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Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov's pioneering work.

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin

Author(s): Oh, SJ; Tataru, D | Abstract: This article is devoted to the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy; and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global scattering solution or a soliton bubbling off. In the last case, the bubbling-off phenomena can happen in one of two ways: (a) in finite time, triggering a finite time blowup; or (b) in infinite time. Our goal here is to first describe the equation and the results, and then to provide an overview of the flow of ideas within their proofs in the above-mentioned four papers.

George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz, again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of Huh and Huh-Katz left open the more general Rota-Welsh conjecture where graphs are generalized to (not necessarily representable) matroids and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh-Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota-Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota-Welsh conjecture would follow from combinatorial analogues of the Hard Lefschetz Theorem and Hodge-Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial Hard Lefschetz Theorem and Hodge-Riemann relations using purely combinatorial arguments. We will survey these developments., Comment: 22 pages. This is an expository paper to accompany my lecture at the 2017 AMS Current Events Bulletin. v2: Numerous minor corrections

This expository paper, based on a Current Events Bulletin talk at the January, 2016 Joint Meetings, introduces the concept of Lyapunov exponents and discusses the role they play in three areas: smooth ergodic theory, Teichm\"uller theory, and the spectral theory of one-frequency Schr\"odinger operators. The inspiration for this paper is the work of 2014 Fields Medalist Artur Avila, and his work in these areas is given special attention., Comment: 27 pages. To appear in the Bulletin of the AMS

Graph theory almost certainly began when, in 1735, Leonhard Euler solved a popular puzzle about bridges. The East Prussian city of Konigsberg (now Kalin- ingrad) occupies both banks of the River Pregel and an island, Kneiphof, which lies in the river at a point where it branches into two parts. There were seven bridges that spanned the various sections of the river, and the problem posed was this: could a person devise a path through Konigsberg so that one could cross each of the seven bridges only once and return home? Long thought to be impossible, the first mathematical demonstration of this was presented by Euler to the members of the Petersburg Academy on August 26, 1735, and written up the following year under the title "Solutio Problematis ad Geometriam Situs Pertinentis (The solution to a problem relating to the geometry of position)" (2) in the proceedings of the Petersburg Academy (the Commentarii). The title page of this volume appears on the cover of this issue of the Bulletin. This story is well-known, and the illustrations in Euler's paper are often repro- duced in popular books on mathematics and in textbooks. Sandifer in (6) claims flatly that "The Konigsberg Bridge Problem is Euler's most famous work," though scholars in other specialties (differential equations, complex analysis, calculus of variations, combinatorics, number theory, physics, naval architecture, music, . . . ) might disagree. N. L. Biggs, E. K. Lloyd, and R. J. Wilson in their history of graph theory (1) clearly view this paper of Euler's as seminal and remark: "The origins of graph theory are humble, even frivolous. Whereas many branches of mathematics were motivated by fundamental problems of calculation, motion, and measurement, the problems which led to the development of graph theory were often little more than puzzles, designed to test the ingenuity rather than to stimulate the imagina- tion. But despite the apparent triviality of such puzzles, they captured the interest of mathematicians, with the result that graph theory has become a subject rich in theoretical results of a surprising variety and depth." Euler provides only a neces- sary condition, not a sufficient condition, for solving the problem. But he does treat more than the original problem by beginning a generalization to two islands and four rivers, as is illustrated in the plate accompanying the original paper (Figure 1). In this renowned paper Euler does not get around to stating the problem until the second page. On the first he states the reason for being interested in the problem— it was an example of a class of problems he attributes to Leibniz as belonging to something Leibniz called "geometry of position." Euler says that "this branch is concerned only with the determination of position and its properties; it does not

