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 Advances in MathematicsJournal Prestige (SJR): 3.027 Citation Impact (citeScore): 2Number of Followers: 12      Subscription journal ISSN (Print) 0001-8708 - ISSN (Online) 1090-2082 Published by Elsevier  [3184 journals]
• Graded analogues of one-parameter subgroups and applications to the
cohomology of GL m n(r)
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Christopher M. Drupieski, Jonathan R. Kujawa We introduce a family Mr;f,η of infinitesimal supergroup schemes, which we call multiparameter supergroups, that generalize the infinitesimal Frobenius kernels Ga(r) of the additive group scheme Ga. Then, following the approach of Suslin, Friedlander, and Bendel, we use functor cohomology to define characteristic extension classes for the general linear supergroup GLm n, and we calculate how these classes restrict along homomorphisms ρ:Mr;f,η→GLm n. Finally, we apply our calculations to describe (up to a finite surjective morphism) the spectrum of the cohomology ring of the r-th Frobenius kernel GLm n(r) of the general linear supergroup GLm n.

• The Frobenius morphism in invariant theory
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Theo Raedschelders, Špela Špenko, Michel Van den Bergh Let R be the homogeneous coordinate ring of the Grassmannian G=Gr(2,n) defined over an algebraically closed field of characteristic p>0. In this paper we give a completely characteristic free description of the decomposition of R, considered as a graded Rp-module, into indecomposables (“Frobenius summands”). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of G and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that R provides a “noncommutative resolution” for Rp when p≥n−2, generalizing a result known to be true for toric varieties.In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirković on Frobenius decompositions for partial flag varieties. We are hopeful that it is an instance of a more general “p-uniformity” principle.

• P 1 { 0 , 1 , , z } +and+a+class+of+relations+among+multiple+zeta+values&rft.title=Advances+in+Mathematics&rft.issn=0001-8708&rft.date=&rft.volume=">Iterated integrals on P 1 ∖ { 0 , 1 , ∞ , z } and a class of
relations among multiple zeta values
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Minoru Hirose, Nobuo Sato In this paper we consider iterated integrals on P1∖{0,1,∞,z} and define a class of Q-linear relations among them, which arises from the differential structure of the iterated integrals with respect to z. We then define a new class of Q-linear relations among the multiple zeta values by taking their limits of z→1, which we call confluence relations (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values. As a good reason for this, we show that confluence relations imply both the regularized double shuffle relations and the duality relations.

• Shape optimization for the Steklov problem in higher dimensions
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Ailana Fraser, Richard Schoen We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in Rn when n≥3. This is in contrast to the situation when n=2, where a result of Weinstock from 1954 shows that the disk uniquely maximizes the first Steklov eigenvalue among all simply connected domains in the plane having the same boundary length. When n≥3, we show that increasing the number of boundary components does not increase the normalized (by boundary volume) first Steklov eigenvalue. This is in contrast to recent results which have been obtained for surfaces and for convex domains.

• Poisson equation on complete manifolds
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Ovidiu Munteanu, Chiung-Jue Anna Sung, Jiaping Wang We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

• A sharp effectiveness result of Demailly's strong openness conjecture
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Qi'an Guan In this article, we establish a sharp effectiveness result of Demailly's strong openness conjecture. We also establish a sharp effectiveness result related to a conjecture posed by Demailly and Kollár.

• Pointlike sets for varieties determined by groups
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): S.J.v. Gool, B. Steinberg For a variety of finite groups H, let H‾ denote the variety of finite semigroups all of whose subgroups lie in H. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to H‾. Our characterization is effective whenever H has a decidable membership problem. In particular, the separation problem for H‾-languages is decidable for any decidable variety of finite groups H. This generalizes Henckell's theorem on decidability of aperiodic pointlikes.

• Trace functions of the parafermion vertex operator algebras
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Chongying Dong, Victor Kac, Li Ren The trace functions for the Parafermion vertex operator algebra associated to any finite dimensional simple Lie algebra g and any positive integer k are studied and an explicit modular transformation formula of the trace functions is obtained.

