| 1. |
- Anderson, Lara B., et al.
(författare)
-
P-1-fibrations in F-theory and string dualities
- 2022
-
Ingår i: Pure and Applied Mathematics Quarterly. - : INT PRESS BOSTON, INC. - 1558-8599 .- 1558-8602. ; 18:4, s. 1264-1354
-
Tidskriftsartikel (refereegranskat)abstract
- In this work, we study F-theory compactifications on manifolds. Such geometries, which we study in both 4- and 6dimensions, are both ubiquitous within the set of Calabi-Yau manifolds and play a crucial role in heterotic/F-theory duality. We discuss the most general formulation of P1-bundles of this type, as well as fibrations which degenerate at higher codimension loci. In the course of this study, we find a number of new phenomena. For example, in both 4- and 6-dimensions we find transitions whereby the base of a P1-bundle can change nature, or "jump ", at certain loci in complex structure moduli space. We discuss the implications of this jumping for the associated heterotic duals. We argue that P1-bundles with only rational sections lead to heterotic duals where the Calabi-Yau manifold is elliptically fibered over the section of the P1 bundle, and not its base. As expected, we see that degenerations of the P1 fibration of the F-theory base correspond to 5-branes in the dual heterotic physics, with the exception of cases in which the fiber degenerations exhibit monodromy. Along the way, we discuss a set of useful formulae and tools for describing F-theory compactifications on this class of Calabi-Yau manifolds.
|
|
| 2. |
- Berman, Robert, 1976
(författare)
-
On the strict convexity of the K-energy
- 2019
-
Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 15:4, s. 983-999
-
Tidskriftsartikel (refereegranskat)abstract
- Let (X, L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1, 1)-forms in c(1)(L), endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X, L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.
|
|
| 3. |
- Berndtsson, Bo, 1950
(författare)
-
Bergman kernels for Paley-Wiener spaces and Nazarov’s proof of the Bourgain-Milman theorem
- 2022
-
Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 18:2, s. 395-409
-
Tidskriftsartikel (refereegranskat)abstract
- We give a general inequality for Bergman kernels of Bergman spaces defined by certain convex weights in Cn. We also discuss how this can be used in Nazarov’s proof of the BourgainMilman theorem, as a substitute for Hörmander’s estimates for the ¯∂-equation.
|
|
| 4. |
- Björner, Anders
(författare)
-
A comparison theorem for f-vectors of simplicial polytopes
- 2007
-
Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 3:1, s. 347-356
-
Tidskriftsartikel (refereegranskat)abstract
- Let f(i)(P) denote the number of i-dimensional faces of a convex polytope P. Furthermore, let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Our main result is that for every simplicial d-polytope P, if f(r) (S (n(1), d)) <= f(r) (P) <= f(r) (C (n(2), d)) for some integers n(1), n(2) and r, then f(s) (S (n(1), d)) <= f(s) (P) <= f(s) (C (n(2), d)) for all s such that r < s. For r = 0 these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain comparison theorem for f-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.
|
|
| 5. |
- Björner, Anders., et al.
(författare)
-
Topological aspects of Boolean functions
- 2024
-
Ingår i: Pure and Applied Mathematics Quarterly. - : International Press of Boston. - 1558-8599 .- 1558-8602. ; 20:3, s. 1029-1063
-
Tidskriftsartikel (refereegranskat)abstract
- We discuss ways in which tools from topology can be used to derive lower bounds for the circuit complexity of Boolean functions.
|
|
| 6. |
- Dahl, Mattias, et al.
(författare)
-
A density theorem for asymptotically hyperbolic initial data satisfying the dominant energy condition
- 2021
-
Ingår i: Pure and Applied Mathematics Quarterly. - : International Press of Boston. - 1558-8599 .- 1558-8602. ; 17:5, s. 1669-1710
-
Tidskriftsartikel (refereegranskat)abstract
- When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, we show that an asymptotically hyperbolic initial data set with non-negative local energy density can be approximated by an initial data set with strictly positive local energy density and a simple structure at infinity, while changing the mass arbitrarily little. This is achieved by suitably modifying the argument used by Eichmair, Huang, Lee and Schoen in the asymptotically Euclidean case.
|
|
| 7. |
- Flodén, Liselott, 1967-, et al.
(författare)
-
Two-scale convergence: Some remarks and extensions
- 2013
-
Ingår i: Pure and Applied Mathematics Quarterly. - : International press of Boston. - 1558-8599 .- 1558-8602. ; 9:3, s. 461-486
-
Tidskriftsartikel (refereegranskat)abstract
- We first study the fundamental ideas behind two-scale conver-gence to enhance an intuitive understanding of this notion. The classicaldefinitions and ideas are motivated with geometrical arguments illustratedby illuminating figures. Then a version of this concept, very weak two-scaleconvergence, is discussed both independently and brie°y in the context ofhomogenization. The main features of this variant are that it works alsofor certain sequences of functions which are not bounded inL2 and atthe same time is suited to detect rapid oscillations in some sequences whichare strongly convergent inL2 . In particular, we show how very weaktwo-scale convergence explains in a more transparent way how the oscilla-tions of the governing coe±cient of the PDE to be homogenized causes thedeviation of theG-limit from the weak L2 NxN-limit for the sequence ofcoe±cients. Finally, we investigate very weak multiscale convergence andprove a compactness result for separated scales which extends a previousresult which required well-separated scales.
|
|
| 8. |
- Karlsson, Anders, et al.
(författare)
-
Linear drift and Poisson boundary for random walks
- 2007
-
Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 3:4, s. 1027-1036
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a nondegenerate random walk on a locally compact group with finite first moment. Then, if there are no nonconstant bounded harmonic functions, all the linear drift comes from a real additive character on the group. As a corollary we obtain a generalization of Varopoulos' theorem that in the case of symmetric random walks, positive linear drift implies the existence of nonconstant bounded harmonic functions. Another consequence is the phenomenon that for some groups (including certain Grigorchuk groups) the drift vanishes for any measure of finite first moment.
|
|
| 9. |
- Khesin, B., et al.
(författare)
-
Curvatures of Sobolev metrics on diffeomorphism groups
- 2013
-
Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 9:2, s. 291-332
-
Tidskriftsartikel (refereegranskat)abstract
- Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, while positive curvature suggests stability. In the first part of the paper we survey what we currently know about the curvature-stability relation in this context and provide detailed calculations for several equations of continuum mechanics associated to Sobolev H0 and H1 energies. In the second part we prove that in most cases (with some notable exceptions) the sectional curvature assumes both signs.
|
|
| 10. |
- Lewis, John L, et al.
(författare)
-
New results for p harmonic functions
- 2011
-
Ingår i: Pure and Applied Mathematics Quarterly. - : International Press of Boston. - 1558-8599 .- 1558-8602. ; 7:2, s. 345-363
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we first discuss new results of the authors concerning a boundary Harnack inequality and Holder continuity up to the boundary for the ratio of two positive pharmonic functions, 1 < p < infinity, which vanish on a portion of a Lipschitz domain. Second we discuss applications of these results to the Martin boundary problem for p harmonic functions and to certain boundary regularity-free boundary problems.
|
|