This paper presents a review and summary of recent research on the boundary value problems for linear ordinary and partial differential equations, with special attention to the investigations of the current authors emphasizing the applications of complex symplectic spaces. In the first part of the previous century, Stone and von Neumann formulated the theory of self-adjoint extensions of symmetric linear operators on a Hilbert space; in this connection Stone developed the properties of self-adjoint differential operators generated by boundary value problems for linear ordinary differential equations. Later, in diverse papers, Glazman, Krein and Naimark introduced certain algebraic techniques for the treatment of appropriate generalized boundary conditions. During the past dozen years, in a number of monographs and memoirs, the current authors of this expository summary have developed an extensive algebraic structure, complex symplectic spaces, with applications to both ordinary and partial linear boundary value problems. As a consequence of the use of complex symplectic spaces, the results offer new insights into the theory and use of indefinite inner product spaces, particularly Krein spaces, from an algebraic viewpoint. For instance, detailed information is obtained concerning the separation and coupling of the boundary conditions at the endpoints of the intervals for ordinary differential operators (see the Balanced Intersection Principle), and the introduction of the generalized boundary conditions over the region for some elliptic partial differential operators (see the Harmonic operator).

This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and Swinnerton-Dyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank.

This paper gives a review of Cherednik’s results on the representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can be associated to every root system. As special cases, they include the Schur functions, the q q -Jacobi polynomials, and certain spherical functions on real and p p -adic symmetric spaces. They have a number of elegant combinatorial properties, which, however, are extremely difficult to prove. In this paper we show that a natural setup for studying these polynomials is provided by the representation theory of Hecke algebras and show how this can be used to prove some of the combinatorial identities for Macdonald polynomials.

The main definition. A parametrized curve γ : I → R is called nondegenerate if for any t ∈ I the vectors γ′(t), . . . , γ(t) are linearly independent. Analogously γ : I → S is called nondegenerate if for any t ∈ I the covariant derivatives γ′(t), . . . , γ(t) span the tangent hyperplane to S at the point γ(t) ( compare with the notion of n-freedom in [G]). Fixing an orientation in R or S we call a nondegenerate curve γ right-oriented if the orientations of γ′, . . . , γ coincide with the given one and left-oriented otherwise. Nondegenerate curves on S are closely related with linear ordinary differential equations of (n + 1)-th order. Such an equation defines two nondegenerate curves on S ⊂ V (n+1)∗, where V (n+1)∗ is the (n + 1)-dimensional vector space dual to the space of its solutions as follows. For each moment t ∈ I we choose the linear hyperplane in V n+1 of all solutions vanishing at t i.e. obtain a unique curve in the projective space P. Raising it to S we obtain a pair of curves; both of them are right-oriented if n is odd and have opposite orientations if n is even (nondegeneracy follows from nonvanishing of its Wronskian). A nondegenerate curve γ : [0, 1] → S defines a monodromy operator M ∈ GLn+1 which maps γ(0), γ ′(0), . . . , γ(0) to γ(1), γ′(1), . . . , γ(1). In the paper [K-O] there is given the complete set of invariants for symplectic leaves of the second Gelfand-Dikii bracket; namely its leaves are enumerated by pairs consisting of 1) monodromy operator, and 2) the connected component of the space of right-oriented curves in the sphere with the given monodromy operator. In this paper we study the number of connected components for closed nondegenerate right-oriented curves (corresponding to the identity monodromy operator). Nondegeneracy is also interesting in connection with the general theory of the hprinciple (see [G]). Let NR (NS) be the space of all nondegenerate closed right-oriented curves in R(S respectively). The question we consider is how to calculate πo(NS) and πo(NR). The first paper studying a similar question is [F]. Later J.Little [L1,L2] studied NS and NR and proved the following (W.Pohl’ conjecture): card(πo(NS)) = 3 and card(πo(NR)) = 2. (The invariant which distinguishes closed nondegenerate curves is an element of π1 of the image of the natural map ν : NR → SOn, where

We survey Mirzakhani's work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of non-experts.

In this paper we survey the major contributions of Abhyankar to the development of the theory of fundamental groups and Galois covers in positive characteristic. We first discuss the current status of four conjectures of Abhyankar about Galois covers in positive characteristic. Then we discuss research directions inspired by Abhyankar’s work, including many open problems.

In this paper we give a short introduction to the different theories of scissors congruence. We begin with classical scissors congruence, which considers equivalence classes of polyhedra under dissection. We then move to multi-dimensional scissors congruence along the lines of McMullen’s polytope algebra and then to the Grothendieck ring of varieties. Tying our discussion together is the question of whether algebraic invariants are sufficient to distinguish scissors congruence classes.

In this paper we take a first step toward understanding representations of cocompact lattices in SO(n,1) into arbitrary Lie groups by studying the deformations of rational representations — see Proposition 5.1 for a rather general existence result. This proposition has a number of algebraic applications. For example, we remark that such deformations show that the Margulis Super-Rigidity Theorem, see [30], cannot be extended to the rank 1 case. We remark also that if Γ ⊂ SO(n,1) is one of the standard arithmetic examples described in Section 7 then Γ has a faithful representation ρ′ in SO(n+1), the Galois conjugate of the uniformization representation, and Proposition 5.1 may be used to deform the direct sum of ρ′ and the trivial representation in SO(n+2) thereby constructing non-trivial families of irreducible orthogonal representations of Γ. However, most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain.

There exist real Banach spaces E such that the norm in E is of class C' away from zero; however, for any p, I < p < oo, the norm in the Lebesgue-Bochner function space LP(E, ,u) is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces. Introduction. The class of Lebesgue-Bochner function spaces, introduced by Bochner and Taylor [4] in 1938, has been found to be of considerable importance in various branches of mathematics, and is discussed at length in Dinculeanu [11], Dunford and Schwartz [12], and Edwards [13]. The study of the geometric properties of the Lebesgue-Bochner function spaces dates back about three decades: Day [8] and McShane [17], respectively, characterized uniform convexity and smoothness of these spaces. In fact, the only known result concerning the smoothness of the Lebesgue-Bochner function spaces is due to McShane, and his result concerns only the directional derivative (Gateaux derivative) of the norm in this class of Banach spaces. Even the Frechet differentiability of the norm has not been considered anywhere. It might be mentioned in this connection that the first systematic study of higher-order differentiability of the norm in a Banach space was made by Kurzweil [15] in 1954. Subsequently, in 1965, Bonic and Frampton [5a] extended Kurzweil's results, and in 1966 [5b] they discussed various categories of smooth Banach manifolds. In 1967, Sundaresan [18] extended some of Kurzweil's results independently. In [5b] and [18], the order of differentiability of the norm in the classical Lp spaces, 1 < p < oo, is obtained; while in Sundaresan [20], the smoothness of the norm in C(X, E) is discussed. For an elegant up-to-date account of smooth Banach spaces, and related concepts one might refer to the lecture notes by S. Yamamuro [23]. This paper contains the first systematic investigation of the higher-order differentiability of the norm function in the LebesgueBochner function spaces Received by the editors April 28, 1973. AMS (MOS) subject classifications (1970). Primary 46E40, 28A45; Secondary 58C20, 28A15.

was defined to be the least integer n for which there is an equivariant map X -+s n. We abbreviate this invariant to co-ind X. In this terminology the classical Borsuk theorem states that co-ind Sn = n. There are also numerous results (for references, see [2]) which among other things relate co-index to the homology of the quotient space X/T. The main purpose of the present note is the computation of the coindex in several examples in which homotopy, rather than homology, considerations are of primary importance. It should be mentioned that A. S. Svarc has also recently studied the application of homotopy theory to equivariant maps [5]; there is a considerable overlap between his work and our previous paper [2]. We consider as in our previous paper the space p(Sn) of paths on Sn which join a given point x to its antipode A(x) = -x together with the natural involution of p(Sn). It is shown that co-ind P(Sn) = n for n : 1, 2, 4 or 8. Next we consider the space V(Sn) of unit tangent vectors to sn, with its involution (the antipodal map on each fibre), and show that co-ind V(Sn) = n for n : 1, 3, or 7 and co-ind V(Sn) = n - 1 for n = 1, 3 or 7. We also compute the co-index of involutions on low dimensional projective spaces. The arguments rely on suspension and Hopf invariant theorems, using particularly the results of J. F. Adams [1] on maps of Hopf invariant one. 2. The space of paths P(Sn). We choose a base point x e Sn and we let P(Sn) denote the space of all paths in Sn which join x to its antipode - x. A fixed point free involution on P(Sn) is given by 17(p)(t) =-p(l - t), where p(t) is a point in P(S"). In this section we show

where 6 = z{d/dz) and where the at and ct are any constants, real or complex, the only restriction being that one of the ct must be equal to unity. Such solutions can be found in a number of places in the literature, f But in attempting to study the logarithmic cases of the problem treated in the above-mentioned paper, the author was unable to find the logarithmic solutions of equation (1) in the literature. I t is the purpose of this paper to present these logarithmic solutions, but for the sake of completeness, the non-logarithmic solutions are also given. The methods used are those of Frobenius.J

Let S be a skewfield. If J and / ' are submodules of 2 such that the nonzero elements of J are the inverse elements of those of J , then J and J' form a "pair of inverse modules." A module admitting an inverse module will be called a /-module and a selfinverse module containing 1 will be called an 5-module. In an earlier paper the author has shown that if S is a (commutative) field of characteristic not equal to 2, then every 5-module is a subfield of S. Only in fields of characteristic 2, nontrivial 5-modules can be found. A corresponding distinction of that characteristic does not hold for skewfields. Even the skewfield of the quaternions contains nontrivial 5-modules, for examples the module generated by 1, J, k. In the present paper some properties of 5-modules and /-modules will be discussed. For example it will be proved that when an 5-module contains the elements a, b and aby it contains all the elements of the skewfield which is generated by a and b. By a similar method it will be shown that finite 5-modules are necessarily Galois-fields.

The scope of the next paper has been explained in §6. As one of the later papers, it assumes a fair familiarity with the machinery of algebraic topology. Let G denote an associative H -space with unit (e.g. a topological group). We will show that the relations between G and a classifying space B G are more readily displayed using a geometric analog of the resolutions of homological algebra. The analogy is quite sharp, the stages of the resolution, whose base is B G , determine a filtration of B G . The resulting spectral sequence for cohomology is independent of the choice of the resolution, it converges to H * B G , and its E 2 term is Ext H ( G ) ( R, R ) ( R = ground ring). We thus obtain spectral sequences of the Eilenberg-Moore type in a simpler and more geometric manner. Geometric resolutions . We shall restrict ourselves to the category of compactly generated spaces. Such a space is Hausdorff and each subset which meets every compact set in a closed set is itself closed (a k -space in the terminology of Kelley). Subspaces are usually required to be closed, and to be deformation retracts of neighborhoods. Let G be an associative H -space with unit e . A right G -action on a space X will be a continuous map X × G → X with xe = x , x (g 1 g 2 ) = (xg 1 )g 2 for all x ∈A X , g 1 , g 2 ∈ G . A space X with a right G -action will be called a G -space.

The extant letters exchanged between Kervaire and Milnor during their collaboration from 1958–1961 concerned their work on the classification of exotic spheres, culminating in their 1963 Annals of Mathematics paper. Michel Kervaire died in 2007; for an account of his life, see the obituary by Shalom Eliahou, Pierre de la Harpe, Jean-Claude Hausmann, and Claude Weber in the September 2008 issue of the Notices of the American Mathematical Society. The letters were made public at the 2009 Kervaire Memorial Conference in Geneva. Their publication in this issue of the Bulletin of the American Mathematical Society is preceded by our commentary on these letters, providing some historical background.

