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We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given., 70 pages, 1 figure

We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of H\"ormander's hypoelliptic regularity theory in an $L^1$ framework which estimates this (degenerate) Fisher information from below by a $W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide range of systems beyond the reach of currently existing mathematically rigorous methods. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE; in this paper we prove that this class includes the Lorenz 96 model in any dimension, provided the additive stochastic driving is applied to any consecutive pair of modes., Comment: 62 pages, updated intro and appendix

In his paper [K], Kashiwara introduced the notion of characteristic cycle for complexes of constructible sheaves on manifolds: let X be a real analytic manifold, and F a complex of sheaves of C-vector spaces on X , whose cohomology is constructible with respect to a subanalytic strati cation; the characteristic cycle CC(F) is a subanalytic, Lagrangian cycle (with in nite support, and with values in the orientation sheaf of X ) in the cotangent bundle T ∗X . The de nition of CC(F) is Morse-theoretic. Heuristically, CC(F) encodes the in nitesimal change of the Euler characteristic of the stalks of F along the various directions in X . It tends to be di cult in practice to calculate CC(F) explicitly for all but the simplest complexes F; on the other hand, the characteristic cycle construction has good functorial properties. The behavior of CC(F) with respect to the operations of proper direct image, Verdier duality, and non-characteristic inverse image of F is well understood [KS]. In this paper, we describe the e ect of the operation of direct image by an open embedding. Combining our result with those that were previously known, we obtain descriptions of CC(Rf∗F) and CC(f∗F) – analogous to those in [KS] – for arbitrary morphisms f : X → Y in the semi-algebraic category, and complexes F with semi-algebraically constructible cohomology. In e ect, this provides an axiomatic characterization of the functor CC, at least in the semi-algebraic context. Our arguments do apply more generally in the subanalytic case, but because statements become quite convoluted, we shall not strive for the greatest degree of generality. As a concrete application, we consider the case of the ag manifold X of a complex semisimple Lie algebra g. Here the Weyl group W of g operates

The main purpose of this paper is to show that an n-dimensional manifold with Ricci curvature greater or equal to (n−1) which is close (in the Gromov– Hausdor topology) to the unit n-sphere has volume close to that of the sphere. This shows the converse of the theorem in [C1]. Namely together with [C1] it shows that an n-manifold with Ricci curvature greater or equal to (n − 1) is close to the sphere if and only if the volume is close to that of the sphere. In particular, by [P], such a manifold is homeomorphic to a sphere. Further, as an application of this and the result of [C1], we prove a Radius Theorem saying that if an n-manifold with Ricci curvature greater or equal to (n − 1) has radius almost equal to ; then the volume is close to that of the sphere. In order to obtain these results we further develop and apply the estimates of [C1]. Whereas the main concern in [C1] were with the large scale geometry the main concern of this paper is with the small scale geometry. Let !n be the volume of the round n-sphere, Sn; with sectional curvature one.

This paper introduces a class of pseudodifferential operators depending on a parameter in a particular way. The main application is a complete expansion of the trace of the resolvent of a Dirac-type operator with nonlocal boundary conditions of the kind introduced by Atiyah, Patodi, and Singer [APS]. This extends the partial expansion in [G2] to a complete one, and extends the complete expansion in [GS 1 ] to the case where the Dirac operator does not have a product structure near the boundary. A secondary application is to obtain a complete expansion of the resolvent of a ~bdo on a compact manifold, essentially reproving a result of Agranovich [Agr]. The resolvent expansion yields immediately an expansion of the trace of the heat kernel, and determines the singularities of the zeta function; moreover, a pseudodifferential factor can be allowed. A major motive for these expansions is to obtain index formulas for elliptic operators; there are many such applications in the physics and geometry literature. The index formula comes from one particular term in the expansion, but each term is a spectral invariant, and they have been used for other purposes as well as for the index. In particular, Branson and Gilkey have a number of papers (e.g. [BG] and [Gi]) analyzing these invariants, and drawing geometric consequences. Interest in the asymptotic behavior of the resolvent goes back to Carleman [C]. More recently, Agmon [Agm] developed it extensively for analytic applications; he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable. This idea was developed in [S1] to analyze the singularities of the zeta function of an elliptic Odo on a compact manifold, and in [$3] to analyze the resolvent of a differential operator with differential boundary conditions. The technique works smoothly for differential operators, producing so-called local invariants, integrals over the underlying

In this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.

