Manfred Einsiedler

Non-archimedean amoebas and tropical varieties

Abstract We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the negative of the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the non-expansive set for a related dynamical system.

Primes in elliptic divisibility sequences.

Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.

Expansive subdynamics for algebraic Z^d-actions

A general framework for investigating topological actions of Zd on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of Rd. Here we completely describe this expansive behavior for the class of algebraic Zd-actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables. We introduce two notions of rank for topological Zd-actions, and for algebraic Zd-actions describe how they are related to each other and to Krull dimension. For a linear subspace of Rd we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.

Distribution of periodic torus orbits on homogeneous spaces

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R). After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most O(� ǫ ) orbits of discriminant ≤ �. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We also give examples of sequences of periodic orbits of this action that fail to become equidistributed, even in higher rank. We give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.

Rigidity properties of \zd-actions on tori and solenoids

We show that Haar measure is a unique measure on a torus or more generally a solenoid X invariant under a not virtually cyclic totally irreducible Zd-action by automorphisms of X such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the nonirreducible case. These results have applications regarding measurable factors and joinings of these algebraic Zd-actions.

Distribution of periodic torus orbits and Duke"s theorem for cubic fields

We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Dukes theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL_3(Z)\SL_3(R)/SO_3(R). In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL_3(Z)\SL_3(R)/SO_3(R) of volume less than V becomes equidistributed as V goes to infinity. The proof combines subconvexity estimates, measure classification, and local harmonic analysis.

MEASURE RIGIDITY AND p-ADIC LITTLEWOOD-TYPE PROBLEMS

The paper investigates various p-adic versions of Little- woods conjecture, generalizing a set-up considered recently by de Mathan and Teulie. In many cases it is shown that the sets of exceptions to these conjectures have Hausdor dimension zero. The proof follows the mea- sure ridigity approach of Einsiedler, Katok and Lindenstrauss.

Badly approximable systems of affine forms, fractals, and Schmidt games

Abstract A badly approximable system of affine forms is determined by a matrix and a vector. We show Kleinbocks conjecture for badly approximable systems of affine forms: for any fixed vector, the set of badly approximable systems of affine forms is winning (in the sense of Schmidt games) even when restricted to a fractal (from a certain large class of fractals). In addition, we consider fixing the matrix instead of the vector where an analog statement holds.

RIGIDITY OF MEASURES -- THE HIGH ENTROPY CASE AND NON-COMMUTING FOLIATIONS

We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures.