1. Pontrjagin's maximum principle [6] is a basic result in deterministic optimal control theory. Analogous results have been obtained for the optimal control of stochastic dynamical systems (see for example the survey by Fleming [3]), and a new approach to such problems, using the martingale theory of Meyer, was made in the paper of Davis and Varaiya [2]. In this paper, by observing that the cost function is a 'semimartingale speciale' (see [5]), we are able to simplify much of [2] and obtain quickly a very general dynamic programming minimum principle.

Introduction. Let 12 C C" be a bounded pseudoconvex domain. Does U have a neighborhood basis consisting of pseudoconvex domains? It is well known that the answer to this question is, in general, "no". But it has been an open problem, at least since 1933 when the fundamental paper [1] of H. Behnke and P. Thullen appeared, whether the answer might be in the affirmative under the additional hypothesis of smoothness of the boundary b£2. The main purpose of this note is to give an example of a bounded pseudoconvex domain £lx C C 2 with smooth boundary that nevertheless does not have a Stein neighborhood basis. Additional hypotheses that guarantee the existence of such a basis are given in [3]. The constructed domain £2X has some more strange properties. In particular, the conjecture of R. 0 . Wells [5, Conjecture 3.1], does not hold for £lx (Theorem 2) and the boundary bQ,1 cannot be described by a smooth function with positive semidefinite Leviform on the whole C at each point p E b£2 r Together with the beautiful example of Kohn and Nirenberg [4] , this domain Q,j shows that bounded smooth domains in C can have quite different analytic properties than strictly pseudoconvex domains. The proofs of the theorems announced in this note will be contained in a later paper of the authors. Definition of £2X. We fix a smooth function X:R —• R + U {0} with the properties X(x) = 0 for x 0 for x > 0, and such that X is "sufficiently" convex. For r > 1 we define a family of smooth functions p : ( C { 0 } ) x C —• R by putting

Corresponding to a morphism f: V W of algebraic varieties (such that dim(V) S dim(W)), we construct a family of subschemes Siq)(f) C V. Wlhen V and W are nonsingular, the Siq), q > 1, induce a filtration of the set of closed points x E V such that the tangent space map dfx: T(V)x -+ T(W)f (x) has rank = dim(V) 1. We prove that if V is a suitably embedded nonsingular projective variety and ir: V -+ pm is a generic projection, then the SiO)(f) and certain fibre products of several of the SOq)(f) are either empty or smooth and of the smallest possible dimension, except in cases where q + 1 is divisible by the characteristic of the ground field. We apply this result to describe explicitly the ring homomorphisms lr*:pm ,(x) 0 VX and (when m > r + 1) to study the local structure of the image V' = r(V) c pm. 0. Introduction. Let k be an algebraically closed field, and let V be a nonsingular projective variety over k. As in [11], we consider an embedding V C P' (projective n-space) and a finite morphism 7r: V pm, where dim(V) dim(J), one objective is to describe the structure of the local rings 0v'y' where y GE V = 7r(V) C P'". (If (A, m) is a local ring, then A is its completion in the m-adic topology.) Using our main results, which are stated in ? 1, this can be done in the case that the map of Zariski tangent spaces d7rx: Tv x -+ Tpm y has rank > dim(V) 1 for all x E 7T-1(y). Details of this are given in ?13. If m = dim(V), the corresponding problem is to describe the local structure of the branch locus 7r(S1(7r)) C Ptm. (S1(1T) is defined in ?2.) I have obtained some results concerning this problem which will appear in another paper. (See also [14].) The results of the present paper were announced in [13]. I would Received by the editors July 25, 1974. AMS (MOS) subject classifications (1970). Primary 141305, 14E15, 14N05.

In a recently published paper J. C. C. McKinsey has pointed out some difficulties which arise from Axiom I of my theory of probability implication. This axiom states the unambiguity of the degree p of a given probability implication (03VP) for the case that the class 0 is not empty, a condition formulated by (o), but postulates ambiguity of p in case of an empty class 0, this condition being formulated by (0). The latter ambiguity is necessary for probability implication because of the relation to Russell's material implication. From the proof published by McKinsey we can infer that this ambiguity has to be restricted to values of p between 0 and 1, limits included, in correspondence with the same restriction holding for the unambiguous degree p of probability in cases of a non-empty class 0, formulated by me in (8, §13). That this general restriction is derivable from Axiom II, 2 is obvious as this axiom contains 0and p as free variables and therefore states the restriction for all classes 0 and all values p. A further objection, which was already indicated in a footnote of McKinsey's paper, has been presented to me in a letter by the referee of this journal, Mr. S. C. Kleene. This objection shows that if the ambiguity of degrees of probability for empty classes 0 is assumed, it can be proved that this ambiguity cannot be restricted to the limits O t o l . This proof is connected with the theorem of addition (Axiom III) which reads III. (03PP).(03qQ).(0.PDQ)3(Br)(03rPVQ)-(r=p+q).

The purpose of this paper is to give full descriptions of the examples announced in [8]. Let n be an integer, n > 2, and p a prime. Let AP denote the p-adic group (see below). Our principal result is the following. Example. There exists a compact metric space XI of dimension n and an action of AP on XI such that dim XI/Ap = n + 2. Such examples are interesting in relation to the important conjecture: If G is a compact group which acts effectively on a (generalized) manifold X, then G is a Lie group. This conjecture is proved for dim X ? 2 in [3] with generalizations given in [1], [9] and [10]. If this conjecture is false, then [4] there must be an example in which G = AP, and in this case [12], [2] dim X/Ap = n + 2. In this sense, the examples discussed here represent first approximations to counter-examples to the conjecture. The approximation is poor inasmuch as the spaces X" contain "large" n-dimensional sets; e.g., the n-skeleton of a triangulated (n+ 2)-manifold. Nor do we know of examples of AP action in which the dimension goes up by 3. It is known [12], [2] that the cohomology dimension cannot go up by more than 3. The authors expect to have more to say on these subjects. The reader might find the example of Kolmogoroff3 [5] a good introduction to the present paper. Kolmogoroff constructs a light open map which raises dimension by one; although not noted in [5], this map is the projection of an action of A2. One can construct analogous examples for all primes p, which as in [5], are based on the non-dimensional-full-valued spaces of Pontrjagin [7]. (A description of the Kolmogoroff example will appear in [11] as an illustration of the "A functor" of ? 2.) The additional complication in the examples described below stems from the following: If the dimension of a compact metric space X is to be decreased by 1 upon the removal of a subset D, D can be chosen to be totally disconnected. In contrast, to achieve a decrease of 2, D must have large components. Thus the individual modifications used in [7] are

Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced by Fomin and Zelevinsky in 2000 in the context of Lie theory, but have since appeared in many other contexts, from Poisson geometry to triangulations of surfaces and Teichmuller theory. In this expository paper we give an introduction to cluster algebras, and illustrate how this framework naturally arises in Teichmuller theory. We then sketch how the theory of cluster algebras led to a proof of the Zamolodchikov periodicity conjecture in mathematical physics.

Probably the most famous of Grothendieck’s contributions to Banach space theory is the result that he himself described as “the fundamental theorem in the metric theory of tensor products”. That is now commonly referred to as “Grothendieck’s theorem” (GT in short), or sometimes as “Grothendieck’s inequality”. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C ¤ -algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of “operator spaces” or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell’s inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by “semidefinite programming’ and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author’s 1986 CBMS notes.

