Showing papers in "Journal of the American Mathematical Society in 2000"

Asymptotics of Plancherel measures for symmetric groups

Abstract: 1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G∧ of irreducible representations of G which assigns to a representation π ∈ G∧ the weight (dim π)/|G|. For the symmetric group S(n), the set S(n)∧ is the set of partitions λ of the number n, which we shall identify with Young diagrams with n squares throughout this paper. The Plancherel measure on partitions λ arises naturally in representation– theoretic, combinatorial, and probabilistic problems. For example, the Plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation [31]. We denote the Plancherel measure on partitions of n by Mn,

A variational principle for domino tilings

Abstract: 1.1. Description of results. A domino is a 1 x 2 (or 2 x 1) rectangle, and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps. In 1961, Kasteleyn [Kal] found a formula for the number of domino tilings of an m x n rectangle (with mn even), as shown in Figure 1 for rm = n = 68. Temperley and Fisher [TF] used a differenit method and arrived at the same result at almost exactly the same time. Both lines of calculation showed that the logarithm of number of tilings, divided by the number of dominos in a tiling (that is, mn/2), converges to 2G/7r 0.58 (here G is Catalan's constant). On the other hand, in 1992 Elkies et al. [EKLP] studied domino tilings of regions they called Aztec diamonds (Figure 2 shows an Aztec diamond of order 48), and showed that the logarithm of the number of tilings, divided by the number of dominos, converges to the smaller number (log 2)/2 0.35. Thus, even though the region in Figure 1 has slightly smaller area than the region in Figure 2, the former has far more domino tilings. For regions with other shapes, neither of these asymptotic formulas may apply. In the present paper we consider simply-connected regions of arbitrary shape. We give an exact formula for the limiting value of the logarithm of the number of tilings per unit area, as a function of the shape of the boundary of the region, as the size of the region goes to infinity. In particular, we show that computation of this limit is intimately linked with an understanding of long-range variations in the local statistics of random domino tilings. Such variations can be seen by comparing Figures 1 and 2. Each of the two tilings is random in the sense that the algorithm [PWI that was used to create it generates each of the possible tilings of the region being tiled with the same probability. Hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

Entire solutions of semilinear elliptic equations in R^3 and a conjecture of De Giorgi

Abstract: This paper is concerned with the study of bounded solutions of semilinear elliptic equations u F u in the whole space R under the assumption that u is monotone in one direction say nu in R n The goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes This type of symmetry question was raised by De Giorgi in who made the following conjecture we quote literally page of DG

Quiver varieties and finite dimensional representations of quantum affine algebras

Abstract: Introduction 145 1. Quantum affine algebra 150 2. Quiver variety 155 3. Stratification of M0 163 4. Fixed point subvariety 167 5. Hecke correspondence and induction of quiver varieties 169 6. Equivariant K-theory 174 7. Freeness 178 8. Convolution 185 9. A homomorphism Uq(Lg)→ KGw×C ∗ (Z(w))⊗Z[q,q−1] Q(q) 192 10. Relations (I) 194 11. Relations (II) 202 12. Integral structure 214 13. Standard modules 218 14. Simple modules 224 15. The Ue(g)-module structure 233 Added in proof 236 References 236

Braid groups are linear

Abstract: The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a faithful representation in the case n=4. In this paper, we show that it is faithful for all n.

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

Abstract: Σ(Q,α) is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities. In particular the set Σ(Q,α) is saturated, i.e., if nσ ∈ Σ(Q,α), then also σ ∈ Σ(Q,α). These results, when applied to a special quiver Q = Tn,n,n and to a special dimension vector, show that the GLn-module Vλ appears in Vμ ⊗ Vν if and only if the partitions λ, μ and ν satisfy an explicit set of inequalities. This gives new proofs of the results of Klyachko ([7, 3]) and Knutson and Tao ([8]). The proof is based on another general result about semi-invariants of quivers (Theorem 1). In the paper [10], Schofield defined a semi-invariant cW for each indecomposable representation W of Q. We show that the semi-invariants of this type span each weight space in SI(Q,α). This seems to be a fundamental fact, connecting semi-invariants and modules in a direct way. Given this fact, the results on sets of weights follow at once from the results in another paper of Schofield [11].

