# Proceedings of the London Mathematical Society

Online ISSN: 0024-6115
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Conference Paper
The weak form of the Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Q<sub>i</sub>(x¯)=0, does not have a solution in the algebraic closure iff 1 is in the ideal generated by the polynomials Q<sub>i</sub>(x¯). We shall prove a lower bound on the degrees of polynomials P<sub>i</sub>(x¯) such that Σ <sub>i</sub> P<sub>i</sub>(x¯)Q<sub>i</sub>(x¯)=1. This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into q-element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count<sub>q</sub> <sup>N</sup>. Ajtai (1988) proved recently that, whenever p, q are two different primes, the propositional formulas Count<sub>q</sub><sup>qn+1 </sup> do not have polynomial size, constant-depth Frege proofs from instances of Count<sub>p</sub><sup>m</sup>, m≠0 (mod p). We give a new proof of this theorem based on the lower bound for the Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This results in an exact characterization of when Count<sub>q</sub> can be proven efficiently from Count<sub>p</sub>, for all p and q

Article
Degree of mobility of a (pseudo-Riemannian) metric is the dimension of the space of metrics geodesically equivalent to it. We describe all possible values of the degree of mobility on a simply connected n-dimensional manifold of lorentz signature. As an application we calculate all possible differences between the dimension of the projective and the isometry groups. One of the main new technical results in the proof is the description of all parallel symmetric (0,2)-tensor fields on cone manifolds of signature $(n-1,2). Article Via a random construction we establish necessary conditions for Lp (lq) inequalities for certain families of operators arising in harmonic analysis. In particular, we consider dilates of a convolution kernel with compactly supported Fourier transform, vector maximal functions acting on classes of entire functions of exponential type, and a characterization of Sobolev spaces by square functions and pointwise moduli of smoothness. 2000 Mathematics Subject Classification 42B25, 42B15, 42B35. Article Peterson varieties are a special class of Hessenberg varieties that have been extensively studied, for example, by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley–Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type An−1, with respect to a natural S1-action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac–Moody flag varieties G/P. Our main results are as follows. First, we identify a computationally convenient basis of H*S1 (Y), which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley–Monk formula for the product of a cohomology-degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both H*S1 (Y) and H*(Y) are generated in degree 2. Finally, by using our Chevalley–Monk formula we give explicit descriptions (via generators and relations) of both the S1-equivariant cohomology ring H*S1 (Y) and the ordinary cohomology ring H*(Y) of the type An−1 Peterson variety. Our methods are both directly from and inspired by those of the GKM (Goresky–Kottwitz–MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work. Article The regular embeddings of complete bipartite graphs$K_{n,n}$in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where$n$is a prime power, obtained in collaboration with Du, Kwak, Nedela and \v{S}koviera, together with results of It\^o, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups. Comment: 32 pages Article In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topology of the U(2,1) and SU(2,1) character varieties of surface groups. The main results are a calculation of the equivariant Poincare polynomials, a Kirwan surjectivity theorem in the non-fixed determinant case, and a description of the action of the Torelli group on the equivariant cohomology of the character variety. This builds on earlier work for stable pairs and rank 2 Higgs bundles. Article In 1993, Schellekens [‘Meromorphic c=24 conformal field theories’, Comm. Math. Phys. 153 (1993) 159–185.] obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras (VOAs) using the theory of framed VOAs and to determine the Lie algebra structures of their weight 1 subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes RM(1, 4)3 and RM(1, 4)⊕ 𝒟(d16+) of length 48. These VOAs correspond to the holomorphic simple current extensions of the lattice type VOAs and . We determine such extensions using a quadratic space structure on the set of all irreducible modules R(W) of W when or As our main results, we construct seven new holomorphic VOAs of central charge 24 in Schellekens' list and obtain a complete list of all Lie algebra structures associated to the weight 1 subspaces of holomorphic framed VOAs of central charge 24. Article We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a