Problems on rigidity of group actions and cocycles

The main topic of discussion in this survey is the geometric and metric properties of geodesic and horocycle flows on surfaces of constant curvature and their multidimensional analogues, the so-called G-induced actions (that is, restrictions to a suitable subgroup F of the natural action of G on a homogeneous space of it). One usually takes as F one-parameter or cyclic subgroups of G and the horospherical subgroups corresponding to them (the details are given in Sect. 1). However, in the course of the exposition of our main topic we sometimes mention results relating to the case when F is an n-parameter subgroup of G or a lattice. This class of dynamical systems consists precisely of group actions admitting an extension to a transitive action of a Lie group. We shall not be considering in our survey (or, at any rate, not systematically) the following themes which are close to our main subject matter: Topological dynamics and metric properties of general group actions (in particular, the work of Zimmer on semisimple group actions (Zimmer 1984); Group automorphisms; Affine transformations of homogeneous spaces; G-induced flows on spaces with an infinite invariant measure. The papers (Hopf 1939), (Sullivan 1981) and the recent book (Nicholls 1989) are devoted to the connections between G-induced flows (and also geodesic flows) and the theory of discrete groups of motions of a hyperbolic space; The actions of subgroups on the double cosets related to geodesic (for n > 2) and horocycle (for n > 3) flows on an n-dimensional manifold of constant negative curvature (see (Gel’fand and Fomin 1952) and, in particular, (Mautner 1957)). Flows of this type in somewhat more general form differing from geodesic and horocycle flows have not been investigated.

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large. The second part of the book first outlines various contributions to linear algebraic groups in the 19th century, due mainly to E. Study, E. Picard, and above all, L. Maurer. After being abandoned for nearly fifty years, the theory was revived by C. Chevalley and E. Kolchin, and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on the work of Chevalley on Lie groups and Lie algebras and of Kolchin on algebraic groups and the Galois theory of differential fields, which put their contributions to algebraic groups in a broader context. Professor Borel brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep understanding of the underlying mathematics. His lifelong involvement and his historical research in the subject area give him a special appreciation of the story of its development.

We establish analogs of the three Bieberbach theorems for a lattice \(\Gamma\) in a semidirect product \(\mathsf{K}\rtimes\mathsf{K}\) where \(\mathsf{S}\) is a connected, simply connected solvable Lie group and \(\mathsf{K}\) is a compact subgroup of its automorphism group. We first prove that the action of \(\Gamma\) on \(\mathsf{S}\) is metrically equivalent to an action of \(\Gamma\) on a supersolvable Lie group. The latter is shown to be determined by \(\Gamma\) itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group \(\Gamma\) as a lattice in a semidirect product \(\mathsf{S}\rtimes\mathsf{F}\) with \(\mathsf{F}\) being a finite group whose order is bounded by a constant only depending on the dimension of \(\mathsf{S}\). This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup \(\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}\) that the quotient \(\mathsf{S}/\Upsilon\) is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.

Weights for symmetric and general linear groups

Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

Abstract

Number-theoretic techniques in the theory of Lie groups and differential geometry

A systematic method is presented whereby any compact Lie group of n-real parameters is dealt with from an infinitesimal approach with the representative matrix method based on a group of inner automorphisms suggested in a previous paper. The group manifold, defined in terms of a metric of group parameters, is identified as a Riemannian one in which these parameters play a role of n curvilinear coordinates. Riemannian geometry is thus valid in the group manifold, and geometric quantities are explicitly calculated in terms of the symmetric or (0) connection by a straightforward application of the ordinary procedure of tensor analysis. A new and simpler method of computing the invariant volume element is presented within this framework. Furthermore, we discuss in detail the group of inner automorphisms for the calculation of the matrix element of finite rotations for any irreducible representation (abbreviated MEFRIR). It is found that our method works very well and yields right and left vector fields together with a set of 2n equations to be satisfied by the MEFRIR. The global properties of the group may, therefore, be obtained as a solution to these equations. Moreover, it provides not only the generalized Maurer-Cartan equations, the Lie structure formulas, two parameter groups of point transformations and the adjoint group, but also two additional nonsymmetric (+) and (−) connections with zero curvature, which do not possess any preassigned metric but possess two absolute parallelisms. Thus, our results on differential geometry completely agree with Cartan and Schouten's. A link between differential geometry and representations is presented by the right and left vector fields which are explicitly calculable in terms of the n parameters and through which geometric quantities, e.g., the Riemann tensor, the Ricci tensor, and the scalar curvature, of any connection are explicitly displayed. A theorem relating both the vector fields to the metric tensors is also included. Finally, the l (rank of the group) invariant differential equations to be satisfied by the MEFRIR are cast in the covariant (or Lie derivative) forms in any connection. Examples of the invariant equations are given for SU(2), SO(3), and SU(3). The two invariant equations of the latter can be cast in terms of the eigenvalues of isospin and hypercharge upon carrying out charge and hypercharge quantizations; in this connection, a new nonrelativistic wave equation to be satisfied by SU(3) multiplets as a generalization of the Schrödinger equation of the symmetric top is also proposed.

A line transversal of a family S of n pairwise disjoint convex objects is a straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely, that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in Rd can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n > 2 pairwise-disjoint convex objects that are induced by lines passing through any fixed point is between K (n − 1, d − 1) and K (n, d − 1), where K (n, d) =d i=0 ( n−1 i ) = (nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d − 1)-spheres in a d-sphere. By a similar argument, we show that the maximum number of connected components of the space of all line transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is K (n, d − 1). Finally, we refute a conjecture of Sharir and Smorodinsky on the number of neighbor pairs in geometric permutations and offer an alternative conjecture which may be a first step towards solving the aforementioned general problem of bounding the number of geometric permutations. ∗ Work on this paper has been partially supported by a grant from the U.S.–Israeli Binational Science Foundation. The first author’s work on this paper has also been supported by NSF Grant CCR-9972568 and NSF ITR Grant CCR-0081964. Part of the work was performed while he was visiting the Mathematical Sciences Research Institute, Berkeley, California. Part of the work on this paper was performed while the second author was a Ph.D student at Tel-Aviv University, under the supervision of Micha Sharir, and later a postdoctoral fellow at the Mathematical Sciences Research Institute, Berkeley, California. 286 B. Aronov and S. Smorodinsky

Participants at an October workshop at the Mathematical Sciences Research Institute in Berkeley, California, learned that a team of mathematicians has proved a deep similarity between a simple form of solitaire and a mathematical tool called random matrices, originally developed to understand the quantum behavior of large atoms. The implications could go well beyond card games to some of the most puzzling patterns in mathematics. Other recent work suggests that the same random matrix key might unlock the most important problem in number theory: proving the Riemann hypothesis, which describes how prime numbers are distributed among other integers.

Let u be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let −λ2 be the corresponding eigenvalue. We consider the problem of estimating max M u in terms of λ, for large λ, assuming ∫ M u 2 = 1. We prove that max M u ≤C M λ(n − 1)/2, which is optimal for some M. Our proof simplifies some of the arguments used before for such problems. We review the ‘wave equation method’ and discuss some special cases which may be handled by more direct methods. *This work was partially supported by NSF grant DMS-9306389 and by the Mathematical Sciences Research Institute, Berkeley.

In 2002, an introductory workshop was held at the Mathematical Sciences Research Institute in Berkeley to survey some of the many directions of the commutative algebra field. Six principal speakers each gave three lectures, accompanied by a help session, describing the interaction of commutative algebra with other areas of mathematics for a broad audience of graduate students and researchers. This book is based on those lectures, together with papers from contributing researchers. David Benson and Srikanth Iyengar present an introduction to the uses and concepts of commutative algebra in the cohomology of groups. Mark Haiman considers the commutative algebra of n points in the plane. Ezra Miller presents an introduction to the Hilbert scheme of points to complement Professor Haiman's paper. Further contributors include David Eisenbud and Jessica Sidman; Melvin Hochster; Graham Leuschke; Rob Lazarsfeld and Manuel Blickle; Bernard Teissier; and Ana Bravo.

This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry. David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.

In the 2012–13 academic year, the Mathematical Sciences Research Institute, Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more, the boundary between them has become increasingly blurred. This was apparent during the MSRI program, where there were a number of joint seminars on subjects of common interest: birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, and tilting theory, to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures, along with expository articles and research papers by participants of the programs. Volume 2 focuses on the most recent research.

Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural non-commutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subellilptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.