The size of the singular set in mean curvature flow of mean-convex sets

Abstract: In this paper, we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point. It is most convenient to describe the results in terms of the level set flow (also called “biggest flow” [I2]) of Chen-Giga-Goto [CGG] and Evans-Spruck [ES]. Under the level set flow, any closed set K in R generates a one-parameter family of closed sets Ft(K) (t ≥ 0) with F0(K) = K. If the boundary of K is a smooth compact hypersurface, then so is the boundary of Ft(K) for t in some interval [0, ), and for such t’s the evolution coincides with motion by mean-curvature as defined classically by partial differential equations and differential geometry (as in [H1]). However, if K is compact, then the boundary of Ft(K) will necessarily become singular at some finite time. Our goal is to show that the singular sets are necessarily quite small. This we do provided the initial set K is compact and mean-convex in the sense that Ft(K) ⊂ interior(K) for all t > 0. In case M = ∂K is a smooth hypersurface, we have the following equivalent characterization of mean-convexity: K is mean-convex if and only if the mean-curvature of M with respect to the inward unit normal is everywhere non-negative. Given a compact set K, let K be the region in spacetime swept out by the Ft(K): K = {(x, t) ∈ R ×R : t ≥ 0, x ∈ Ft(K)}. (∗) A point X = (x, t) in the boundary of K is called a regular point if (1) X has a neighborhood in which K is a smooth manifold-with-boundary, and (2) the tangent plane to ∂K at X is not horizontal (i.e., is not R × [0]). Note that if X = (x, t) is a regular point, then in a neighborhood of x, Ft(K) is a smooth manifold-with-boundary in R. A point X = (x, t) in ∂K with t > 0 that is not a regular point is called a singular point.

The enumerative geometry of K3 surfaces and modular forms

Abstract: We prove the conjectures of Yau-Zaslow and Gottsche concerning the number curves on K3 surfaces. Specifically, let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system |C| for any g and n satisfying C^2=2g+2n-2. When g=0, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Gottsche has generalized their conjecture to arbitrary g in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to P^2 blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus g constrained to g points are also given in terms of quasi-modular forms.

Generalized group characters and complex oriented cohomology theories

Abstract: Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex.

Percolation in the hyperbolic plane

Abstract: The purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi-isometric to the hyperbolic plane.

Definable sets, motives and p-adic integrals

Abstract: 0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) of its Zp-rational points. For every n in N, there is a natural map πn : X(Zp)→ X(Z/p) assigning to a Zp-rational point its class modulo p. The image Yn,p of X(Zp) by πn is exactly the set of Z/p-rational points which can be lifted to Zp-rational points. Denote by Nn,p the cardinality of the finite set Yn,p. By a result of the first author [7], the Poincare series

Topology of symplectomorphism groups of rational ruled surfaces

Abstract: Let M be either S 2 S 2 or the one point blow-up CP 2 # CP 2 of CP 2 . In both cases M carries a family of symplectic forms ! , where > 1 determines the cohomology class [! ]. This paper calculates the rational (co)homology of the group G of symplectomorphisms of (M; ! ) as well as the rational homotopy type of its classifying space BG . It turns out that each group G contains a nite collection Kk; k = 0; : : : ; ‘ = ‘( ), of nite dimensional Lie subgroups that generate its homotopy. We show that these subgroups \asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as ! 1. However, for each xed there is essentially one nonvanishing product that gives rise to a \jumping generator" w in H (G ) and to a single relation in the rational cohomology ring H (BG ). An analog of this generator w was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg{Witten theory. Our methods involve a close study of the space of ! -compatible almost complex structures on M.

Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion

Abstract: Consider a compact Lie group and a closed subgroup. Generalizing a result of Klyachko, we give a necessary and sufficient criterion for a coadjoint orbit of the subgroup to be contained in the projection of a given coadjoint orbit of the ambient group. The criterion is couched in terms of the ``relative'' Schubert calculus of the flag varieties of the two groups.

Metric and isoperimetric problems in symplectic geometry

Abstract: Introduction 411 1. Some basic results in symplectic topology 413 2. Capacity and symplectic reduction 414 3. Volume estimates for Lagrange submanifolds 416 3.1. The case of R. 417 3.2. Deformations of the zero-section in cotangent bundles 419 3.3. Generalization to the case of CP 422 4. An application to billiards 423 5. Geometry of convex sets and periodic orbits 425 6. Compensated compactness and closure of the symplectic group 426 Appendix A. A summary of symplectic geometry through generating functions 427 Appendix B. John’s ellipsoid 428 References 430

An Eulerian-Lagrangian approach for incompressible fluids: Local theory

Abstract: We study a formulation of the incompressible Euler equations in terms of the inverse Lagrangian map. In this formulation the equations become a first order advective nonlinear system of partial differential equations.

Large character sums

Abstract: Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n infinity and q -> infinity (q is the size of the finite field).

The product replacement algorithm and Kazhdan’s property (T)

Abstract: A problem of great importance in computational group theory is to generate (nearly) uniformly distributed random elements in a finite groupG. A good example of such an algorithm should start at any given set of generators, use no prior knowledge of the structure of G, and in a polynomial number of group operations return the desired random group elements (see [Bb2]). Then these random elements can be used further to determine the structure of G. In a pioneer paper [Bb1] Babai found such an algorithm which provably generates (nearly) uniformly distributed random elements in O(log |G|) group multiplications, too many for practical applications. A different heuristic, the “product replacement algorithm”, was designed by Leedham-Green and Soicher [LG], and studied by Celler et al. in [CLMNO]. In spite of the fact that very little theoretical justification was known, practical experiments showed excellent performance. So, it quickly became the most popular “practical” algorithm to generate random group elements, and was included in the two most frequently used group algebra packages GAP ([Sc]) and MAGMA ([BCP]). A systematic and quantitative approach was carried out by Diaconis and SaloffCoste [DS1], [DS2] (see also [Bb2], [CG]), but their results did not reveal the mystery of the truly outstanding performance of the algorithm. The aim of this paper is to propose a conceptual explanation based on Kazhdan’s property (T) from representation theory of Lie groups and to improve some of the previous estimates on the running time. The product replacement algorithm works as follows ([CLMNO]): Given a finite group G, let Γk(G) be the set of k-tuples (g) = (g1, . . . , gk) of elements of G such that 〈g1, . . . , gk〉 = G. We call elements of Γk(G) the generating k-tuples. Given a generating k-tuple (g1, . . . , gk), define a move on it in the following way: Choose uniformly a pair (i, j), such that 1 ≤ i 6= j ≤ k, then apply one of the following four operations with equal probability:

Linear algebraic groups and countable Borel equivalence relations

Abstract: This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces (i.e., Polish spaces equipped with their Borel structure). In mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants. Frequently these objects can be viewed as elements of a standard Borel space X and the equivalence turns out to be a Borel equivalence relation E on X. A complete classification of X up to E consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). For this to be of any interest both I and c must be explicit or definable and as simple and concrete as possible. The theory of Borel equivalence relations studies the set-theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems.

Syzygies of abelian varieties

Abstract: Let A be an ample line bundle on an abelian variety X (over an algebraically closed field). A theorem of Koizumi ([Ko], [S]), developing Mumford's ideas and results ([M]), states that if m > 3 the line bundle L = AO' embeds X in projective space as a projectively normal variety. Moreover, a celebrated theorem of Mumford ([M2]), slightly refined by Kempf ([K4]), asserts that the homogeneous ideal of X is generated by quadrics as soon as m > 4. Such results turn out to be particular cases of a statement, conjectured by Rob Lazarsfeld, concerning the minimal resolution of the graded algebra RL = 0% Ho(X, L?h) over the polynomial ring SL DO SymhHO (X, L). The purpose of this paper is to prove Lazarsfeld's conjecture. To put such matters into perspective, it is useful to review the case of projective curves. A classical theorem of Castelnuovo states that a curve X, embedded in projective space by a complete linear system ILI, is projectively normal as soon as deg L > 2g(X) + 1, and a theorem of Mattuck, Fujita and Saint-Donat states that if deg L > 2g(X) + 2, then the homogeneous ideal of X is generated by quadrics. Green ([G1]) unified, re-interpreted and generalized these results to a statement about syzygies. Specifically, given a (smooth) projective variety X and a very ample line bundle L on X, a minimal resolution of RL as a graded SL-module (notation as above) looks like

A point set whose space of triangulations is disconnected

Abstract: In this paper we explicitly construct a triangulation of a 6-dimensional point configuration of 324 points which admits no geometric bistellar operations (or flips, for short). This triangulation is an isolated element in the graph of triangulations of the point configuration. It has been a central open question in polytope combinatorics in the last decade whether point configurations exist for which this graph is not connected (see, e.g., [37, Question 1.2] and [48, Challenge 3]). We also construct a 234-dimensional polytope with 552 vertices whose graph of triangulations has an isolated element. Our construction is likely to have an impact in algebraic geometry too, via the connections between lattice polytopes and toric varieties [21, 23, 31, 43]. For example, in [2, Section 2] and [24, Section 4] the different authors study algebraic schemes based on the poset of subdivisions of an integer point configuration. The connectivity of these schemes and of the graph of triangulations are equivalent. See Section 4.3, in particular Corollary 4.9. The graph of triangulations is also related to the Baues poset, which appears in oriented matroid theory, zonotopal tilings and hyperplane arrangements, so our result has implications in these areas. ∗ Let A be a finite point set in the real affine space R. A polyhedral subdivision of A is a geometric polyhedral complex with vertices in A which covers the convex hull of A. If all the cells are simplices, then it is a triangulation. More combinatorial definitions are convenient if A is not in convex position, i.e. if some element of A is not a vertex of the convex hull. See Definitions 4.1 and 1.1 for details, and also [6], [21, Chapter 7], [36], [47, Chapter 9], or the monograph in preparation [14]. There are at least the following three ways to give a structure to the collection of all triangulations of a point configuration A: (A) Flips. Geometric bistellar operations, or flips, are the minimal changes which can be made in a triangulation of A to produce a new one (see Definition 1.3). A particular case is the familiar diagonal edge flip in two-dimensional triangulations, of frequent use in computational geometry and geometric combinatorics.

The “hot spots” conjecture for domains with two axes of symmetry

Abstract: Consider a convex planar domain with two axes of symmetry. We show that the maximum and minimum of a Neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only. We deduce J. Rauch's "hot spots" conjecture in the following form. If the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. In fact the maximum point reaches the boundary in finite time if the boundary has positive curvature. Results of this type have already been proved by Bafiuelos and Burdzy [BB] using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions. We introduce here a new technique based on deformation of the domain. An advantage of our method is that it works even in the case of multiple eigenvalues. On the way toward our results, we prove monotonicity properties for Neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the Dirichlet problem of independent interest.

Convergence and finite determination of formal CR mappings

Abstract: In this paper, we study the convergence and finite determination of formal holomorphic mappings of (C , 0) taking one real submanifold into another. By a formal (holomorphic) mapping H : (C , 0) → (C , 0), we mean an N -vector H = (H1, . . . , HN ), where each Hj is a formal power series in N indeterminates with no constant term. If M and M ′ are real smooth submanifolds through 0 in C defined near the origin by ρ(Z, Z) = 0 and ρ′(Z, Z) = 0 respectively, where ρ and ρ′ are vector valued smooth defining functions, then we say that a formal mapping H : (C , 0)→ (C , 0) sends M into M ′ if the vector valued power series ρ′(H(Z), H(Z)) is a (matrix) multiple of ρ(Z, Z). For real-analytic hypersurfaces, we shall prove the following. Theorem 1. Let M and M ′ be real-analytic hypersurfaces through the origin in C , N ≥ 2. Assume that neither M nor M ′ contains a nontrivial holomorphic subvariety through 0. Then any formal mapping H : (C , 0)→ (C , 0) sending M into M ′ is convergent. The condition that M ′ above does not contain a nontrivial holomorphic subvariety is necessary (see Remark 2.3). As a corollary we obtain the following characterization.