In this paper, we study certain infinite dimensional representations of the general linear group GL2 over a local field K, in vector spaces over K. These representations are closely related to the theory of analytic functions on the p-adic upper half plane over K. They were originally introduced by Morita [Mor] , who studied them intensively in the case when the characteristic of K is zero. Morita 's work has subsequently been greatly extended by Schneider and Stuhler [SS] and Schneider [S] to representations of GL,(K) and higher p-adic symmetric spaces. In this paper, we re-consider some of Morita 's representations of GL2 from the point of view of integral and modular representation theory. It seems quite natural to study p-adic representations by reducing them modp, and we supply a technique for doing this. In particular, suppose that Ydenotes the Bruhat Tits tree of PGLz(K). We consider the representations of PGL2(K) on the harmonic functions on W of e v e n weight that is, the harmonic functions on the edges of J taking values in even symmetric powers of the standard representation of GL2. In this paper, we find an invariant integral lattice in these representations, and we describe the modular representations of PGL2(K) obtained by reducing this lattice modp. The modular representations obtained from Morita 's representations are quite simple and natural objects, although they seem a bit peculiar at first. Suppose that F is the residue field of K, that V is a vector space over F, and that p :GL2(F) ~ End(V) is a finite-dimensional (modular) representation. Notice that p may be viewed in a natural way as a modular representation of GL/(R), where R is the valuation ring of K. We show that a typical infinite dimensional representation of Morita's, when reduced mod p, has a filtration whose factors are isomorphic to "I"~CL2~R~WJ.T~n~L2~)t~ Note that there are only finitely many irreducible, finite dimensional modular representations of GLz(F); the appearance of these finitely many

The latter is a sort of 'L-indistinguishability' result concerning the (presumably transcendental) Petersson norms-squared of Hecke eigenfunctions. The general assertion is essential in discussion of special values of L-functions obtained as inner products. Incidental to the proof of this fact, we obtain a very short proof of the important (known) result that the space of holomorphic cuspforms of such weight and with respect to a principal congruence subyroup is spanned by those with rational Fourier coefficients (for weights as above). Rather than starting from the theory of canonical models, we begin with consideration of the arithmetic of the Fourier coefficients of Siegel's Eisenstein series: this is a relatively elementary issue, amenable to direct calculation (although, ironically, the best reference currently available seems to be [H3], wherein canonical models results are invoked). By contrast, previous proofs of results concerning rationality properties of Fourier coefficients have relied essentially upon the theory of canonical models: the paradigms are the two papers [Sh2] and [Sh3] of Shimura, which depend upon the sequence of his papers culminating in [Shl] which developed the necessary theory of canonical models.

Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image of a finite interval under a Lipschitz mapping. Recall that for a connected set F c C, F is a rectifiable curve (not necessarily simple) if and only if l(F) < ~ , where l(-) denotes one dimensional Hausdorff measure. This classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge (but not necessarily each vertex!) at most twice. If K is a finite set then we are essentially reduced to the classical Traveling Salesman Problem (TSP): Compute the length of the shortest Hami l ton ian cycle which hits all points of K. This is the same, up to a constant multiple, as asking for the inf imum of l(F) where F is a curve, K c F. (Such a F is called a spanning tree in TSP theory.) For infinite sets K, we cannot hope in general to have K be a subset of a Jordan curve. What we should therefore look at is connected sets which conta in K. Let Fmi n be the shortest (minimal) spanning tree. Then we cannot possibly solve our problem for sets K of infinite cardinality if we cannot find F, I(F) < C O/(Fmin) , for any finite set K. (Here and throughout the paper C, Co, C1, c o , etc. denote various universal constants.) While there are several algorithms for computing l(Fml.), these algorithms work for finite graphs satisfying the triangle inequality, and do not use the Euclidean properties of K. (See [13] for an excellent discussion of some of these algorithms.) Therefore these methods cannot solve our problem for general infinite K. We present a method which is a minor modification of a well-known algorithm ("Farthest Insert ion" see [13]) which yields a F with I(F) < C O l(Fmi,). The Farthest Insert ion algorithm has been extensively studied with large numerical calculations on computers, and is experimentally good in the sense that the F produced satisfy I(F) < C O l(F,,,i,) for all examples which have

After the Plateau problem had been solved by Douglas and Rad6 by finding a possibly branched disk type minimal immersion bounded by a given Jordan curve in ~N, soon more general problems were proposed. On one hand, the existence of minimal surfaces of higher topological structure was investigated by Douglas I-D], Courant [C], and Shiffman [Sh 1]. On the other hand, the problem of finding unstable minimal surfaces, and, more generally, developing a Morse theory for minimal surfaces of disk type was attacked by Morse-Tompkins [MT1] and Shiffman [Sh2]. A natural question then was to develop a Morse theory for minimal surfaces of arbitrary topological structure. While Morse-Tompkins in their paper [MT2] only treated a very special case, namely annulus type surfaces, of what their title "Unstable minimal surfaces of higher topological structure" claims, Shiffman confronted the case of genus 0 and arbitrary connectivity [Sh3]. These classical papers were not in every respect satisfactory. The investigations of Douglas and Courant on the higher genus problem were recently critisized by Tromba [T 1]; however, Luckhaus I-L] was able to carry out a systematic reworking of the arguments of Courant and Shiffman. The original approaches of Morse-Tompkins, Shiffman moreover severely suffer from the fact that these authors work in the C~ instead of the more natural H l"2-topology. In the C~ Dirichlet's integral (whose critical points parametrize the sought-after minimal surfaces) is not differentiable and no intrinsic notion (i.e. depending only on the surface) of Morse index can be defined. This makes it impossible to decide in the work of Morse-Tompkins and Shiffman whether the Morse relations for minimal surfaces reflect a property of the surfaces

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such ‘exotic’ components in moduli spaces of $$\mathrm {SO}(p,q)$$ -Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group $$\mathrm {SO}(p,q)$$ . Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for $$\mathrm {SO}(2,q)$$ , with $$q\geqslant 4$$ ).

In this paper, we show that the total area of two distinct surfaces with Gaussian curvature equal to 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least $$4 \pi $$ . In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We refer to this lower bound of total area as the Sphere Covering Inequality. The inequality and its generalizations are applied to a number of open problems related to Moser–Trudinger type inequalities, mean field equations and Onsager vortices, etc, and yield optimal results. In particular, we prove a conjecture proposed by Chang and Yang (Acta Math 159(3–4):215–259, 1987) in the study of Nirenberg problem in conformal geometry.

Andr\'e recently gave a beautiful proof of Hochster's direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where "discriminants" acquire all $p$-power roots. In this paper, we explain a quicker proof of Hochster's conjecture that circumvents the perfectoid Abhyankar lemma; instead, we prove and use a quantitative form of Scholze's Hebbarkeitssatz (the Riemann extension theorem) for perfectoid spaces. The same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de Jong., Comment: 12 pages, comments welcome

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group Ham c (M) of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction onM. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic toM×D2 which are symplectically ruled overD2. When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided thatM is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory ofJ-holomorphic curves in arbitraryM.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongstall paths, not only the homotopic ones) under even more restrictive conditions onM, for example whenM is exact and convex or of dimension 2. The new difficulty is caused by the possibility that there are non-trivial and very short loops in Ham c (M). When such lengthminimizing paths do exist, we can extend the Bialy-Polterovich calculation of the Hofer norm on a neighbourhood of the identity (Cl-flatness).

The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such "spectral collapse," however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erd\H{o}s and Szekeres., Comment: to appear in Inventiones mathematicae

The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes., Comment: Substantial revision to improve the readability of the paper

This paper proceeds with the study of the $${\mathcal {C}}^0$$ -symplectic geometry of smooth submanifolds, as initiated in Humiliere et al. (Duke Math J 164(4), 767–799, 2015) and Opshtein (Ann Sci Ec Norm Super 42(5), 857–864, 2009), with the main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic homeomorphism may preserve and squeeze codimension 4 symplectic submanifolds ( $${\mathcal {C}}^0$$ -flexibility), while this is impossible for codimension 2 symplectic submanifolds ( $${\mathcal {C}}^0$$ -rigidity). We also discuss $${\mathcal {C}}^0$$ -invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-Gromov $${\mathcal {C}}^0$$ -rigidity type question for submanifolds, which we solve in many cases. Our main technical tool is a quantitative h-principle result in symplectic geometry.

In his Bombay lecture ([15]), R.Thom used some mysterious geometric argument to prove his Theorem 3. Recently, S. Koike and W. Kucharz have discovered some astonishing phenomenon on jets ([7]). In this paper, we shall use Whitney's (a,b)-regularity conditions and the Trotman tS-regularity condition to clarify Thom's intuitive ideas, thus yielding some geometric characterizations of sufficiency of jets; we are then able to explain the Koike-Kucharz phenomenon. Recall that the problem of sufficiency is to determine how to omit higher ordered terms in a Taylor expansion. Let d,.(n, p), or simply d~t,.l, denote the set of all Cr-mappings f: (IR",0)-~(IRP,0). Given an r-jet zEJ"(n,p), and geoe~+~, s>0 , with jt~)(g)=z, we call g an 8~r+q-realization of z z is C~ (resp. vsufficient) in Eta+. q if for any two So~r+~l-realizations g~ ,g2, there exists a local homeomorphism h, g~=g2oh (resp. the germs of gi-~(0) and g~-l(0) at 0 are homeomorphic). A number of criteria and characterizations, for the cases s=0 , 1, have been found in the past (See References); they are of analytic nature a type of Lojasiewicz inequality in each case. The characterizations in this paper are geometric, expressing various types of transversality. Consider the following conditions on a jet ze,l"(n, p).