Diagonalizable flows on locally homogeneous spaces and number theory

Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm-Loewner Evolution.We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of non-singular Calabi�Yau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems.During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the paper, we overview the major components of the resulting theory (conic duality and primal-dual interior point polynomial time algorithms), outline the extremely rich �expressive abilities� of conic quadratic and semidefinite programming and discuss a number of instructive applications.This is a short survey of recent developments in one of the oldest areas of ergodic theory, sometimes called the spectral theory of dynamical systems. We mainly discuss the spectral realization problem in the rich class of all invertible measure preserving dynamical systems, a �behavior� of different spectral invariants in natural subclasses of dynamical systems, and a complete solution of Rokhlin�s problem on homogeneous spectrum in ergodic theory.In this paper we survey some results on the structure of noncommutative rings. We focus particularly on nil rings, Jacobson radical rings and rings with finite Gelfand�Kirillov dimension.We describe a general method of studying prevalent properties of diffeomorphisms of a compact manifold M, where by prevalent we mean true for Lebesgue almost every parameter a in a generic finite-parameter family {fa} of diffeomorphisms on M. Usually a dynamical property P can be formulated in terms of properties Pn of trajectories of finite length n. Let P be such a dynamical property that can be expressed in terms of only periodic trajectories. The first idea of the method is to discretize M and split the set of all possible periodic trajectories of length n for the entire family {fa} into a finite number of approximating periodic pseudotrajectories. Then for each such pseudotrajectory, we estimate the measure of parameters for which it fails Pn. This bounds the total parameter measure for which Pn fails by a finite sum over the periodic pseudotrajectories of length n. Application of Newton interpolation polynomials to estimate the measure of parameters that fail Pn for a given periodic pseudotrajectory of length n is the second idea. We outline application of these ideas to two quite different problems: � Growth of number of periodic points for prevalent diffeomorphisms (Kaloshin�Hunt). � Palis� conjecture on finititude of number of �localized� sinks for prevalent surface diffeomorphisms (Gorodetski�Kaloshin).The sixth of Hilbert�s famous 1900 list of twenty-three problems is a programmatic call for the axiomatization of physical sciences. Contrary to a prevalent view this problem was naturally rooted at the core of Hilbert�s conception of what axiomatization is all about. The axiomatic method embodied in his work on geometry at the turn of the twentieth-century originated in a preoccupation with foundational questions related with empirical science, including geometry and other physical disciplines at a similar level. From all the problems in the list, the sixth is the only one that continually engaged his efforts over a very long period, at least between 1894 and 1932.Our goal is to survey some of the main advances which took place recently in the study of the geometry of projective or compact Kahler manifolds: very efficient new transcendental techniques, a better understanding of the geometric structure of cones of positive cohomology classes and of the deformation theory of Kahler manifolds, new results around the invariance of plurigenera and in the minimal model program.Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston�s characterization of rational functions with finite post-critical set. In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpi´nski carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting.Generalizing and synthesizing earlier work on the model theory of valued difference fields and on the model theory of valued fields with analytic structure, we prove Ax�Kochen� Er�ov style relative completeness and relative quantifier elimination theorems for a theory of valuation rings with analytic and difference structure. Specializing our results to the case of W[F alg p ], the ring of Witt vectors of the algebraic closure of the field with p elements, given together with the relative Frobenius and the Tate algebras as analytic structure, we develop a model theoretic account of Buium�s p-differential functions. In so doing, we derive a uniform p-adic version of the Manin�Mumford conjecture.Amodular tensor category provides the appropriate data for the construction of a threedimensional topological field theory. We describe the following analogue for two-dimensional conformal field theories: a 2-category whose objects are symmetric special Frobenius algebras in amodular tensor category and whose morphisms are categories of bimodules. This 2-category provides sufficient ingredients for constructing all correlation functions of a two-dimensional rational conformal field theory. The bimodules have the physical interpretation of chiral data, boundary conditions, and topological defect lines of this theory.All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner�s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting.We will state an equivariant foliated version of the classical Brouwer PlaneTranslation Theorem and will explain how to apply this result to the study of homeomorphisms of surfaces. In particular we will explain why a diffeomorphism of a closed oriented surface of genus . 1 that is the time-one map of a time dependent Hamiltonian vector field has infinitely many periodic orbits. This gives a positive answer in the case of surfaces to a more general question stated by C. Conley. We will give a survey of some recent results on homeomorphisms and diffeomorphisms of surfaces and will explain the links with the improved version of Brouwer�fs theorem.We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous spaces, particularly their invariant measures, and present some number theoretic and spectral applications. Entropy plays a key role in the study of theses invariant measures and in the applications.This paper is a survey of recent results on greedy approximations with regard to bases. The theory of greedy approximations is a part of nonlinear approximations. The standard problem in this regard is the problem of m-term approximation where one fixes a basis and seeks to approximate a target function by a linear combination of m terms of the basis. When the basis is a wavelet basis or a basis of other waveforms, then this type of approximation is the starting point for compression algorithms. We are interested in the quantitative aspects of this type of approximation. Introducing the concept of best m-term approximation we obtain a lower bound for the accuracy of any method providing m-term approximation. It is known that a problem of simultaneous optimization over many parameters (like in best m-term approximation) is a very difficult problem. We would like to have an algorithm for constructing m-term approximants that adds at each step only one new element from the basis and keeps elements of the basis obtained at the previous steps. The primary object of our discussion is the Thresholding Greedy Algorithm (TGA) with regard to a given basis. The TGA, applied to a function f , picks at the mth step an element with the mth biggest coefficient (in absolute value) of the expansion of f in the series with respect to the basis. We show that this algorithm is very good for a wavelet basis and is not that good for the trigonometric system. We discuss in detail the behavior of the TGA with regard to the trigonometric system. We also discuss one example of an algorithm from a family of very general greedy algorithms that works in the case of a redundant system instead of a basis. It turns out that this general greedy algorithm is very good for the trigonometric system.The trajectories of a vector field in 3-space can be very entangled; the flow can swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be very complicated. In this talk, I will survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part will be devoted to vorticity, helicity, and asymptotic cycles for flows. The second part will deal with various notions of rotation and spin for surface diffeomorphisms. Finally, I will describe the important example of the geodesic flow on the modular surface, where the linking between geodesics turns out to be related to well-known arithmetical functions.We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems.Grigory Perelman has been awarded the Fields Medal for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture.We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors with calculation of their fundamental group and cocycle superrigidity for actions with applications to orbit equivalence ergodic theory.A famous theorem of Szemeredi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemeredi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green�Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemeredi�s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.A survey of the role of the complex of curves in recent work on 3-manifolds and mapping class groups.Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebro-geometric problems, such as the derived equivalence conjecture and description of T. Bridgeland�s space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a t-structure on the derived category of the resolution. The intriguing fact that the same t-structure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory.From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewise-linear objects that take over the role of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the problems in classical (real and complex) geometry.The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk Iwill showa panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory.This is an introduction to Iwasawa theory and its generalizations. We discuss some main conjectures and related subjects.Cooperation means a donor pays a cost, c, for a recipient to get a benefit b. In evolutionary biology, cost and benefit are measured in terms of fitness. While mutation and selection represent the main �forces� of evolutionary dynamics, cooperation is a fundamental principle that is required for every level of biological organization. Individual cells rely on cooperation among their components. Multi-cellular organisms exist because of cooperation among their cells. Social insects are masters of cooperation. Most aspects of human society are based on mechanisms that promote cooperation. Whenever evolution constructs something entirely new (such as multi-cellularity or human language), cooperation is needed. Evolutionary construction is based on cooperation. I will present five basic principles for the evolution of cooperation, which arise in the theories of kin selection, direct reciprocity, indirect reciprocity, graph selection and group selection.The P versus NP question distinguished itself as the central question of theoretical computer science nearly four decades ago. The quest to resolve it, and more generally, to understand the power and limits of efficient computation, has led to the development of computational complexity theory. While this mathematical discipline in general, and the P vs. NP problem in particular, have gained prominence within the mathematics community in the past decade, it is still largely viewed as a problem of computer science. In this paper I shall try to explain why this problem, and others in computational complexity, are not only mathematical problems but also problems about mathematics, faced by the working mathematician. I shall describe the underlying concepts and problems, the attempts to understand and solve them, and some of the research directions this led us to. I shall explain some of the important results, as well as the major goals and conjectures which still elude us. All this will hopefully give a taste of the motivations, richness and interconnectedness of our field. I shall conclude with a few non computational problems, which capture P vs. NP and related computational complexity problems, hopefully inviting more mathematicians to attack them as well. I believe it important to give many examples, and to underlie the intuition (and sometimes, philosophy) behind definitions and results. This may slow the pace of this article for some, in the hope to make it clearer to others.We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac�s Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in terms of loop algebras of Kac�Moody Lie algebras, and explain how it is proved using Hall algebras. We discuss applications to the problem of describing the possible conjugacy classes of sums and products of matrices in known conjugacy classes.A selection of aspects of the theory of bounded cohomology is presented. The emphasis is on questions motivating the use of that theory as well as on some concrete issues suggested by its study. Specific topics include rigidity, bounds on characteristic classes, quasification, orbit equivalence, amenability.We discuss criteria for the algebraicity of a formal subscheme V� in the completionX�P at some rational point P of an algebraic variety X over some field K. In particular we consider the case where K is a function field or a number field, and we discuss applications concerning the algebraicity of leaves of algebraic foliations, algebraic groups, absolute Tate cycles, and the rationality of germs of formal functions on a curve over a number field.Symplectic field theory (SFT) attempts to approach the theory of holomorphic curves in symplectic manifolds (also called Gromov-Witten theory) in the spirit of a topological field theory. This naturally leads to new algebraic structures which seems to have interesting applications and connections not only in symplectic geometry but also in other areas of mathematics, e.g. topology and integrable PDE. In this talk we sketch out the formal algebraic structure of SFT and discuss some current work towards its applications.Separate wall plates disposed perpendicularly to one another and with a seal between the abutting surfaces of the wall plates, form the mold cavity in a continuous casting mold. Each wall plate has a device for oscillating it in the radial direction of the mold cavity and another device for oscillating it in the plane of its contacting surface with the casting passing through the mold cavity. The wall plates are enclosed by a frame with abutment strips extending between the frame and the outer surface of the plates. Further, elastic elements are articulated to and extend between the wall plates and the frame. The elastic elements are adjustable for adapting to the tapering action of the casting as it moves through the mold.This paper reviews several mathematical results for partial differential equations modelling chemotaxis. In particular, questions like singularity formation for the Keller�Segel model and continuation of the solutions beyond the blow-up time will be discussed. Some of the open problems that remain for the Keller�Segel model as well as some new mathematical problems arising in the study of chemotaxis problems will be discussed.In this article I survey the descent method of Ginzburg, Rallis and Soudry and its main applications to the Langlands functorial lift of automorphic, cuspidal, generic representations on a classical group to (appropriate) GLn, and to establishing a local Langlands reciprocity law for (split) SO2n+1 (joint work with D. Jiang). The descent method arises when we consider certain residues of special cases of a family of global integrals, attached to pairs of automorphic, cuspidal representations, one on a classical group G and one on GLn. The last part of this article focuses on the caseG = SOm (split), and the progress made in a joint work with S. Rallis, towards establishing, via the converse theorem, the functorial lift from any automorphic, cuspidal representation on G to GL2[m2 ].We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed. Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined.Many physical systems can be modelled by nonconvex variational problems regularized by higher-order terms. Examples include martensitic phase transformation, micromagnetics, and the Ginzburg�Landau model of nucleation. We are interested in the singular limit, when the coefficient of the higher-order term tends to zero. Our attention is on the internal structure of walls, and the character of microstructure when it forms. We also study the pathways of thermally-activated transitions, modeled via the minimization of action rather than energy. Our viewpoint is variational, focusing on matching upper and lower bounds.The Hardy�Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov�s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the Hardy�Littlewood method has been generalised to obtain, for example, an asymptotic for the number of 4-term arithmetic progressions of primes less than N.We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2, . . . , n was obtained by Vershik�Kerov and (almost) by Logan�Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions.