Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

Some New Observations on Interpolation in the Spectral Unit Ball

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ω _n , n ≥ 2. We begin by showing that a known necessary condition for the existence of a -interpolant ( here being the unit disc in ), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem – one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of Ω _n , n ≥ 2. This work is supported in part by a grant from the UGC under DSA-SAP, Phase IV.     

Liftings of distributive lattices by locally matricial algebras with respect to the Id c functor

We study representations of distributive -lattices, considered as join-semilattices, by semilattices of finitely generated two-sided ideals of locally matricial algebras over a field k, aiming to find a functorial solution of the problem. We find simple examples of a finite subcategory of the category Ld of distributive -lattices and of a subcategory of L_d corresponding to a partially ordered class which cannot be lifted with respect to the Id_c functor. On the other hand, we prove that there is such a lifting of every diagram in L_d or of a subcategory L_d1 of L_d whose objects are all distributive -lattices and whose morphisms are -embeddings. This paper is dedicated to Walter Taylor. Received February 8, 2005; accepted in final form August 11, 2005. The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by INTAS project 03-51-4110, the grant GAUK 448/2004/B-MAT, and the post-doctoral grant GAČR 201/03/P140.     

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes études Scientifiques for their hospitality.     

Genus one polyhedral surfaces,spaces of quadratic differentials on tori and determinants of Laplacians

We prove a formula for the determinant of the Laplacian on an arbitrary compact polyhedral surface of genus one. This formula generalizes the well-known Ray–Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic surface is also considered. We study the determinant of the Laplacian as a functional on the moduli space of meromorphic quadratic differentials with L simple poles and L simple zeros and derive formulas for variations of this functional with respect to natural coordinates on . We give also a new proof of Troyanov’s theorem stating the existence of a conformal flat conical metric on a compact Riemann surface of arbitrary genus with a prescribed divisor of conical points.     

The coarse Lipschitz geometry of \ell_p\oplus\ell_q

We show that for 1 < p < ∞ with p ≠ 2 the space L _p (0,1) is not uniformly homeomorphic to . We also show that if 1 < p < 2 < q < ∞ the space has unique uniform structure, answering a question of Johnson, Lindenstrauss and Schechtman (Geom. Funct. Anal. 6:430–470, 1996). The first author was supported by NSF grant DMS-0555670 and the second author was supported by NSF grant DMS-0701097.     

Equational theories as congruences of enriched monoids

The problem of characterizing the lattices of equational theories is still unsolved. In this paper we describe a class of monoids enriched by two unary operations and show that a lattice L is a lattice of equational theories if and only if L is isomorphic to a lattice of congruences of some enriched monoid belonging to . The author was supported by INTAS grant 03-51-4110 and The Alexander von Humboldt Foundation.     

<Emphasis Type="Italic">L</Emphasis>-functions of symmetric products of the Kloosterman sheaf over Z

The classical n-variable Kloosterman sums over the finite field F _p give rise to a lisse -sheaf Kl _n+1 on , which we call the Kloosterman sheaf. Let L _p (G _{m, F p} , Sym ^k Kl _n+1, s) be the L-function of the k-fold symmetric product of Kl _n+1. We construct an explicit virtual scheme X of finite type over Spec Z such that the p-Euler factor of the zeta function of X coincides with L _p (G _{m, F p} , Sym ^k Kl _n+1, s). We also prove similar results for and . The research of L. Fu is supported by the NSFC (10525107).     

Second order average estimates on local data of cusp forms

We specify sufficient conditions for the square modulus of the local parameters of a family of GL_n cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n ≦ 4. As an application, we show that Rankin-Selberg L-functions on , for n_i ≦ 4, satisfy the standard convexity bound. Received: 21 June 2005     

Codimension two singularities for representations of extended Dynkin quivers

Let M and N be two representations of an extended Dynkin quiver such that the orbit of N is contained in the orbit closure and has codimension two. We show that the pointed variety is smoothly equivalent to a simple surface singularity of type A _n , or to the cone over a rational normal curve.     

Read-Once Branching Programs, Rectangular Proofs of the Pigeonhole Principle and the Transversal Calculus

We investigate read-once branching programs for the following search problem: given a Boolean m × n matrix with m > n, nd either an all-zero row, or two 1s in some column. Our primary motivation is that this models regular resolution proofs of the pigeonhole principle , and that for m > n ² no lower bounds are known for the length of such proofs. We prove exponential lower bounds (for arbitrarily large m!) if we further restrict this model by requiring the branching program either to finish one row of queries before asking queries about another row (the row model) or put the dual column restriction (the column model).Then we investigate a special class of resolution proofs for that operate with positive clauses of rectangular shape; we call this fragment the rectangular calculus. We show that all known upper bounds on the size of resolution proofs of actually give rise to proofs in this calculus and, inspired by this fact, also give a remarkably simple rectangular reformulation of the Haken–Buss–Turán lower bound for the case m n ². Finally we show that the rectangular calculus is equivalent to the column model on the one hand, and to transversal calculus on the other hand, where the latter is a natural proof system for estimating from below the transversal size of set families. In particular, our exponential lower bound for the column model translates both to the rectangular and transversal calculi.* Part of the work was done while this author was visiting Special Year on Logic and Algorithms at DIMACS, Princeton. Also supported by Russian Basic Research Foundation grant 96-01-01222. Part of this work was done while on sabbatical leave at the Institute for Advanced Study and Princeton University, Princeton. This work was supported by USA-Israel BSF grant 92-00106 and by a Wolfson research award administered by the Israeli Academy of Sciences, as well as a Sloan Foundation grant. This work was supported in part by National Science Foundation and DARPA under grant CCR-9627819, and by USA-Israel BSF grant 92-00106.     

Closed characteristics on non-compact hypersurfaces in $${\mathbb {R}^{2n}}$$

Viterbo demonstrated that any (2n − 1)-dimensional compact hypersurface of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space (, ω). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces coming from mechanical Hamiltonians, and prove an analogue of Viterbo’s result. The main result provides a strong connection between the top half homology groups H _i (M), i = n, . . . , 2n − 1, and the existence of closed characteristics in the non-compact case (including the compact case). J. B. van den Berg is supported by NWO VENI grant 639.031.204. R. C. Vandervorst and F. Pasquotto are supported by NWO VIDI grant 639.032.202. This research is also partially supported by the RTN project ‘Fronts-Singularities’.     

Bivariate Function Spaces and the Embedding of Their Marginal Spaces

For a probability space we denote the marginal measures of , defined on Σ and Λ respectively, by and . If ρ is a function norm defined on marginal function norms ρ₁ and ρ₂ are defined on and . We find conditions which guarantee L_{ρ 1} + L_{ρ 2} to be embedded in L_ρ as a closed subspace. The problem is encountered in Statistics when estimating a bivariate distribution with known marginals. We find a condition which, applied to the binormal distribution in L², improves some known conditions.     

Regular Resolution Lower Bounds For The Weak Pigeonhole Principle

We prove that any regular resolution proof for the weak pigeon hole principle, with n holes and any number of pigeons, is of length , (for some global constant > 0).* Research supported by NSF grant CCR-9820831, US-Israel BSF grant 98-00349, and an NSERC grant. Research supported by US-Israel BSF grant 98-00349, and NSF grant CCR-9987077.     

Lower bounds on the canonical height associated to the morphism $${\phi(z)= z^d+c}$$

A lower bound is obtained of the canonical height associated to the morphism evaluated at wandering points α. The lower bound is of the form Ch(c), for some constant C depending on the the number of primes of bad reduction for , and the degree of the number field . The author was supported by a PDF grant from NSERC.     

The Number Of Orientations Having No Fixed Tournament

Let T be a fixed tournament on k vertices. Let D(n,T ) denote the maximum number of orientations of an n-vertex graph that have no copy of T. We prove that for all sufficiently (very) large n, where t_k−1(n) is the maximum possible number of edges of a graphon n vertices with no K_k, (determined by Turán’s Theorem). The proof is based on a directed version of Szemerédi’s regularity lemma together with some additional ideas and tools from Extremal Graph Theory, and provides an example of a precise result proved by applying this lemma. For the two possible tournaments with three vertices we obtain separate proofs that avoid the use of the regularity lemma and therefore show that in these cases already holds for (relatively) small values of n. * Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.     

On the asymptotic convergence of sequences of analytic functions

We consider sequences {f _n } of analytic self mappings of a domain and the associated sequence {Θ _n } of inner compositions given by . The case of interest in this paper concerns sequences {f _n } that converge assymptotically to a function f, in the sense that for any sequence of integers {n _k } with n ₁ < n ₂ < ... one has that locally uniformly in Ω. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Ω. Received: 16 December 2006     

A Generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups

The classical prime geodesic theorem (PGT) gives an asymptotic formula (as x tends to infinity) for the number of closed geodesics with length at most x on a hyperbolic manifold M. Closed geodesics correspond to conjugacy classes of π₁(M) = Γ where Γ is a lattice in G = SO(n,1). The theorem can be rephrased in the following format. Let be the space of representations of into Γ modulo conjugation by Γ. is defined similarly. Let be the projection map. The PGT provides a volume form vol on such that for sequences of subsets {B _t }, satisfying certain explicit hypotheses, |π⁻¹(B _t )| is asymptotic to vol(B _t ). We prove a statement having a similar format in which is replaced by a free group of finite rank under the additional hypothesis that n = 2 or 3.      

Fractional time-dependent Schrödinger equation on the Heisenberg group

Let Δ be the Kohn sublaplacian on the Heisenberg group , . In this paper we estimate the L ²-norm of the local maximal function of the unitary group of operators generated by L, by the Sobolev W _γ,ε -norm for some γ > 0 and for all ε > 0. Research supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. The first author was also supported by the MNiSW research grant N201 012 31/1020.     

Central simple algebras with involution: a geometric approach

Let k be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free PGL _n -varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of , where τ is the automorphism of PGL _n given by τ (h) = (h ⁻¹)^transpose. Basic properties of an involution are described in terms of the action of on the associated variety, and in particular in terms of the stabilizer in general position for this action.