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• 1. Andersen, Henning Haahr
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Category O for quantum groups2015Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 17, nr 2, s. 405-431Artikkel i tidsskrift (Fagfellevurdert)

We study the BGG-categories O-q associated to quantum groups. We prove that many properties of the ordinary BGG-category O for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when q is a complex root of unity. Here we prove a tensor decomposition for simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for O and for finite-dimensional U-q-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in O-q. As a consequence, we also recover the known result that the generic quantum case behaves like the classical category O.

• 2.
Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay.
Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
L2 well-posedness of boundary value problems and the Kato square root problem for parabolic systems with measurable coefficients2016Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863Artikkel i tidsskrift (Fagfellevurdert)

We introduce a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space. This involves a parabolic Dirac operator at the boundary. We allow for measurable time dependence and some transversal dependence in the coefficients. We obtain layer potential representations for solutions in some classes and prove new well-posedness and perturbation results. As a byproduct, we prove for the first time a Kato estimate for the square root of parabolic operators with time dependent coefficients. This considerably extends prior results obtained by one of us under time and transversal independence. A major difficulty compared to a similar treatment of elliptic equations is the presence of non-local fractional derivatives in time.

• 3.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Dipartimento di Matematica "F. Casorati", Università di Pavia. Dipartimento di Matematica "F. Brioschi", Politecnico di Milano.
Boundary Estimates for Certain Degenerate and Singular Parabolic Equations2016Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, nr 2, s. 381-424Artikkel i tidsskrift (Fagfellevurdert)

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplace equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S-T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

• 4.
University of Washington, Seattle, USA.
University of Missouri, Columbia, USA. Instituto de Ciencias Matematicas, Madrid, Spain. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori. University of Washington, Seattle, USA.
A new characterization of chord-arc domains2017Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, nr 4, s. 967-981Artikkel i tidsskrift (Fagfellevurdert)

We show that if Ω⊂Rn+1, n≥1, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior corkscrew conditions, and an interior Harnack chain condition. We discuss some implications of this result for theorems of F. and M. Riesz type, and for certain free boundary problems.

• 5.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Rational symplectic field theory over Z_2 for exact Lagrangian cobordisms2008Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 10, nr 3, s. 641-704Artikkel i tidsskrift (Fagfellevurdert)

We construct a version of rational symplectic ﬁeld theory for pairs (X, L), where X is an exact symplectic manifold, where L ⊂ X is an exact Lagrangian submanifold with components subdivided into k subsets, and where both X and L have cylindrical ends. The theory associates to (X, L) a Z-graded chain complex of vector spaces over Z_2 , ﬁltered with k ﬁltration levels. The corresponding k -level spectral sequence is invariant under deformations of (X, L) and has the following property: if (X, L) is obtained by joining a negative end of a pair (X, L) to a positive end of a pair (X, L), then there are natural morphisms from the spectral sequences of (X, L) and of (X ,L) to the spectral sequence of (X, L). As an application, we show that if \Lambda ⊂ Y is a Legendrian submanifold of a contact manifold then the spectral sequences associated to (Y × R, \Lambda_s × R), where Y × R is the symplectization of Y and where \Lambda_s ⊂ Y is the Legendrian submanifold consisting of s parallel copies of \Lambda subdivided into k subsets, give Legendrian isotopy invariants of \Lambda.

• 6.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri. Inst Mittag Leffler, Aurav 17, S-18260 Djursholm, Sweden.
Univ So Calif, Los Angeles, CA 90089 USA. Tokyo Inst Technol, Meguro Ku, Tokyo 1528551, Japan.
Legendrian knots and exact Lagrangian cobordisms2016Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, nr 11, s. 2627-2689Artikkel i tidsskrift (Fagfellevurdert)

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X, L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L subset of X which agrees with cylinders over Legendrian links Lambda(+) and Lambda (-) at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of Lambda(+) to that of Lambda (-) .We give a gradient flow tree description of the DGA maps for certain pairs (X, L), which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

• 7.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik. University of Warwick, Coventry, England.
Unique Bernoulli g-measures2012Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 14, nr 5, s. 1599-1615Artikkel i tidsskrift (Fagfellevurdert)

We improve and subsume the conditions of Johansson and O¨ berg [18] and Berbee [2]for uniqueness of a g-measure, i.e., a stationary distribution for chains with complete connections.In addition, we prove that these unique g-measures have Bernoulli natural extensions. In particular,we obtain a unique g-measure that has the Bernoulli property for the full shift on finitely manystates under any one of the following additional assumptions.

(1)P1n=1(varn log g)2 < 1,(2) For any fixed ✏ > 0,P1n=1 e−(1/2+✏)(var1 log g+···+varn log g) = 1,(3) varn log g = o(1/pn) as n!1.

That the measure is Bernoulli in the case of (1) is new. In (2) we have an improved version ofBerbee’s [2] condition (concerning uniqueness and Bernoullicity), allowing the variations of log gto be essentially twice as large. Finally, (3) is an example that our main result is new both foruniqueness and for the Bernoulli property.We also conclude that we have convergence in the Wasserstein metric of the iterates of theadjoint transfer operator to the g-measure.

• 8.
University of Kentucky, Lexington, KY, USA.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Quasi-linear PDEs and low-dimensional sets2018Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 20, nr 7, s. 1689-1746Artikkel i tidsskrift (Fagfellevurdert)

In this paper we establish new results concerning boundary Harnack inequalities and the Martin boundary problem, for non-negative solutions to equations of \$p\$-Laplace type with variable coefficients. The key novelty is that we consider solutions which vanish only on a low-dimensional set \$\Sigma\$ in \$\mathbb R^n\$ and this is different compared to the more traditional setting of boundary value problems set in the geometrical situation of a bounded domain in \$\mathbb R^n\$ having a boundary with (Hausdorff) dimension in the range \$[n-1,n)\$. We establish our  quantitative and scale-invariant estimates in the context of low-dimensional Reifenberg flat sets.

• 9. Lewis, John L
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik.
On the dimension of p-harmonic measure in space2013Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 15, nr 6, s. 2197-2256Artikkel i tidsskrift (Fagfellevurdert)

Let Omega subset of R-n, n >= 3, and let p, 1 < p < infinity, p not equal D 2, be given. In this paper we study the dimension of p-harmonic measures that arise from nonnegative solutions to the p-Laplace equation, vanishing on a portion of partial derivative Omega, in the setting of delta-Reifenberg flat domains. We prove, for p >= n, that there exists (delta) over tilde = (delta) over tilde (p, n) > 0 small such that if Omega is a delta-Reifenberg flat domain with delta < <(delta)over tilde>, then p-harmonic measure is concentrated on a set of sigma-finite Hn-1-measure. We prove, for p >= n, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p-harmonic measure is always less than n - 1. We also prove that if 2 < p < n, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is less than n - 1, while if 1 < p < 2, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is larger than n - 1. Furthermore, perturbing off the case p = 2; we derive estimates for the Hausdorff dimension of p-harmonic measure when p is near 2.

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