Roy Wagner

Tail estimates for sums of variables sampled by a random walk

We prove tail estimates for variables of the form ∑if(Xi), where (Xi)i is a sequence of states drawn from a reversible Markov chain, or, equivalently, from a random walk on an undirected graph. The estimates are in terms of the range of the function f, its variance, and the spectrum of the graph. The purpose of our estimates is to determine the number of chain/walk samples which are required for approximating the expectation of a distribution on vertices of a graph, especially an expander. The estimates must therefore provide information for fixed number of samples (as in Gillmans [4]) rather than just asymptotic information. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein-and Bennett-type inequalities, as well as an inequality for sub-Gaussian variables.

Finite high-order games and an inductive approach towards Gowers's dichotomy

Abstract We present the notion of finite high-order Gowers games, and prove a statement parallel to Gowerss Combinatorial Lemma for these games. We derive ‘quantitative’ versions of the original Gowers Combinatorial Lemma and of Gowerss Dichotomy, which place them in the context of the recently introduced infinite dimensional asymptotic theory for Banach spaces.

Gowers’ dichotomy for asymptotic structure

In this paper Gowers’ dichotomy is extended to the context of weaker forms of unconditionality, most notably asymptotic unconditionality. A general dichotomic principle is demonstrated; a Banach space has either a subspace with some unconditionality property, or a subspace with a corresponding ‘proximity of subspaces’ property. 0. Notation In this paper, unless stated otherwise, all spaces will be infinite dimensional. Once we have a basis {xi}i=1 in a space X , we define a finite support vector to be a vector of the form ∑m i=n aixi. The range of a finite support vector x will be the smallest interval [n,m] such that x may be written as ∑m i=n aixi. We write x < y for two finite support vectors x and y if the range of x ends before the range of y begins (i.e. supp(x) = [n1,m1], supp(y) = [n2,m2] and m1 < n2). We say that the vectors {yi}i=1 are consecutive if y1 < y2 < · · · < yk. A block subspace of X (with respect to a basis) is a subspace generated by a basic sequence of consecutive finite support vectors. Finally, define an H.I. (Hereditarily Indecomposable) space to be a Banach space, in which two infinite dimensional subspaces have zero angle between them (i.e. for every ε > 0 and for all Y, Z ⊂ X , subspaces, there are vectors y ∈ Y , z ∈ Z, ‖y‖ = ‖z‖ = 1, ‖y − z‖ < ε). This also means that the span of any two disjoint infinite dimensional closed subspaces is not a closed subspace of X . The existence of such a space was recently proved in [GM].

On the extremality of Hofer's metric on the group of Hamiltonian diffeomorphisms

Let M be a closed symplectic manifold, and let | | be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if | | is dominated from above by the L-Infinity-norm, and | | is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if | | is, additionally, not equivalent to the L-Infinity-norm, then the induced Finsler metric on the group of Hamiltonian diffeomorphisms on M vanishes identically.

Mathematical Variables as Indigenous Concepts

This paper explores the semiotic status of algebraic variables. To do that we build on a structuralist and post‐structuralist train of thought going from Mauss and Lévi‐Strauss to Baudrillard and Derrida. We import these authors’ semiotic thinking from the register of indigenous concepts (such as mana), and apply it to the register of algebra via a concrete case study of generating functions. The purpose of this experiment is to provide a philosophical language that can explore the openness of mathematical signs to reinterpretation, and bridge some barriers between philosophy of mathematics and critical approaches to knowledge.

Some Remarks on a Lemma of Ran Raz

In this note we will review a Lemma published by Ran Raz in [R], and suggest improvements and extensions. Raz’ Lemma compares the measure of a set on the sphere to the measure of its section with a random subspace. Essentially, it is a sampling argument. It shows that, in some sense, we can simultaneously sample a function on the entire sphere and in a random subspace.

For Some Histories of Greek Mathematics

This paper argues for the viability of a different philosophical point of view concerning classical Greek geometry. It reviews Reviel Netzs interpretation of classical Greek geometry and offers a Deleuzian, post-structural alternative. Deleuzes notion of haptic vision is imported from its art history context to propose an analysis of Greek geometric practices that serves as counterpoint to their linear modular cognitive narration by Netz. Our interpretation highlights the relation between embodied practices, noisy material constraints, and operational codes. Furthermore, it sheds some new light on the distinctness and clarity of Greek mathematical conceptual divisions.

A Historically and Philosophically Informed Approach to Mathematical Metaphors

This article discusses the concept of mathematical metaphor as a tool for analyzing the formation of mathematical knowledge. It reflects on the work of Lakoff and Núñez as a reference point against which to rearticulate a richer notion of mathematical metaphor that can account for actual mathematical evolution. To reach its goal this article analyzes historical case studies, draws on cognitive research, and applies lessons from the history of metaphors in philosophy as analyzed by Derrida and de Man.

Wronski’s Infinities

This article interprets Józef Maria Hoëné Wronski’s (1776–1853) use of actual infinities in his mathematical work. The interpretation places this usage, which undermined Wronski’s acceptance as a mathematician, in his contemporary mathematical and philosophical context and in the context of his own sociopolitical-philosophical project.

Funeral Rites, Queer Politics

This paper builds on Jean Genet’s funeral rites and on psychoanalytic theory to construct a model for a psycho-political technique. The paper then confronts the model with contemporary political struggles in the contexts of the Israeli-Palestinian conflict and of AIDS. We find that the technique we articulate is not only identity shattering (hence queer), but also contingent, temporary and unstable. Moreover, this psycho-political technique operates below micro-politics: it depends on distributed, fissured subjectivities; it therefore cannot be adapted to a political discourse where fully fledged subject positions are supposed. In some sense, we admit, this unstable ‘nano-politics’ is bound to eventually fail. We then discuss what might be the telos of, and motivation to engage in, such queer, pre-subjective, psycho-political technique. Executioner adoration syndrome A well-known Jewish hereditary disease Don’t get excited. Genet’s Funeral Rites was published 52 years ago. It is a sinister and bitter book. But since sinister and bitter days are here now, it is fitting, so it seems, to consider it still. The novel’s bitterness is inevitable. Jean, the narrator, has lost his lover, resistance hero Jean D., in the final days of the liberation of Paris from Nazi occupation. How can one not be left with a bitter taste having devoured him — the dearest and only lover who ever loved me? Indeed, as Jean assures us, to eat a youngster shot on the barricades, to devour a young hero, is no easy thing (17–18). But before we conclude our grief by feasting on the dead, I would like to discuss what it is that may happen between the time a love is lost, and the time when we will have incorporated it into our minds and flesh. What happens in Genet’s Funeral Rites between loss of love and incorporation of the dead happens across an elaborate complex of identifications, I thank Yehonatan Alshekh, Dr. Lyat Friedman, and Prof. Adi Ophir for their comments on earlier drafts of this paper. Talkback no. 121 to Yitzhak Frankenthal, “Arafat, get well soon”, Ynet, October 30, 2004, http://www.ynet.co.il/articles/0,7340,L-2997192,00.html (Hebrew). Page numbers in parentheses refer to Jean Genet, Funeral Rites (Bernard Frechtman, trans., Grove Press, 1969). I apply the convention of using boldface to mark quotations. Italics in quotations are always in the original.