• Topology of posets with special partial matchings
• Abstract: Publication date: 25 May 2019Source: Advances in Mathematics, Volume 348Author(s): Nancy Abdallah, Mikael Hansson, Axel Hultman Special partial matchings (SPMs) are a generalisation of Brenti's special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti's zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniavsky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.

• Poincaré duality for Cuntz–Pimsner algebras
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Adam Rennie, David Robertson, Aidan Sims We present a new approach to Poincaré duality for Cuntz–Pimsner algebras. We provide sufficient conditions under which Poincaré self-duality for the coefficient algebra of a Hilbert bimodule lifts to Poincaré self-duality for the associated Cuntz–Pimsner algebra.With these conditions in hand, we can constructively produce fundamental classes in K-theory for a wide range of examples. We can also produce K-homology fundamental classes for the important examples of Cuntz–Krieger algebras (following Kaminker–Putnam) and crossed products of manifolds by isometries, and their non-commutative analogues.

• A differential complex for CAT(0) cubical spaces
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): J. Brodzki, E. Guentner, N. Higson In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to C⁎-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on finite dimensional CAT(0)-cubical spaces.

• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Gelu Popescu Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball Bn, n=(n1,…,nk)∈Nk, which is a noncommutative analogue of the scalar polyball (Cn1)1×⋯×(Cnk)1. The Bohr radius Kmh(Bn) (resp. Kh(Bn)) associated with the multi-homogeneous (resp. homogeneous) power series expansions of the free holomorphic functions are the main objects of study in this paper. We extend a theorem of Bombieri and Bourgain for the disc D:={z∈C: z

• Adiabatic groupoid and secondary invariants in K-theory
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Vito Felice Zenobi In this paper we define new K-theoretic secondary invariants attached to a Lie groupoid G. The receptacle for these invariants is the K-theory of Cr⁎(Gad∘) (where Gad∘ is the adiabatic deformation G restricted to the interval [0,1)). Our construction directly generalises the cases treated in [29], [30], in the setting of the Coarse Geometry, to more involved geometrical situations, such as foliations. Moreover we tackle the problem of producing a wrong-way functoriality between adiabatic deformation groupoid K-groups associated to transverse maps. This extends the construction of the lower shriek map in [6]. Furthermore we prove a Lie groupoid version of the Delocalized APS Index Theorem of Piazza and Schick. Finally we give a product formula for secondary invariants.

• Multiplicativity of the idempotent splittings of the Burnside ring and the
G-sphere spectrum
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Benjamin Böhme We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.

• Relative Hard Lefschetz theorem for fans
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Kalle Karu We prove the Relative Hard Lefschetz theorem and the Relative Hodge-Riemann bilinear relations for combinatorial intersection cohomology sheaves on fans.

• Measure complexity and Möbius disjointness
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Wen Huang, Zhiren Wang, Xiangdong Ye In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity. We then apply this result to a number of situations, including certain systems whose invariant measure don't all have discrete spectrum.

• Reproducing kernel functions and asymptotic expansions on Jordan-Kepler
manifolds
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Miroslav Engliš, Harald Upmeier We study the complex geometry of generalized Kepler manifolds, defined in Jordan theoretic terms, introduce Hilbert spaces of holomorphic functions defined by radial measures, and find the complete asymptotic expansion of the corresponding reproducing kernels for Kähler potentials, both in the flat and bounded setting.

• Unified theory for finite Markov chains
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): John Rhodes, Anne Schilling We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup S. Our methods use geometric finite semigroup theory via the Karnofsky–Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to P(n), the set of all subsets of an n element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or R-trivial bands) by Brown and Diaconis, extensions to R-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky–Rhodes expansion of the right Cayley graph of S in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded S. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded S and the original S. Analyzing the normal forms also provides an estimate on the mixing time.

• Examples of Mori dream surfaces of general type with p g  = 0
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): JongHae Keum, Kyoung-Seog Lee In this paper we study effective, nef and semiample cones of minimal surfaces of general type with pg=0. We provide examples of minimal surfaces of general type with pg=0 and 2≤K2≤9 which are Mori dream spaces. On these examples we also give explicit description of their effective cones with all negative curves. We also present non-minimal surfaces of general type with pg=0 that are not Mori dream surfaces.

• Tilt-stability, vanishing theorems and Bogomolov-Gieseker type
inequalities
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Hao Sun We investigate the tilt-stability of stable sheaves on projective varieties with respect to certain tilt-stability conditions depends on two parameters constructed by Bridgeland [11] (see also [1], [6], [5]). For a stable sheaf, we give effective bounds of these parameters such that the stable sheaf is tilt-stable. These allow us to prove new vanishing theorems for stable sheaves and an effective Serre vanishing theorem for torsion free sheaves. Using these results, we also prove Bogomolov-Gieseker type inequalities for the third Chern character of a stable sheaf on P3.

• Almost sure local well-posedness and scattering for the 4D cubic nonlinear
Schrödinger equation
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Benjamin Dodson, Jonas Lührmann, Dana Mendelson We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in Hxs(R4) with 13

• Inductive limits of semiprojective C -algebras
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Hannes Thiel We prove closure properties for the class of C⁎-algebras that are inductive limits of semiprojective C⁎-algebras. Most importantly, we show that this class is closed under shape domination, and so in particular under shape and homotopy equivalence. It follows that the considered class is quite large. It contains for instance the stable suspension of any nuclear C⁎-algebra satisfying the UCT and with torsion-free K0-group. In particular, the stabilized C⁎-algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives.

• Triangulated categories of relative 1-motives
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Simon Pepin Lehalleur The category DA1(S) of relative cohomological 1-motives is the localizing subcategory of the triangulated category DA(S):=DAe´t(S,Q) of relative Voevodsky motives with rational coefficients over a scheme S which is generated by cohomological motives of curves over S. We construct and study a candidate for the standard motivic t-structure on DA1(S) (for S noetherian, finite-dimensional and excellent). We show this t-structure is non-degenerate and relate its heart MM1(S) with Deligne 1-motives over S; in particular, when S is regular, the category of Deligne 1-motives embeds in MM1(S) fully faithfully. We also study the inclusion of DA1(S) into the larger category DAcoh(S) of relative cohomological motives on S, and prove that its right adjoint ω1 preserves compact objects.

• Lech's inequality, the Stückrad–Vogel conjecture, and uniform
behavior of Koszul homology
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Patricia Klein, Linquan Ma, Pham Hung Quy, Ilya Smirnov, Yongwei Yao Let (R,m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set {l(M/IM)e(I,M)}I=m is bounded below by 1/d!e(R‾) where R‾=R/Ann(M). Moreover, when Mˆ is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stückrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

• H -calculus for semigroup generators on BMO
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Timothy Ferguson, Tao Mei, Brian Simanek We prove that the negative generator L of a semigroup of positive contractions on L∞ has bounded H∞(Sη)-calculus on the associated Poisson semigroup-BMO space for any angle η>π/2, provided L satisfies Bakry-Émery's Γ2≥0 criterion. Our arguments only rely on the properties of the underlying semigroup and work well in the noncommutative setting. A key ingredient of our argument is a type of quasi monotone properties for the subordinated semigroup Tt,α=e−tLα,0

• Fundamental groups of locally connected subsets of the plane
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Gregory Conner, Curtis Kent We show that every homomorphism from the fundamental group of a one-dimensional Peano continuum to the fundamental group of a planar Peano continuum is induced by a continuous map up to conjugation. The set of points at which a planar Peano continuum X is not locally simply connected, which we denote by B(X), is determined solely by the algebraic structure of its fundamental group. Furthermore, we demonstrate how to reconstruct the topological structure of B(X) using only the subgroup lattice of its fundamental group.

• Wiener's lemma along primes and other subsequences
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Christophe Cuny, Tanja Eisner, Bálint Farkas Inspired by subsequential ergodic theorems, we study the validity of Wiener's lemma and the extremal behavior of a measure μ on the unit circle via the behavior of its Fourier coefficients μˆ(kn) along subsequences (kn). We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity sequences, return times sequences and strongly sweeping out sequences as well as measures on R. We also present consequences for orbits of operators and of C0-semigroups on Hilbert and Banach spaces extending the results of Goldstein [34] and Goldstein, Nagy [36]. The results are complemented by some open questions and indication of interesting research directions.