Higher rank case of Dwork's conjecture

Abstract: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0894-0347(XX)0000-0 arXiv:math/0005309v1 [math.NT] 9 May 2000 HIGHER RANK CASE OF DWORK’S CONJECTURE DAQING WAN Dedicated to the memory of Bernard Dwork 1. Introduction In this series of two papers, we prove the p-adic meromorphic continuation of the pure slope L-functions arising from the slope decomposition of an overconvergent F- crystal, as conjectured by Dwork [6]. More precisely, we prove a suitable extension of Dwork’s conjecture in our more general setting of σ-modules, see section 2 for precise definitions of the various notions used in this introduction. Our main result is the following theorem. Theorem 1.1. Let X be a smooth affine variety defined over a finite field F q of characteristic p > 0. Let (M, φ) be a finite rank overconvergent σ-module over X/F q . Then, for each rational number s, the pure slope s L-function L s (φ, T ) attached to φ is p-adic meromorphic everywhere. The proof of this theorem will be completed in two papers. In the present higher rank paper, we introduce a reduction approach which reduces Theorem 1.1 to the special case when the slope s (s = 0) part of φ has rank one and the base space X is the simplest affine space A n . This part is essentially algebraic. It depends on Monsky’s trace formula, Grothendieck’s specialization theorem, the Hodge-Newton decomposition and Katz’s isogeny theorem. In our next paper [23], we will handle the rank one case over the affine space A n . The rank one case is very much analytic in nature and forces us to work in a more difficult infinite rank setting, generalizing and improving the limiting approach introduced in [19]. Dwork’s conjecture grew out of his attempt to understand the p-adic analytic variation of the pure pieces of the zeta function of a variety when the variety moves through an algebraic family. To give an important geometric example, let us con- sider the case that f : Y → X is a smooth and proper morphism over F q with a smooth and proper lifting to characteristic zero. Berthelot’s result [1] says that the relative crystalline cohomology R i f crys,∗ Z p modulo torsion is an overconvergent F-crystal M i over X. Applying Theorem 1.1, we conclude that the pure L-functions arising from these geometric overconvergent F-crystals M i are p-adic meromorphic. In particular, this implies the existence of an exact p-adic formula for geometric p-adic character sums and a suitable p-adic equi-distribution theorem for the roots of zeta functions. For more detailed arithmetic motivations and further open prob- lems, see the expository papers [20][21]. 1991 Mathematics Subject Classification. Primary 11G40, 11S40; Secondary 11M41, 14G15. Key words and phrases. L-functions, p-adic meromorphic continuation, σ-modules. This work was partially supported by NSF. c American Mathematical Society

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

Abstract: 1.1. Classical r-matrices. In the early eighties, Belavin and Drinfeld [BD] classified nonskewsymmetric classical r-matrices for simple Lie algebras. It turned out that such r-matrices, up to isomorphism and twisting by elements from the exterior square of the Cartan subalgebra, are classified by combinatorial objects which are now called Belavin-Drinfeld triples. By definition, a Belavin-Drinfeld triple for a simple Lie algebra g is a triple (Γ1,Γ2, T ), where Γ1,Γ2 are subsets of the Dynkin diagram Γ of g, and T : Γ1 → Γ2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition: if α ∈ Γ1, then there exists k such that T k−1(α) ∈ Γ1 but T (α) / ∈ Γ1. The r-matrix corresponding to such a triple is given by a certain explicit formula. These results generalize in a straightforward way to semisimple Lie algebras. In [S], the third author generalized the work of Belavin and Drinfeld and classified classical nonskewsymmetric dynamical r-matrices for simple Lie algebras. It turns out that they have an even simpler classification: up to gauge transformations, they are classified by generalized Belavin-Drinfeld triples, which are defined as the usual Belavin-Drinfeld triples but without any nilpotency condition. The dynamical rmatrix corresponding to such a triple is given by a certain explicit formula. As before, these results can be generalized to semisimple Lie algebras.

Interpolating hereditarily indecomposable Banach spaces

Abstract: A Banach space X is said to be Hereditarily Indecomposable (HI) if for any pair of closed subspaces Y , Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace (Throughout this section by the term “subspace” we mean a closed infinite-dimensional subspace of X ) The HI spaces form a new and, as we believe, fundamental class of Banach spaces The celebrated example of a Banach space with no unconditional basic sequence, due to W Gowers and B Maurey ([GM]), is the first construction of a HI space It is easily seen that every HI space does not contain any unconditional basic sequence Actually, the concept of HI spaces came after W Johnson’s observation that this was a property of the Gowers-Maurey example To describe even further the peculiar structure of a HI space, we recall an alternative definition of such a space A Banach space X is a HI space if and only if for every pair of subspaces Y , Z and e > 0 there exist y ∈ Y , z ∈ Z with ||y|| = ||z|| = 1 and ||y − z|| < e Thus, HI spaces are structurally irrelevant to classical Banach spaces, in particular to Hilbert spaces Other constructions of HI spaces already exist We mention Argyros and Deliyanni’s construction of HI spaces which are asymptotic ` spaces ([AD2]), V Ferenczi’s example of a uniformly convex HI space ([F2]) and HI modified asymptotic ` spaces contained in [ADKM] Other examples of Banach spaces which are HI or which have a HI subspace are given in [G1], [H], [OS1] The construction of such a space requires several steps and it uses two fundamental ideas The first is Tsirelson’s recursive definition of saturated norms ([Ts]) and the second is Maurey-Rosenthal’s construction of weakly null sequences without unconditional basic subsequences ([MR]) An important ingredient in the GowersMaurey construction is the Schlumprecht space This is a Tsirelson type Banach space which is arbitrarily distortable and has been used in the solution of important problems Thus beyond its use in the constructions of HI spaces it plays a central role in the solution of the distortion problem for Hilbert spaces ([OS]) The essential difference between Schlumprecht and Tsirelson spaces became more transparent in [AD2] where the mixed Tsirelson spaces were introduced It is natural to expect that HI spaces share special and interesting properties not found in the previously known Banach spaces Indeed, the following theorem is proven in

Mating Siegel quadratic polynomials

Abstract: 1.1. Mating: Definitions and some history. Mating quadratic polynomials is a topological construction suggested by Douady and Hubbard [Do2] to partially parametrize quadratic rational maps of the Riemann sphere by pairs of quadratic polynomials. Some results on matings of higher degree maps exist, but we will not discuss them in this paper. While there exist several, presumably equivalent, ways of describing the construction of mating, the following approach is perhaps the most standard. Consider two monic quadratic polynomials fi and f2 whose filled Julia sets K(fi) are locally-connected. For each fi, let (Di denote the conformal isomorphism between the basin of infinity C -,. K(f,) and C -. ED, with i(00) = 00 and V(oo) = 1. These Bottcher maps conijugate the polynomials to the squaring map:

Rank one case of Dwork's conjecture

Abstract: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0894-0347(XX)0000-0 arXiv:math/0005308v1 [math.NT] 9 May 2000 RANK ONE CASE OF DWORK’S CONJECTURE DAQING WAN 1. Introduction In the higher rank paper [17], we reduced Dwork’s conjecture from higher rank case over any smooth affine variety X to the rank one case over the simplest affine space A n . In the present paper, we finish our proof by proving the rank one case of Dwork’s conjecture over the affine space A n , which is called the key lemma in [17]. The key lemma had already been proved in [16] in the special case when the Frobenius lifting σ is the simplest q-th power map σ(x) = x q . Thus, the aim of the present paper is to treat the general Frobenius lifting case. Our method here is an improvement of the limiting method in [16]. It allows us to move one step further and obtain some explicit information about the zeros and poles of the unit root L-function. As in [16], to handle the rank one case, we are forced to work in the more difficult infinite rank setting, see section 2 for precise definitions of the various basic infinite rank notions. Let F q denote the finite field of characteristic p > 0. Our main result of this paper is the following theorem. Theorem 1.1. Let φ be a nuclear overconvergent σ-module over the affine n-space A n /F q , ordinary at the slope zero side. Let φ unit be the unit root (slope zero) part of φ. Assume that φ unit has rank one. Let ψ be another nuclear overconvergent σ-module over A n /F q . Then for each integer k, the L-function L(ψ ⊗ φ ⊗k unit , T ) is p-adic meromorphic. Furthermore, the family L(ψ ⊗ φ ⊗k T of L-functions unit parametrized by integers k in each residue class modulo (q − 1) is a strong family of meromorphic functions with respect to the p-adic topology of k. A finite rank σ-module is automatically nuclear. Thus, Theorem 1.1 includes the key lemma of [17] over A n as a special case. The basic ideas of the present paper are the same as the limiting approach in [16]. The details are, however, quite different. In the simplest q-th power Frobenius lifting case, one has the fundamental Dwork trace formula available, which is completely explicit for uniform estimates. This makes it easy to extend the Dwork trace formula to infinite rank setting. It also makes it possible to see the various analytic subtleties involved in a concrete case. As a result, we were able to prove analytically optimal results in [16]. For a general Frobenius lifting (even over the simplest affine n-space A n as we shall work in this paper), one has to use the much more difficult Monsky trace formula which is a generalization of Dwork’s trace formula. Thus, the first task of this paper is to extend the Monsky trace formula to infinite rank setting and to make it sufficiently 1991 Mathematics Subject Classification. Primary 11G40, 11S40; Secondary 11M41, 14G15. Key words and phrases. L-functions, Fredholm determinants, p-adic meromorphic continua- tion, nuclear σ-modules and Banach modules. This work was partially supported by NSF. c American Mathematical Society

The spectra of nonnegative integer matrices via formal power series

Abstract: An old problem in matrix theory is to determine the n-tuples of complex numbers which can occur as the spectrum of a matrix with nonnegative entries (see [BP94, Chapter 4] or [Min88, Chapter VII]). Authors have studied the case where the ntuple is comprised of real numbers [Bor95, Cia68, Fri78, Kel71, Per53, Sal72, Sou83, Sul49], the case where the matrices under consideration are symmetric [Fie74, JLL96], and the general problem [Joh81, LM99, LL79, Rea94, Rea96, Wuw97]. Various necessary conditions and sufficient conditions have been provided, but a complete characterization is known for real n-tuples only for n ≤ 4 [Kel71, Sul49] and for complex n-tuples only for n ≤ 3 [LL79]. Motivated by symbolic dynamics, Boyle and Handelman refocused attention on the nonzero part of the spectrum by making the following “Spectral Conjecture” [BH91, BH93] (see also [Boy93, §8] and [LM95, Chapter 11]). Below, a matrix A is primitive if all entries of A are nonnegative and for some n, all entries of A are strictly positive. Also,

The averaging lemma

Abstract: Averaging lemmas arise in the study of regularity of solutions to nonlinear transport equations. The present paper shows how techniques from Harmonic Analysis, such as wavelet decompositions, maximal functions, and interpolation, can be used to prove averaging lemmas and to establish their sharpness. Let f(x, v) be a real-valued function defined on R × Ω, where Ω is a bounded domain in R. In applications Ω is a set of velocity vectors. Associated to f , we have the velocity average