nine-dimensional vector space over an algebraically closed field of characteristic different from 2 and 3. To a general such element, one can naturally associate an Abelian surface$X$, which is embedded in eight-dimensional projective space. We study the combinatorial structure of this embedding and explicitly recover the genus 2 curve whose Jacobian variety is$X$. We also classify the types of degenerations of$X$that can occur. Taking the union over all Abelian surfaces in Heisenberg normal form, we get a five-dimensional variety which is a birational model for a genus$2$analog of Shioda's modular surfaces. We find determinantal set-theoretic equations for this variety and present some additional equations which conjecturally generate the radical ideal. Article The random ordered graph is the up to isomorphism unique countable homogeneous linearly ordered graph that embeds all finite linearly ordered graphs. We determine the reducts of the random ordered graph up to first-order interdefinability. Article In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in$\mathbb{R}^n$and null holomorphic curves in$\mathbb{C}^n$for any$n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in$\mathbb{R}^n$which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of$\mathbb{R}^n$which extend continuously up to the boundary; for$n\ge 5$we find embeddings with these properties. Article We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various$p$-adic analytic and adèlic profinite groups of type$\mathsf {A}_2$. This has consequences for the representation zeta functions of arithmetic groups$\Gamma \subset \mathbf {H}(k)$, where$k$is a number field and$\mathbf {H}$is a$k$-form of$\mathsf {SL}_3$: assuming that$\Gamma $possesses the strong congruence subgroup property, we obtain precise, uniform estimates for the representation growth of$\Gamma $. Our results are based on explicit, uniform formulae for the representation zeta functions of the$p$-adic analytic groups$\mathsf {SL}_3(\mathfrak {o})$and$\mathsf {SU}_3(\mathfrak {o})$, where$\mathfrak {o}$is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral$\mathfrak {p}$-adic$3\times 3$matrices in$\mathsf {gl}_3(\mathfrak {o})$and$\mathsf {gu}_3(\mathfrak {o})$, where$\mathfrak {o}$is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form$\mathsf {SL}_3(\mathfrak {o})$,$\mathsf {SU}_3(\mathfrak {o})$,$\mathsf {GL}_3(\mathfrak {o})$, and$\mathsf {GU}_3(\mathfrak {o})$, arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of$\mathfrak {o}$is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe$p$-adic analogues of ‘Ennola duality’. Article The moduli space of principally polarised abelian 4-folds can be compactified in several different ways by toroidal methods. Here we consider in detail the Igusa compactification and the (second) Voronoi compactification. We describe in both cases the cone of nef Cartier divisors. The proof depends on a detailed description of the Voronoi compactification, which makes it possible to proceed by induction, using the known description of the nef cone for compactifications of A3. The Igusa compactification has a non-Q-factorial singularity, which is resolved by a single blow-up: this resolution is the Voronoi compactification. The exceptional divisor E is a toric Fano variety (of dimension 9): the other boundary divisor, D, corresponds to degenerations with corank∼1. After imposing a level structure in order to avoid certain technical complications, we show that the closure of D in the Voronoi compactification maps to the Voronoi compactification of A3. The toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed. This inductive process is only valid for the Voronoi compactification: the result for the Igusa compactification is deduced from the Voronoi compactification. 2000 Mathematics Subject Classification 14K10, 14E99, 14M25. Article The level curves of an analytic function germ can have bumps (maxima of Gaussian curvature) at unexpected points near the singularity. This phenomenon is fully explored for$f(z,w)\in {\mathbb {C}}\{z,w\}$, using the Newton–Puiseux infinitesimals and the notion of gradient canyon. Equally unexpected is the Dirac phenomenon: as$c \rightarrow 0$, the total Gaussian curvature of$f=c$accumulates in the minimal gradient canyons, and nowhere else. Our approach mimics the introduction of polar coordinates in Analytic Geometry. Article We study the moduli space ℳ of torsion-free G2-structures on a fixed compact manifold M7, and define its associated universal intermediate Jacobian 𝒥. We define the Yukawa coupling and relate it to a natural pseudo-Kähler structure on 𝒥. We consider natural Chern-Simons-type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang-Mills connections), and also deformed Donaldson-Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in 𝒥 by means of G2-analogues of Abel-Jacobi maps. Article We study finiteness problems