• Whittaker coinvariants for GL(m n)
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Jonathan Brundan, Simon M. Goodwin Let Wm n be the (finite) W-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra glm n(C). In this paper we study the Whittaker coinvariants functor, which is an exact functor from category O for glm n(C) to a certain category of finite-dimensional modules over Wm n. We show that this functor has properties similar to Soergel's functor V in the setting of category O for a semisimple Lie algebra. We also use it to compute the center of Wm n explicitly, and deduce consequences for the classification of blocks of O up to Morita/derived equivalence.

• Quantum Grothendieck ring isomorphisms, cluster algebras and
Kazhdan-Lusztig algorithm
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): David Hernandez, Hironori Oya We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories CQ,Bn and CQ,A2n−1 of finite-dimensional representations of quantum affine algebras of types Bn(1) and A2n−1(1), respectively. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at t=1 to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002: the multiplicities of simple modules in standard modules in CQ,Bn are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

• A summation formula for triples of quadratic spaces
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Jayce R. Getz, Baiying Liu Let V1,V2,V3 be a triple of even dimensional vector spaces over a number field F equipped with nondegenerate quadratic forms Q1,Q2,Q3, respectively. LetY⊂∏i=1Vi be the closed subscheme consisting of (v1,v2,v3) on which Q1(v1)=Q2(v2)=Q3(v3). Motivated by conjectures of Braverman and Kazhdan and related work of Lafforgue, Ngô, and Sakellaridis we prove an analogue of the Poisson summation formula for certain functions on this space.

• Crossed products of nuclear C-algebras and their traces
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Timothy Rainone, Christopher Schafhauser This work addresses the Blackadar–Kirchberg Question (BKQ) as it pertains to group and reduced crossed product C⁎-algebras. The BKQ asks if every stably finite C⁎-algebra admits the matricial field (MF) property. We characterize the MF property for a class of reduced crossed product C⁎-algebras and for their traces. We prove that every ordered abelian group is, in a sense, MF, and show that there is no K-theoretic obstruction to an affirmative answer to the BKQ. Using classification techniques and induced K-theoretic dynamics, we give an affirmative answer to the BKQ for reduced crossed products of AH-algebras of real rank zero by free groups. Combining our results with recent progress in Elliott's Classification Program shows that the reduced crossed product of a separable, simple, unital, nuclear, UCT, and monotracial C⁎-algebra by the free group is always MF. We also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties. By appealing to a result of Ozawa, Rørdam, and Sato, we show that semidirect products of discrete amenable groups by free groups admit MF reduced group C⁎-algebras.

• Uniqueness of Coxeter structures on Kac–Moody algebras
• Abstract: Publication date: 30 April 2019Source: Advances in Mathematics, Volume 347Author(s): Andrea Appel, Valerio Toledano Laredo Let g be a symmetrizable Kac–Moody algebra, and Uħg the corresponding quantum group. We showed in [1], [2] that the braided Coxeter structure on integrable, category O representations of Uħg which underlies the R-matrix actions arising from the Levi subalgebras of Uħg and the quantum Weyl group action of the generalized braid group Bg can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R-matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in [3] to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of Uħg. Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g.

• Sofic mean length
• Abstract: Publication date: Available online 30 January 2019Source: Advances in MathematicsAuthor(s): Hanfeng Li, Bingbing Liang Given a length function L on the R-modules of a unital ring R, for each sofic group Γ we define a mean length for every locally L-finite RΓ-module relative to a bigger RΓ-module. We establish an addition formula for the mean length.We give two applications. The first one shows that for any unital left Noetherian ring R, RΓ is stably direct finite. The second one shows that for any ZΓ-module M, the mean topological dimension of the induced Γ-action on the Pontryagin dual of M coincides with the von Neumann–Lück rank of M.

• The Diederich–Fornæss index I: For domains of non-trivial index
• Abstract: Publication date: Available online 23 November 2018Source: Advances in MathematicsAuthor(s): Bingyuan Liu We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich–Fornæss index defined in 1977. This connects the Diederich–Fornæss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the β-worm domain is of index π/(2β). It is the first time that a precise non-trivial Diederich–Fornæss index in Euclidean spaces is obtained. This finding also indicates that the Diederich–Fornæss index is a continuum in (0,1], not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations.

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