Definable equivalence relations and zeta functions of groups

We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math. So

Uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sums

This thesis contains some new results on the uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sumsComment: Doctoral thesis, University of Lill

Valued fields, Metastable groups

We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort $\Gamma$) if every type over a sufficiently rich base structure can be viewed as part of a $\Gamma$-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from $\Gamma$, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF.Comment: 48 pages. Minor corrections and improvements following a referee repor

A history of Galois fields

This paper stresses a specific line of development of the notion of finite field, from Évariste Galois’s 1830 “Note sur la théorie des nombres,” and Camille Jordan’s 1870 Traité des substitutions et des équations algébriques, to Leonard Dickson’s 1901 Linear groups with an exposition of the Galois theory. This line of development highlights the key role played by some specific algebraic procedures. These intrinsically interlaced the indexations provided by Galois’s number-theoretic imaginaries with decompositions of the analytic representations of linear substitutions. Moreover, these procedures shed light on a key aspect of Galois’s works that had received little attention until now. The methodology of the present paper is based on investigations of intertextual references for identifying some specific collective dimensions of mathematics. We shall take as a starting point a coherent network of texts that were published mostly in France and in the U.S.A. from 1893 to 1907 (the “Galois fields network,” for short). The main shared references in this corpus were some texts published in France over the course of the 19th century, especially by Galois, Hermite, Mathieu, Serret, and Jordan. The issue of the collective dimensions underlying this network is thus especially intriguing. Indeed, the historiography of algebra has often put to the fore some specific approaches developed in Germany, with little attention to works published in France. Moreover, the “German abstract algebra” has been considered to have strongly influenced the development of the American mathematical community. Actually, this influence has precisely been illustrated by the example of Elliakim Hasting Moore’s lecture on “abstract Galois fields” at the Chicago congress in 1893. To be sure, this intriguing situation raises some issues of circulations of knowledge from Paris to Chicago. It also calls for reflection on the articulations between the individual and the collective dimensions of mathematics. Such articulations have often been analysed by appealing to categories such as nations, disciplines, or institutions (e.g., the “German algebra,” the “Chicago algebraic research school”). Yet, we shall see that these categories fail to characterize an important specific approach to Galois fields. The coherence of the Galois fields network had underlying it some collective interest for “linear groups in Galois fields.” Yet, the latter designation was less pointing to a theory, or a discipline, revolving around a specific object, i.e. Gln(Fpn) (p a prime number), than to some specific procedures. In modern parlance, general linear groups in Galois fields were introduced in this context as the maximal group in which an elementary abelian group (i.e., the multiplicative group of a Galois field) is a normal subgroup. The Galois fields network was actually rooted on a specific algebraic culture that had developed over the course of the 19th century. We shall see that this shared culture resulted from the circulation of some specific algebraic procedures of decompositions of polynomial representations of substitutions

Definable equivalence relations and zeta functions of groups

The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald.Peer reviewedPostprin

The phenomenological and discursive practice of place in lifestyle migration: a case study of Stanthorpe, Queensland

Lifestyle migration, popularly known as ‘tree-’ or ‘sea-change’ in Australia, is a phenomenon increasingly depicted in various media including films, television shows, books, blogs and magazines. Much of the previous research on lifestyle migration has been anthropological and sociological in approach, and has not fully examined the important links to the media so prevalent in late modernity. In contrast, this research is positioned within cultural studies and employs a cultural, materialist and phenomenological approach. Using three methods, textual analysis, interviews and researcher reflections, it examines how selected texts depict lifestyle migration, influence the lived experience, and impact identity. To pursue these foci, textual and discursive analyses of recent lifestyle migration media were conducted. These analyses were combined with semi-structured interviews with 12 lifestyle migrants—using Stanthorpe, Queensland as a case study—and personal reflections of the researcher, also a lifestyle migrant, to offer a fuller, hermeneutical analysis. The central argument of this thesis is that texts create and are created by imagined worlds which influence people to make life decisions that then impact their necessarily emplaced identity. I argue that employing a cultural studies orientation to the field of lifestyle migration, and engaging methods such as discourse analysis, interviews and personal reflections, produces new understandings that reflect the growing importance of cultural texts on the decisions made in our day-to-day lives. These understandings would not be possible using one method alone. This approach both extends and deepens the field of lifestyle migration research and situates it within existing social imaginaries and popular discourse central to the late modern experience

Still got the blues - no time but here, no place but now: geographies of art & activism in the work of Clyde Woods.

The fight against displacement and dispossession in the increasingly inequitable process of reterritorialization is arguably the most pressing struggle of the modern city – and it is articulated through cultural production as much as through political debate and economic development. Yet research that is at the nexus of critical, urban, and cultural studies and which explores the resistant potential of countercultural production rather than the hegemonic impact of mainstream cultural production is underdeveloped. Clyde Woods’s blues epistemology offers a guide to reading cultural production of marginalized groups as maps of meaning that offer alternative visions of the past, the present, and the future. However, research that substantially leverages the blues epistemology has been limited, is rarely conducted within the field of urban studies, and tends to approach Woods’s multifaceted blues in a piecemeal fashion. This dissertation explores extant blues epistemic research as critical, urban, cultural studies and it asks what methodological themes are necessary for nuanced, blues-based social investigation, how we might perform that investigation with a consistent emphasis on radical change, and how that call for change might be refocused by putting Clyde Woods’s socio-spatial sensibilities directly into conversation with Herbert Marcuse’s consistently utopic thinking. The resulting papers reposition Woods’s work to a broader audience, revitalizing and operationalizing blues epistemologies as they amplify the (harmonic and melodic) voices of marginalized groups and illustrate how their cultural production has grappled with inequality and has sought to keep that struggle at the forefront of the urban imagination