for isogeny classes of abelian varieties over an algebraic function field K in one variable over the field of complex numbers. In particular, we construct explicitly a non-isotrivial absolutely simple abelian fourfold X over a certain K such that the isogeny class of X×X contains infinitely many mutually non-isomorphic principally polarized abelian varieties. (Such examples do not exist when the ground field is finitely generated over its prime subfield.) Article We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over$\ZZ_p^d$-extensions of function fields ramifying at a finite set of places. Article We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd prime p. All these formulations are equivalent and hold if Iwasawa's \mu-invariant vanishes. Under mild hypotheses, we use this to prove non-abelian generalizations of Brumer's conjecture, the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture provided that \mu = 0. Article We prove that the jacobian of a hyperelliptic curve$y^2=(x-t)h(x)$has no nontrivial endomorphisms over an algebraic closure of the ground field$K$of characteristic zero if$t \in K$and the Galois group of the polynomial$h(x)$over$K$is "very big" and$deg(h)$is an even number >8. (The case of odd$deg(h)>3$follows easily from previous results of the author.) Article A family$A_\alpha$of differential operators depending on a real parameter$\alpha\ge 0$is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum$\sigma_{a.c.}$of the operator$A_\alpha$and its multiplicity for all values of the parameter. The spectrum of$A_0$is purely a.c. and admits an explicit description. It turns out that for$\alpha<\sqrt 2$one has$\sigma_{a.c.}(A_\alpha)= \sigma_{a.c.}(A_0)$, including the multiplicity. For$\alpha\ge\sqrt2$an additional branch of absolutely continuous spectrum arises, its source is an auxiliary Jacobi matrix which is related to the operator$A_\alpha$. This birth of an extra-branch of a.c. spectrum is the exact mathematical expression of the effect which was interpreted by Smilansky as irreversibility. Article We study positive solutions of equation (E1)$-\Delta u +u^p|\nabla u|^q= 0$($0\leq p$,$0\leq q\leq 2$,$p+q>1$) and (E2)$-\Delta u +u^p + |\nabla u|^q =0$($p>1$,$1\lt q\leq 2$) in a smooth bounded domain$\Omega \subset \mathbb {R}^N$. We obtain a sharp condition on$p$and$q$under which, for every positive, finite Borel measure$\mu$on$\partial \Omega$, there exists a solution such that$u=\mu$on$\partial \Omega$. Furthermore, if the condition mentioned above fails, then any isolated point singularity on$\partial \Omega$is removable, namely, there is no positive solution that vanishes on$\partial \Omega$everywhere except at one point. With respect to (E2), we also prove uniqueness and discuss solutions that blow up on a compact subset of$\partial \Omega$. In both cases, we obtain a classification of positive solutions with an isolated boundary singularity. Finally, in Appendix A a uniqueness result for a class of quasilinear equations is provided. This class includes (E1) when$p=0$but not the general case. Article The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian measure at infinity, in particular, focusing on the limiting absorption principle. As a result, we will obtain the absolute continuity of the Laplace-Beltrami operator. Article We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt (5) and SL 2(5) is a Beauville group. In particular, we settle a conjecture of Bauer, Catanese and Grunewald which asserts that all non-abelian finite quasisimple groups except for the alternating group Alt (5) are Beauville groups. Article Let$G$be a finite simple group of Lie type, and let$\pi_G$be the permutation representation of$G$associated with the action of$G$on itself by conjugation. We prove that every irreducible representation of$G$is a constituent of$\pi_G$, unless$G=PSU_n(q)$and$n$is coprime to$2(q+1)$, where precisely one irreducible representation fails. Let St be the Steinberg representation of$G$. We prove that a complex irreducible representation of$G$is a constituent of the tensor square$St\otimes St$, with the same exceptions as in the previous statement. Article We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,\mu) is generically turbulent. Article Let$G$be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let$X$be a rational nilpotent$G$-space. In this paper we analyze the homotopy type of the homotopy fixed point set$X^{hG}$, and the natural injection$k\colon X^G\hookrightarrow X^{hG}$. We show that if$X$is elliptic, that is, it has finite dimensional rational homotopy and cohomology, then each path component of$X^{hG}$is also elliptic. We also give an explicit algebraic model of the inclusion$k$based on which we can prove, for instance, that for$G$a torus,$\pi_*(k)$is injective in rational homotopy but, often, far from being a rational homotopy equivalence. Article We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence. Article We show that several important normal subgroups$\Gamma$of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action$\Gamma \curvearrowright X$is stably$OE$-superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from [CIK13] we deduce that any free, ergodic, probability measure preserving action of almost any surface braid group is stably$W^*$-superrigid, i.e., it can be completely reconstructed from its von Neumann algebra. Article We study the topological variant of Rokhlin dimension for topological dynamical systems (X,{\alpha},Z^m) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C*-algebraic level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias. In particular, it implies under these conditions that the transformation group C*-algebra has finite nuclear dimension. Generalizing a result of Yonatan Gutman, we show that free Z^m-actions on finite dimensional spaces satisfy a strengthened version of the so-called marker property, which yields finite Rokhlin dimension for said actions. Article We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong "infiniteness" condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional. Article We consider semigroup actions on the interval generated by two attracting maps. It is known that if the generators are sufficiently$C^2$-close to the identity, then the minimal set coincides with the whole interval. In this article, we give a counterexample to this result under the$C^1$-topology. Article We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain regular Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the stable Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby–Smith twisted actions in the group case. As an application we describe twisted étale groupoid C*-algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sieben's twisted actions essentially correspond to twisted étale groupoids with topologically trivial twists. Article Let G be a finite p-group and k be a field of characteristic p>0. We show that G has a non-linear faithful action on a polynomial ring U of dimension n=log p(|G|) such that the invariant ring UG is also polynomial. This contrasts with the case of linear and graded group actions with polynomial rings of invariants, where the classical theorem of Chevalley–Shephard–Todd and Serre requires G to be generated by pseudo-reflections. Our result is part of a general theory of ‘trace surjective G-algebras’, which, in the case of p-groups, coincide with the Galois ring extensions in the sense of Chase, Harrison and Rosenberg [‘Galois theory and Galois cohomology of commutative rings’, Mem. Amer. Math. Soc. 52 (1965) 15–33]. We consider the dehomogenized symmetric algebra Dk, a polynomial ring with non-linear G-action, containing U as a retract and we show that DGk is a polynomial ring. Thus, U turns out to be universal in the sense that every trace surjective G-algebra can be constructed from U by ‘forming quotients and extending invariants’. As a consequence we obtain a general structure theorem for Galois extensions with given p-group as Galois group and any prescribed commutative k-algebra R as invariant ring. This is a generalization of the Artin–Schreier–Witt theory of modular Galois field extensions of degree ps. Article A subgroup of an amenable group is amenable. The$C^*$-algebra version of this fact is false. This was first proved by M.-D. Choi who proved that the non-nuclear$C^*$-algebra$C^*_r(\ZZ_2*\ZZ_3)$is a subalgebra of the nuclear Cuntz algebra${\cal O}_2$. A. Connes provided another example, based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same. In fact he proved that certain of the$C^*$-algebras studied by J. Cuntz and W. Krieger [10] can be constructed naturally as crossed product algebras. For example if the group$\Gamma$acts simply transitively on a homogeneous tree of finite degree with boundary$\Omega$then$\cross$is a Cuntz-Krieger algebra. Such trees may be regarded as affine buildings of type$\widetilde A_1$. The present paper is devoted to the study of the analogous situation where a group$\G$acts simply transitively on the vertices of an affine building of type$\widetilde A_2$with boundary$\O$. The corresponding crossed product algebra$\cross$is then generated by two Cuntz-Krieger algebras. Moreover we show that$\cross$is simple and nuclear. This is a consequence of the facts that the action of$\G$on$\O$is minimal, topologically free, and amenable. Article We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah-Segal. Article The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the$p$-adic$L$-function of a rational modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the classical modular symbol attached to a Bianchi modular form and critical values of its$L$-function, which then allows us to construct its$p$-adic$L$-function via overconvergent methods. Article We show that all p-adic quintic forms in at least n>4562911 variables have a non-trivial zero. We also derive a new result concerning systems of cubic and quadratic forms. Article This paper provides foundations for studying p-adic deformations of arithmetic eigenpackets, that is, of systems of Hecke eigenvalues occurring in the cohomology of arithmetic groups with coefficients in finite-dimensional rational representations. The concept of ‘arithmetic rigidity’ of an arithmetic eigenpacket is introduced and investigated. An arithmetic eigenpacket is said to be ‘arithmetically rigid’ if (modulo twisting) it does not admit a p-adic deformation containing a Zariski dense set of arithmetic specializations. The case of GL(n) and ordinary eigenpackets is worked out, leading to the construction of a ‘universal p-ordinary arithmetic eigenpacket’. Tools for explicit investigation into the structure of the associated eigenvarieties for GL(n) are developed. Of note is the purely algebraic Theorem 5.1, which keeps track of the specializations of the universal eigenpacket. We use these tools to prove that known examples of non-selfdual cohomological cuspforms for GL(3) are arithmetically rigid. Moreover, we conjecture that, in general, arithmetic rigidity for GL(3) is equivalent to non-selfduality. Article The geometric conjecture developed by the authors in [1,2,3,4] applies to the smooth dual Irr(G) of any reductive p-adic group G. It predicts a definite geometric structure - the structure of an extended quotient - for each component in the Bernstein decomposition of Irr(G). In this article, we prove the geometric conjecture for the principal series in any split connected reductive p-adic group G. The proof proceeds via Springer parameters and Langlands parameters. As a consequence of this approach, we establish strong links with the local Langlands correspondence. One important feature of our approach is the emphasis on two-sided cells in extended affine Weyl groups. Article We study the quantum cohomology of adjoint and coadjoint homogeneous spaces. The product of any two Schubert classes does not involve powers of the quantum parameter greater than 2. With the help of the quantum to classical principle, we give presentations of the quantum cohomology algebras. These algebras are semi-simple for adjoint non-coadjoint varieties and some properties of the induced strange duality are shown. Article This paper considers$N\times N$matrices of the form$A_\gamma =A+ \gamma B$, where$A$is self-adjoint,$\gamma \in C$and$B$is a non-self-adjoint perturbation of$A$. We obtain some monodromy-type results relating the spectral behaviour of such matrices in the two asymptotic regimes$|\gamma |\to\infty$and$|\gamma |\to 0$under certain assumptions on$B$. We also explain some properties of the spectrum of$A_\gamma$for intermediate sized$\gamma$by considering the limit$N\to\infty$, concentrating on properties that have no self-adjoint analogue. A substantial number of the results extend to operators on infinite-dimensional Hilbert spaces. Article Let H0, H be a pair of self-adjoint operators for which the standard assumptions of the smooth version of scattering theory hold true. We give an explicit description of the absolutely continuous (a.c.) spectrum of the operator 𝒟θ=θ(H)−θ(H0) for piecewise continuous functions θ. This description involves the scattering matrix for the pair H0, H, evaluated at the discontinuities of θ. We also prove that the singular continuous spectrum of 𝒟θ is empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the ‘thresholds’ of the a.c. spectrum of 𝒟θ. Our approach relies on the construction of ‘model’ operators for each jump of the function θ. These model operators are defined as certain symmetrized Hankel operators (SHOs) which admit explicit spectral analysis. We develop the multichannel scattering theory for the set of model operators and the operator θ(H)−θ(H0). As a by-product of our approach, we also construct the scattering theory for general SHOs with piecewise continuous symbols. Article For almost all interval exchange maps T_0, with combinatorics of genus g>=2, we construct affine interval exchange maps T which are semi-conjugate to T_0 and have a wandering interval. Article In this paper we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on a given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graph of a basic representation is characterized as the set of all reduced proper Young walls. The character of a basic representation can be computed easily by counting the number of colored blocks that have been added to the ground-state wall. 2000 Mathematical Subject Classification: 17B37, 17B65, 81R50, 82B23. Article Let$\mu$be a probability measure on$\mathbb{R}^n$with a bounded density$f$. We prove that the marginals of$f$on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely-invariant extremal inequalities for certain averages