Hillel Furstenberg’s 1981 article in the Bulletin gives an elegant introduction to the interplay between dynamics and number theory, summarizing the major developments that occurred in the few years after his landmark paper [21]. The field has evolved over the past thirty years, with major advances on the structural analysis of dynamical systems and new results in combinatorics and number theory. Furstenberg’s article continues to be a beautiful introduction to the subject, drawing together ideas from seemingly distant fields. Furstenberg’s article [21] gave a general correspondence between regularity properties of subsets of the integers and recurrence properties in dynamical systems, now dubbed the Furstenberg Correspondence Principle. He then showed that such recurrence properties always hold, proving what is now referred to as the Multiple Recurrence Theorem. Combined, these results gave a new proof of Szemeredi’s Theorem [45]: if S ⊂ Z has positive upper density, then S contains arbitrarily long arithmetic progressions. This proof lead to an explosion of activity in ergodic theory and topological dynamics, beginning with new proofs of classic results of Ramsey Theory and ultimately leading to significant new combinatorial and number theoretic results. The full implications of these connections have yet to be understood. The approach harks back to the earliest results on recurrence, in the measurable setting and in the topological setting. A measure preserving system is a quadruple (X,B, μ, T ), where X denotes a set, B is a σalgebra on X, μ is a probability measure on (X,B), and T : X → X is a measurable transformation such that μ(T−1(A)) = μ(A) for all A ∈ B. Poincare Recurrence states that if (X,B, μ, T ) is a measure preserving system and A ∈ B with μ(A) > 0, there exists n ∈ N such that μ(A ∩ T−nA) > 0. A (topological) dynamical system is a pair (X,T ), where X is a compact metric space and T : X → X is a continuous map. One can show that any such topological space admits a Borel, probability measure that preserves T . In particular, Poincare Recurrence implies recurrence in the topological setting. Birkhoff [13] gave a direct proof of

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? No, according to counterexamples by K. Kuperberg and others. On the other hand there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes’s proof, and the bigger picture into which it fits. Taubes’s proof of the Weinstein conjecture is the culmination of a large body of work, both by Taubes and by others. In an attempt to make this story accessible to nonspecialists, much of the present article is devoted to background and context, and Taubes’s proof itself is only partially explained. Hopefully this article will help prepare the reader to learn the full story from Taubes’s paper [62]. More exposition of this subject (which was invaluable in the preparation of this article) can be found in the online video archive from the June 2008 MSRI hot topics workshop [44], and in the article by Auroux [5]. Below, in §1–§3 we introduce the statement of the Weinstein conjecture and discuss some examples. In §4–§6 we discuss a natural strategy for approaching the Weinstein conjecture, which proves it in many but not all cases, and provides background for Taubes’s work. In §7 we give an overview of the big picture surrounding Taubes’s proof of the Weinstein conjecture. Readers who already have some familiarity with the Weinstein conjecture may wish to start here. In §8–§9 we recall necessary material from Seiberg-Witten theory. In §10 we give an outline of Taubes’s proof, and in §11 we explain some more details of it. To conclude, in §12 we discuss some further results and open problems related to the Weinstein conjecture. 1. Statement of the Weinstein conjecture The Weinstein conjecture asserts that certain vector fields must have closed orbits. Before stating the conjecture at the end of this section, we first outline its origins. This is discussion is only semi-historical, because only a sample of the relevant works will be cited, and not always in chronological order. 1.1. Closed orbits of vector fields. Let Y be a closed manifold (in this article all manifolds and all objects defined on them are smooth unless otherwise stated), 2000 Mathematics Subject Classification. 57R17,57R57,53D40. Partially supported by NSF grant DMS-0806037.

A classical problem in algebraic geometry is to describe quantities that are invariants under birational equivalence as well as to determine some convenient birational model for each given variety, a minimal model. One such quantity is the ring of objects which transform like a tensor power of a differential of top degree, known as the canonical ring. The histories of the existence of minimal models and the finite generation of the canonical ring are intertwined; minimal models and canonical rings constitute the major building blocks for the birational classification of algebraic varieties. In this paper we will discuss some of the ideas involved, recent advances on the existence of minimal models, some applications, and the (algebraic-geometric proof of the) finite generation of the canonical ring. These results have been long standing conjectures in algebraic geometry.