In this paper we give a full description of the spectrum of the Ruelle-Perron-Frobenius operator acting on the Banach space of Holder continuous functions on a subshift of finite type (Theorem 1). These results are then used to extend the meromorphic domain of generalised zeta functions (Theorem 2). The most important application of these results is to the domain of the Smale zeta function for Axiom A flows (Theorem 3). In the course of this paper we settle questions raised by Ruelle and Sunada.

A map germ f is said to be determined of order k if f is smoothly equivalent to its Taylor polynomial of order k, and any g with the same Taylor polynomial of order k is smoothly equivalent to f A map germ is finitely determined if it is determined of some order k. Given a finitely determined germ f a question immediately arises as to the order of the Taylor polynomial which determines f. The only previous estimate for finitely determined non stable germs is due to Mather and the estimate is astronomical [2]. This paper provides a low order estimate of the order of deter- mination in terms of the power of the maximal ideal contained by tf (O(n)) + o)f (O(p)) and p the dimension of the target space. (1.4) Theorem. Suppose f is in Eo(n,p) and tf(O(n))+mf(O(p)) contains mkO(f), k > O; then f is k(p + 1) determined. (Recall that iff is finitely determined tf(O(n)) + ~of (O(p)) must contain m k O(f) for some k.) Another variant of the generalized Malgrange preparation theorem is used as a tool in the proof. (2.3) Theorem. (Generalized Preparation Theorem). Let f: (IR", 0)-* (~P, 0), g: (~,, 0)--~ (~P', 0) be Coo map germs. Let A be a finitely generated (f, g)* Cp+p, module. If A/g* (rap,) A is finite dimensional as an lR vector space, then A is finitely generated as a g*(Cp,) module. The notation used in this paper is essentially the same as John Mather's papers on stability of C ~ mappings: d is the group of coordinate changes in source and target. Eo(n, p) stands for orign preserving map germs from either ~" to IR p or from I~" to IE p which are either Coo or analytic in the first case or analytic in the second case. C, stands for Coo germs f: (IR", 0)~ IR". Iff is in Eo(n, p), tf(O(n)) is the set of p tuples of germs obtained by applying the differential of f to the set of n tuples of germs in E, ; o)f(O(p)) is the set of p

In this paper we obtain a curvature estimate for embedded minimal surfaces in a three-dimensional manifold of positive Ricci curvature in terms of the geometry of the ambient manifold and the genus of the minimal surface. It should be mentioned that there are two main points in our result: One is the absence of a stability assumption and the other is the requirement of being embedded. Most known curvature estimates require the stability assumption, and once the stability assumption is dropped, many of these known results cease to be valid. (See [SS] and [An] for another example of a curvature estimate without the assumption of stability.) The embeddedness condition is rather subtle because of the way it enters in our proof. Our proof depends on the eigenvalue estimate and the area bound due to the first author and Wang [CW] which require embeddedness in an essential way. Present knowledge indicates that closed embedded minimal surfaces in S 3 are rare, while immersed surfaces are more plentiful. For example, only a finite number of minimal embeddings of a given genus are known. On the other hand, Otsuki [O] constructed infinitely many immersed minimal tori with arbitrarily large area. We obtain the curvature estimate indirectly by proving the smooth compactness theorem (Theorem 1). Theorem 1 has many interesting consequences. It shows that the set of conformal structures that can be realized on a minimal embedding in S 3 is a compact subset of the moduli space. This is in contrast with the result of Bryant [B] who showed that every Riemann surface is conformally and minimally immersed in S 4. In view of our compactness result and the scarcity of examples, it is very tempting to conjecture that there are only finitely many embedded minimal surfaces (up to rigid motion) in S 3 for each fixed genus. Throughout this paper, manifold means manifold without boundary unless explictly stated otherwise. When we say a sequence M i of surfaces converges to a

