix form an ultra log-concave sequence.
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| 8. |
- Caffarelli, L. A., et al.
(författare)
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Regularity of a free boundary with application to the Pompeiu problem
- 2000
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Ingår i: Annals of Mathematics. - : JSTOR. - 0003-486X .- 1939-8980. ; 151:1, s. 269-292
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Tidskriftsartikel (refereegranskat)abstract
- In the unit ball B(0, 1), let u and Omega (a domain in R-N) salve the following overdetermined problem: Delta u = chi(Omega) in B(0, 1), 0 is an element of partial derivative Omega, u = \del u\ = 0 in B(0, 1) \ Omega, where chi(Omega) denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of Omega does not develop cusp singularities at the origin then we prove partial derivative Omega is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.
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| 9. |
- Ekedahl, Torsten, et al.
(författare)
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Recovering the good component of the Hilbert scheme
- 2014
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Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 179:3, s. 805-841
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Tidskriftsartikel (refereegranskat)abstract
- We give an explicit construction, for a flat map X -> S of algebraic spaces, of an ideal in the n'th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length n subschemesof the locusof n distinct points.This generalizes Haiman's corresponding result for the affine complex plane.However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory.In the nonflat case we obtain a similar result by replacing the n'th symmetric product by the n'th divided power product.
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| 10. |
- Faber, Carel, et al.
(författare)
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Hodge integrals, partition matrices, and the lambda g conjecture
- 2003
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Ingår i: Annals of Mathematics. - : Annals of Mathematics. - 0003-486X .- 1939-8980. ; 157:1, s. 97-124
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Tidskriftsartikel (refereegranskat)abstract
- We prove a closed formula for integrals of the cotangent line classes against the top Chern class of the Hodge bundle on the moduli space of stable pointed curves. These integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. An analysis of several natural matrices indexed by partitions is required.
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