# Proceedings of the London Mathematical Society

Print ISSN: 0024-6115

## .css-10d2qir{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-flex:1;-webkit-flex-grow:1;-ms-flex-positive:1;flex-grow:1;-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;}Articles

Lower bounds on Hilbert's Nullstellensatz and propositional proofs
• Conference Paper
• Full-text available

December 1994

· P. Beame

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· J. Krajicek

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· P. Pudlak
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

Degree of mobility for metrics of Lorentzian signature and parallel (0,2)-tensor fields on cone manifolds

December 2012

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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). Necessary conditions for vector-valued operator inequalities in harmonic analysis, arXiv:math.CA/0504030 April 2005 · 17 Reads 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. A positive Monk formula in the S 1 -equivariant cohomology of type A Peterson varieties, arXiv:0908.3517 August 2009 · 111 Reads 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. Jones, G.A.: Regular embeddings of complete bipartite graphs: classification and enumeration. Proc. Lond. Math. Soc. 101(3), 427-453 November 2009 · 70 Reads 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 Cohomology of U(2,1) representation varieties of surface groups September 2011 · 18 Reads 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. Quadratic spaces and holomorphic framed vertex operator algebras of central charge 24 October 2010 · 60 Reads 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. Alternating trilinear forms on a 9-dimensional space and degenerations of (3,3)-polarized Abelian surfaces January 2013 · 50 Reads 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. The 42 reducts of the random ordered graph September 2013 · 64 Reads 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. Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1. March 2015 · 49 Reads 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. Similarity classes of integral$p$-adic matrices and representation zeta functions of groups of type$\mathsf{A}_2$October 2014 · 39 Reads 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’. The nef cone of toroidal compactifications of${\cal A}_4$April 2002 · 21 Reads 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. A'Campo curvature bumps and the Dirac phenomenon near a singular point June 2012 · 55 Reads 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. Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps September 2007 · 53 Reads 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. Isogeny classes of abelian varieties over function fields May 2005 · 35 Reads 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.) Equivariant Iwasawa theory and non-abelian Stark-type conjectures September 2011 · 21 Reads 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. Families of absolutely simple hyperelliptic jacobians May 2008 · 46 Reads 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.) On the absolutely continuous spectrum in a model of irreversible quantum graph May 2005 · 34 Reads 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. Elliptic equations with nonlinear absorption depending on the solution and its gradient September 2014 · 35 Reads 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. Limiting absorption principle on manifolds having ends with various measure growth rate limits June 2006 · 37 Reads 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. Generation of finite simple groups with an application to groups acting on Beauville surfaces October 2010 · 71 Reads 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. Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type September 2012 · 43 Reads 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. C*-Algebras Arising from Group Actions on the Boundary of a Triangle Building February 2013 · 37 Reads 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  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  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.

Universal twist in Equivariant K-theory for proper and discrete actions

February 2012

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