of$f$on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann--Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on$\mu$, any marginal$\pi_E(\mu)$, or a small perturbation thereof, satisfies a nearly optimal small-ball probability. Article Consider the generalized flag manifold$G/B$and the corresponding affine flag manifold$\mathrm{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in$\mathrm{Fl}_G$to construct certain affine Gromov-Witten invariants of$\mathrm{Fl}_G$, and to obtain a family of affine quantum Chevalley operators$\Lambda_0,\ldots , \Lambda_n$indexed by the simple roots in the affine root system of$G$. These operators act on the cohomology ring$H^*(\mathrm{Fl}_G)$with coefficients in$\mathbb{Z}[q_0, \ldots,q_n]$. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for$G= \mathrm{SL}_n(\mathbb{C})$. The first quantum ring is a deformation of the subalgebra of$H^*(\mathrm{Fl}_G)$generated by divisors. The second ring, denoted$QH^*_{\mathrm{aff}}(G/B)$, deforms the ordinary quantum cohomology ring$QH^*(G/B)$by adding an affine quantum parameter$q_0$. We prove that$QH^*_{\mathrm{aff}}(G/B)$is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. We further develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of$QH^*_{\mathrm{aff}}(G/B)$by generators and relations. For$G$of Lie types$A_n - D_n$or$E_6$, we prove that the ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of$G\$.

Article
In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps into any Kaehler manifolds and harmonic maps into Kaehler manifolds with strongly semi-negative curvature respectively. These monotonicity inequalities enable us to derive some holomorphicity and Liouville type results for these pluriharmonic maps and harmonic maps. We also use the stress-energy tensors to investigate the holomorphic extension problem of CR maps.

Article
Consider a compact K\"ahler manifold X with a simple normal crossing divisor D, and define Poincar\'e type metrics on X\D as K\"ahler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincar\'e type K\"ahler metric on X\D implies the existence of a constant scalar curvature (resp. an extremal) K\"ahler metric, possibly of Poincar\'e type, on every component of D. We also show that when the divisor is smooth, the constant scalar curvature/extremal metric on X\D is asymptotically a product near the divisor.

Article
We investigate the case of the Kahler-Ricci flow blowing down disjoint exceptional divisors with normal bundle O(-k) to orbifold points. We prove smooth convergence outside the exceptional divisors and global Gromov-Hausdorff convergence. In addition, we establish the result that the Gromov-Hausdorff limit coincides with the metric completion of the limiting metric under the flow. This improves and extends the previous work of the authors. We apply this to P^1-bundles which are higher-dimensional analogues of the Hirzebruch surfaces. In addition, we consider the case of a minimal surface of general type with only distinct irreducible (-2)-curves and show that solutions to the normalized Kahler-Ricci flow converge in the Gromov-Hausdorff sense to a Kahler-Einstein orbifold.

Article
In this paper we develop a theory of stable homotopy 2-groups for spectra which is compatible with the smash product on the structured models for the stable homotopy category, therefore the homotopy 2-groups of a ring spectrum form a 2-ring, etc. For instance, the primary algebraic model of a ring spectrum R is the ring π*R of homotopy groups. We introduce the secondary model π*, *R which has the structure of a secondary analogue of a ring. The homology of π*, *R is π*R and triple Massey products in π*, *R coincide with triple Toda brackets in π*R. We also describe the secondary model π*, *Q of a commutative ring spectrum Q from which we derive the cup-one square operation in the graded commutative ring π*Q. As an application we obtain for each ring spectrum R new derivations of the ring π*R.

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We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the preprojective algebra of Q. This generalizes the K-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. We construct an action of the preprojective CoHA on the A-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when A is the intersection theory. We give a shuffle algebra description of this CoHA in terms of the underlying formal group law of A. For the quiver with potential of Ginzburg, we compare the critical CoHA defined by Kontsevich-Soibelman with the preprojective CoHA. As applications, we obtain a shuffle description of the Yangians, and an embedding of the positive part of the Yangian into the critical CoHA.

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• Imperial College London
• University of Western Australia
• University of Oxford
• University of North Texas
• Brigham Young University - Provo Main Campus