In a recent paper [H], we proved the "Mock Theta Conjectures". These are identities, stated by Ramanujan in his "lost notebook" JR2, pp. 19-20], involving two of the 5th order mock 0-functions. In fact, Ramanujan gave one such identity for each of the ten 5th order functions; as shown in [A-G1], these ten identities are all equivalent to the two proved in I-HI. In this paper, we prove the corresponding identities for Ramanujan's 7th order mock 0-functions, and show how they are related to rank differences of partitions. To state our results, we first introduce some standard notation: If q and x are complex numbers with [ql < 1 and n is a nonnegative integer, then

Let G be a connected real semi-simple Lie group with finite center. One of the main problems in harmonic analysis is to determine the unitary spectrum of G. In this paper we treat this question in the case when real rank of G is I. Although the answer was known for G of classical type previously ([2], [5], [7], [15]), we have redone this work sometimes giving simpler arguments. With the arbitrary rank case in mind we have tried to deal with the problem of classifying unitary representations in as systematic a way as possible pro- ceeding by induction on the dimension of G. We have concentrated on the case when G is linear and has a compact Cartan subgroup, the other cases being known already. As an application we also give a list of the unitary representations that contribute to the L2-index formula for the Dirac operator with coefficients. As is apparent from the calculations in [19], this is intimately connected with one of our main techniques for determining the unitary spectrum, the Dirac in- equality. We will deal with some related problems in a future paper.

Connes [6] and Haagerup [11] established the relation between von Neumann algebras (abbreviate: VNA) and selfdual, homogenous and orientated cones in Hilbert space. The ideal center ofa selfdual cone in a Hilbert space is the hermitian part of an abelian VNA [3]. This gives rise to direct integral decompositions of selfdual cones with respect to von Neumann subalgebras of the VNA generated by the ideal center (see w Theorem 3.17). The existence of a "central" decomposition has been shown in [18], too. The purpose of the present paper is the study of the properties of such decompositions (w 3). Especially, we determine the facial structure of a direct integral of selfdual cones (w 3, Proposition 3.5, Lemma 3.6, 3.7, Corollary 3.8). It follows rough ly spokenthat all properties of selfdual cones, which are expressed by means of facial projections, hold for a direct integral of selfdual cones iff they hold for the fibres (w 3, Corollary 3.9, 3.10, Remark 3.11). Further, this is used to study the connection between direct integrals of VNA and the associated selfdual, homogenous and orientated cones, i.e. the associated standard forms (w Proposition 3.14, Theorem 3.15). Another consequence is the decomposibility of certain linear maps between direct integrals of VNA (w Theorem 4.1). Direct integrals of standard forms of VNA have also been considered in [13, 15] and [21]. But the approach to direct integrals of standard forms of VNA given there is based on the decomposition theory for left Hilbert algebras, which isn't used in this paper. The paper is subdivided into four sections. In the preliminaries (w 1) we recall basis facts on selfdual cones in Hilbert space and the terminology needed here. w 2 is devoted to the theory of direct sums. In w 3 and w 4 we develop the decomposition theory for selfdual cones and present as mentioned above applications to von Neumann algebra theory.

In a previously published paper [9], the authors proved an approximation theorem for geodesically convex functions on Riemannian manifolds (Theorem 2 of [91). Applications of this theorem to complex function theory on noncompact K~ihler manifolds have been announced in [8, II and III] . The main result of the present paper is an application of this approximation theorem to Riemannian geometry proper. To describe this principal theorem, let M be a Riemannian manifold and let Z(M) be the closure of the set of all C ~176 subharmonic functions in C~ the algebra of continuous functions on M equipped with the compact open topology. (A C ~ function f is subharmonic if Af>O, where the sign

This paper represents an detailed version of our paper [12] presented at the Hawaii Conference on the Geometry of the Laplace Operator, March 1979. In this paper we present the solution to the following problem posed by Singer in [8], w 4, concerning elliptical operators on PL-maifolds: "If M is a PL-manifold, the L-polynomials are still well defined (Thom [13]) from which one can still define the rational Pontrjagin classes. The Hirzebruch signature theorem still holds. Is there an associated elliptic operator (as in the smooth case) whose index is the signature? On what spaces does it operate? Does it have a symbol and where does the symbol lie?"

Here, arrow (i) refers to our previous papers [9, 12] and [13], in which we have shown that a fixed primitive cusp form f of Sk(F I(N)) has congruences with other cusp forms of Sk(F~(N)) modulo the special value at s = k of a certain zeta function off . In this paper, we will construct the second arrow for the cusp form f associated with a Hecke character of an imaginary quadratic field K, and consequently obtain the third. Namely, in w167 6 we will show, as in the works of Coates-Wiles [3] and Robert [23], that the split primes of K which divide this special value are irregular in an appropriate sense. In this case, the zeta function o f f mentioned above is a Hecke L-function of K. Such a criterion of irregularity for Hurwitz numbers has already been obtained in [3] and [233. However, our result together with those of [3] and [23] does not cover all the L-values of Hecke L-functions of K whose algebraicity was given by Damerell [5] (for details, see below). Our method is analogous to that of Ribet [21] (see also Wiles [35]), who used congruences between cusp forms and Eisenstein series to prove the non-vanishing of certain eigenspaces (relative to the action of Gal(Q/Q)) of the p-primary part of the class group of the field of p-th roots of unity. By adopting this approach, we will obtain such an information of eigenspaces again in our case (see Theorem 0.1 below). To be more specific, let 2 be a Hecke character of K satisfying

Let X = G/K be a Riemannian symmetric space of non-compact type, and let D(X) be the algebra of invariant differential operators on X. In a previous paper [1] we developed a theory of asymptotic expansions for joint eigenfunctions of D(X) of at most exponential growth. In the present paper we show that local asymptotic data determine the eigenfunctions completely. We also develop a theory of asymptotic expansions "along walls". Both results are of importance for the theory of the discrete series of a semisimple symmetric space. Let a be a maximal abelian split subspace of the Lie algebra 9 of G, and denote by a* its complexified dual. Let E be the restricted root system of a in g, and W the associated Weyl group. Then Harish-Chandra's isomorphism D ( X ) ~ S(a) w determines a bijection 2~--~Z~ from a*/W to the set D(X) ^ of algebra homomorphisms D ( X ) ~ C . Given 2ca* , let ga(X) denote the joint eigenspace of functions f e C o~ (X) satisfying

In the present paper we shall give evidence to support the claim (Conjecture I below and (1.3)) that every elliptic curve A/o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of A. In particular, we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of Stickelberger elements ad p-adic L-functions attached to A. In addition, if Conjecture I is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the Q-isogeny class of A distinguished by modular considerations. For most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of X I (N) (Definition 1.1). However, to justify our view of these parametrizations as being canonical, we begin here with a more intrinsic definition. Recall that Shimura ([19], Chap. 6; see w 1 of this paper) has defined a compatible system of canonical models of modular curves {Xs, SeS~}, where 5 p is a certain collection of open subgroups of the group GL(2, Az) over the finite adeles A I of Q. We define the adelic upper half-plane to be the pro-variety )~=lL_m Xs and give )~ the Q-structure induced by the s field of modular functions whose q-expansions at the 0-cusp have coefficients in Q. A modular parametrization of A is a Q-morphism ~: ) ( ~ A which sends

is called the momentum mapping of the Hamiltonian T-action. Given (1.1), the condition (1.2) just means that T acts along the fibers of J. For the basic definitions and properties of non-commutative Hamiltonian group actions, see [AM]. The results of this paper can easily be extended to Hamiltonian actions of arbitrary compact connected Lie groups, by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping. For some more details, see the remarks at the end of Sect. 2. We will assume throughout this paper that the momentum map is proper, that is J I(U) is compact for each compact subset U of t*. Now let r be a regular value of J, that is T,,J: TmM-~t* is surjective for all meYe=J 1(4 ). Then Ye is a smooth submanifold of M, compact because

Introduction Around 1900 Hilbert, Hecke ([7]), Blumenthal ([2]) and others started the study of certain 2-dimensional complex spaces, which are closely related to the classification of special types of 2-dimensional abelian varieties. These complex spaces can easily be described. In fact, let n be a natural number, n > 1, which is square free and let K = Q q/n). Then, if o K is the ring of algebraic integers of K, the group SLz(oK) operates in a natural way on ~ • ~ and on ~ • ~ where ~ is the upper and ~ the lower half plane of C. The quotients of See • ~ and Jg x J g by the action of SL2(ov,) are the 2-dimensional complex spaces mentioned above. If the field K has a unit of negative norm, then the two actions on ~ • ~ and ~ • J g are isomorphic. This is true if n is a prime congruent 1 rood 4, the only case we shall consider in this paper. Therefore, from now on we assume that K= Q (l,/p), where p is a prime congruent 1 mod 4. The complex space Jg x .)~/SLz(oK) can be compactified by means of a finite number of points, called the cusps, to a compact 2-dimensional complex space. After resolving the cusps and also the quotient singularities on ~ x ~ / S L 2 (OK) , both in a canonical, explicit way, a nonsingular compact complex surface Y(R) is obtained, which in fact is an algebraic surface. The field of meromorphic (i. e. rational) functions on Y(p) is isomorphic to the field of meromorphic functions of • ~ / S L 2 (OK). On the other hand, although no complete classification of algebraic surfaces is known, there exists a rough classification in several classes (for most of which the surfaces contained in that class can be classified completely, at least in principle). In big outline this classification was already known to the italian school, but its precise formulation (this time also covering the non-algebraic case) and many of the proofs involved are due to Kodaira. Now the question considered in this paper is the following: where are the surfaces Y(p) to be placed in the rough classification of algebraic surfaces?

In order to treat certain "non-symmetric" potential theories, It6 I-4] and Bliedtner [1] have generalized Beurling and Deny's theory of Dirichlet spaces by replacing the inner product in the Dirichlet space with a bilinear form defined on a real, regular functional space. This bilinear form, called a Dirichlet form, is supposed to be continuous and coercive, and furthermore to satisfy a "contraction"-condition. When the underlying space is a locally compact abelian group, there is a complete characterization of translation invariant Dirichlet spaces in terms of real, negative definite functions on the dual group. The main purpose of the paper is to obtain an analogous characterization (Theorem 3.7) of translation invariant Dirichlet forms, in terms of complex, negative definite functions on the dual group. The contents of the paper may be summarized in the following way: In w 1 we study positive closed sesquilinear forms on an abstract complex Hilbert space, and using the Lax-Milgram theorem, it is shown, that with every such form is associated a resolvent. The theory is specialized in w 2 to sesquilinear forms on the Hilbert space L 2 (X, ~), where some relations with normal contractions are discussed. In particular an example of a positive closed form fl is given, such that the unit contraction operates with respect to fl, but not with respect to the form fl*, which is adjoint to ft. However, in the case of a translation invariant, positive closed form fl on L 2 (G), where G is a locally compact abelian group, it is shown in w 3, that the unit contraction operates with respect to fl if and only if it operates with respect to fl*, and we finish by establishing the above mentioned characterization of translation invariant Dirichlet forms.

For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem. In this paper we consider the physical dimensions $d=2,3$ and show that if we perturb any given smooth initial data in $H^{s_c}$ norm, then the corresponding solution can have infinite $H^{s_c}$ norm instantaneously at $t>0$. The constructed solutions are unique and even $C^{\infty}$-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems., 119 pages

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves in SL_2(C) that are related to Bryant surfaces. We establish a basic structure theorem for the set of all A-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that A is irreducible and is not contained in any hyperplane, we show that every A-immersion can be approximated by A-embeddings; this holds in particular for null curves in C^3. If in addition A-{0} is an Oka manifold, then A-immersions are shown to satisfy the Oka principle, including the Runge and the Mergelyan approximation theorems. Another version of the Oka principle holds when A admits a smooth Oka hyperplane section. This lets us prove in particular that every open Riemann surface is biholomorphic to a properly embedded null curve in C^3., Inventiones Math., in press

In this paper we obtain estimates for certain transcendence measures of an entire function $f$. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial $P(z,w)$ in ${\Bbb C}^2$ along the graph of $f$. These inequalities provide, in turn, estimates for the number of zeros of the function $P(z,f(z))$ in the disk of radius $r$, in terms of the degree of $P$ and of $r$. Our estimates hold for arbitrary entire functions $f$ of finite order, and for a subsequence $\{n_j\}$ of degrees of polynomials. But for special classes of functions, including the Riemann $\zeta$-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values $f(E)$, in terms of the size of the set $E$., Comment: 40 pages

In this paper, we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface. More precisely, let \(Pr:\mathcal{H}\textit{omeo}(M)\to\mathcal{M}\mathcal{C}(M)\) denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface M of genus g>5. We show that there is no homomorphism \(\mathcal{E}:\mathcal{M}\mathcal{C}(M)\to\mathcal{H}\textit{omeo}(M)\), such that \(Pr\circ\mathcal{E}\) is the identity. This answers a question by Thurston (see [11]).

The theory of Nichols algebras of diagonal type is known to be closely related to that of semisimple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semisimple Lie algebra. They give rise to the definition of a Brandt groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding. Key Words: Brandt groupoid, Hopf algebra, pseudo-reflections, Weyl group

In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of {\it closed} one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of {\it strong homotopy Lie algebroid} (or {\it $L_\infty$-algebroid}) and associate a canonical isomorphism class of strong homotopy Lie algebroids to each pre-symplectic manifold $(Y,\omega)$ and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy Lie algebroid. The formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the {\it Maurer-Cartan equation} (or the {\it master equation}) of the strong homotopy Lie algebroid, and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic $A$-branes. We provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation., Comment: 60 pages, This is the final version that will appear in Invent. Math

This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand making essential use of the results of the first two parts., TeX reformatting of v1, now 62pp. (Thanks Adrian.)

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kaehler metric (which is hyperkaehler if and only if c_1(X)=0), and that if c_1(X), 29 pages, 3 figures, related to math.DG/0105263, improved exposition, new asymptotically complex hyperbolic examples

This paper describes the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. The conformally invariant powers of the Laplacian arise as residues of the scattering matrix and Branson's Q-curvature in even dimensions as a limiting value. The integrated Q-curvature is shown to equal a multiple of the coefficient of the logarithmic term in the renormalized volume expansion.

In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the K\"ahler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functionals in case of Kaehler manifolds. On K\"ahler-Einstein manifold with positive scalar curvature, if the initial metric has positive bisectional curvature, we prove that these functionals have a uniform lower bound, via the effective use of Tian's inequality. Consequently, we prove the following theorem: Let $M$ be a K\"ahler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler Ricci flow will converge exponentially fast to a K\"ahler-Einstein metric with constant bisectional curvature. Such a result holds for K\"ahler-Einstein orbifolds.

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmonic polynomial on the Cartesian square of the Cartan subalgebra, that transforms under an irreducible representation of the Weyl group. In the special case of sl_n, the conjugacy classes of principal nilpotent pairs and the irreducible representations of the Symmetric group, S_n, are both parametrised (in a compatible way) by Young diagrams. In general, our theory provides a natural generalization to arbitrary Weyl groups of the classical construction of simple S_n-modules in terms of Young's symmetrisers.

In 1955, Berger [Ber] gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. While this list was stated to be complete in the case of metric connections, the situation in the general case remained unclear. The representations which are missing from this list are called exotic. In this paper, we use twistor techniques to detect an infinite series of candidates for exotic holonomies. We then develop a general method for constructing torsion-free affine connections with prescribed holonomy which is based onequivariant deformations of a certain class of linear Poisson structures. When applied to the new series, this method yields an exhaustive description of all torsion-free connections with these holonomies, and hence not only proves the existence of such connections, but also allows us to deduce some striking facts about their local and global behaviour.

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant. 2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.) 3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnecessary.

In recent years pseudo-holomorphic curves on almost complex manifolds have played an important role in connection with the developments in symplectic geometry, notably on account of Gromov's work [8] in 1985. In particular, the existence of compact pseudo-holomorphic curves of the type of S 2 or the closed disc have been established on particular manifolds. In this paper we study, unrelated from symplectic geometry, non-compact pseudo-holomorphic curves on an almost complex torus (T2n,J). These curves carry a complex structure and therefore represent Riemann surfaces. In our case they are of the type of ~ or a cylinder ~* = IE\(0). In particular we are interested in the question of the persistence of these curves under perturbation of the almost complex structure. This question has similar features as that of the persistence of invariant tori in classical mechanics, as initiated by Kolmogorov [2,15,18]. However, in contrast with that theory we are dealing with nonlinear elliptic systems of partial differential equations of Cauchy-Riemann type. Our results provide global solutions for such systems.

The main objectives of this paper are: (1) to construct an Euler product from a Hecke eigenform of half-integral weight; (2) to prove its meromorphic continuation to the whole plane; (3) to examine its relationship with the Fourier coefficients of the form; and (4) to apply these results to the investigations of Eisenstein series of half-integral weight. The automorphic forms are holomorphic ones, considered on the metaplectic cover M~, of G~,, where G n = Sp(n,F) with a totally real algebraic number field F. The first three are generalizations of the corresponding results in the elliptic and Hilbert modular cases obtained in [$2] and [$6]. At the same time all four closely parallel similar results in the case of integral weight in [$9] and [S10]. To describe our results, let us take for simplicity F to be Q, in which case the weight k is half an odd positive integer. We consider a holomorphic form f of weight k, with respect to a principal congruence subgroup of Sp(n,Z) of level N, which is an eigenform in the sense that f iT(m) = 2 ( m ) f with a complex number 2(m), where N is a multiple of 4 and T(m) is a Hecke operator defined for each positive integer m in a natural way. As our first main result we shall prove (Theorem 4.4) that

To any dynamical system equipped with a metric, we associate a class of “well approximable” sets. In the case of an expanding rational map of the Riemann sphere acting on its Julia set, we estimate and in some cases compute the Hausdorff dimension of the associated “well approximable” sets. The methods used show a clear link between distortion properties and the type of results obtained in this paper, via ergodic theory and ubiquity.