This document is intended as a (slightly expanded) writeup of my (anticipated) talk at the AMS Current Events Bulletin in New Orleans, January 2007. It is a brief report on the work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture [5]. It is not intended in any sense for specialists and is, indeed, aimed at readers without any specific background either in measure theory, dynamics or number theory. Any reader with any background in ergodic theory will be better served by consulting either the original paper, or one of the surveys written by those authors: see [7] and [12].

Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

Euler’s work on elliptic integrals is a milestone in the history of algebraic geometry. The founders of calculus understood that some algebraic functions could be integrated using elementary functions (logarithms and inverse trigonometric functions). Euler realized that integrating other algebraic functions leads to genuinely different functions, elliptic integrals. These functions are not something ugly. As Abel discovered, their inverses are doubly periodic functions on the complex plane. What we now call elliptic curves (algebraic curves of genus 1) take their name from elliptic integrals. Although these curves had been studied earlier, indeed in great depth by Fermat, it is Euler’s analysis that clarifies the key points: elliptic curves are fundamentally different from rational curves, and not only in a negative way. They have a richer symmetry, the famous group structure possessed by an elliptic curve. This paper considers two main themes in algebraic geometry descended from Euler’s work: integrals of algebraic functions (in fancier terms, Hodge theory) and birational geometry. In section 1, we reach a major open problem of algebraic geometry: which representations of the fundamental group are summands of the cohomology of some family of algebraic varieties? Or, equivalently: which linear differential equations can be solved by integrals of algebraic functions? One might not expect any good answer to these questions, but in fact there are two promising approaches (the Simpson and Bombieri-Dwork conjectures). Section 2, more elementary, gives an introduction to birational geometry. I hope to explain the significance of the problem of finite generation of the canonical ring, which has just been solved. Thanks to Carlos Simpson for his comments on an earlier version.

In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.

By the early 1950’s algebraic topology had reached great heights with Serre’s thesis and the calculations in the seminar of Henri Cartan of the cohomology of spaces with one nonzero homotopy group in terms of Steenrod operations. There was also the appearance of the new characteristic classes of vector bundles, Pontryagin classes (Z coefficients) and Chern-Weil classes (coefficients in R) joining those of Stiefel Whitney (Z mod 2 coefficients). Rene Thom absorbed all this structure, made vigorous use of it, and added a geometric perspective that combined to revolutionize topology, manifold theory, and algebraic geometry. For the unexpected and fertile results in bordism (closed manifolds mod boundaries of manifolds) of the 1954 paper [1], Thom received the Fields Medal at Edinburgh in 1958. Many more applications of Thom’s ideas came even later.

Since the nineteenth century, when Boltzmann formalized the concepts of kinetic equations, their range of application has been considerably extended. First introduced as a means to unify various perspectives on fluid mechanics, they are now used in plasma physics, semiconductor technology, astrophysics, biology.... They all are characterized by a density function that satisfies a Partial Differential Equation in the phase space. This paper presents some of the simplest tools that have been devised to study more elaborate (coupled and nonlinear) problems. These tools are basic estimates for the linear first order kinetic-transport equation. Dispersive effects allow us to gain time decay, or space-timeLpL^pintegrability, thanks to Strichartz-type inequalities. Moment lemmas gain better velocity integrability, and macroscopic controls transform them into spaceLpL^pintegrability for velocity integrals. These tools have been used to study several nonlinear problems. Among them we mention for example the Vlasov equations for mean field limits, the Boltzmann equation for collisional dilute flows, and the scattering equation with applications to cell motion (chemotaxis). One of the early successes of kinetic theory has been to recover macroscopic equations from microscopic descriptions and thus to be able theoretically to compute transport coefficients. We also present several examples of the hydrodynamic limits, the diffusion limits and especially the recent derivation of the Navier-Stokes system from the Boltzmann equation, and the theory of strong field limits.

This paper is a handyman’s manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups.