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Rectifiable Sets, Densities and Tangent Measures
The characterization of rectifiable sets through the existence of densities is a pearl of geometric measure theory. The difficult proof, due to Preiss, relies on many beautiful and deep ideas and novel techniques. Some of them have already proven useful in other contexts, whereas others have not yet been exploited. These notes give a simple and short presentation of the former and provide some perspective of the latter. This text emerged from a course on rectifiability given at the University of Zurich. It is addressed both to researchers and students; the only prerequisite is a solid knowledge in standard measure theory. The first four chapters give an introduction to rectifiable sets and measures in Euclidean spaces, covering classical topics such as the area formula, the theorem of Marstrand and the most elementary rectifiability criterions. The fifth chapter is dedicated to a subtle rectifiability criterion due to Marstrand and generalized by Mattila, and the last three focus on Preiss' result. The aim is to provide a self-contained reference for anyone interested in an overview of this fascinating topic.
$Q$-Valued Functions Revisited
In this memoir the authors revisit Almgren's theory of $Q$-valued functions, which are functions taking values in the space $\mathcal{A}_Q(\mathbb{R}^{n})$ of unordered $Q$-tuples of points in $\mathbb{R}^{n}$. In particular, the authors: give shorter versions of Almgren's proofs of the existence of $\mathrm{Dir}$-minimizing $Q$-valued functions, of their Holder regularity, and of the dimension estimate of their singular set; propose an alternative, intrinsic approach to these results, not relying on Almgren's biLipschitz embedding $\xi: \mathcal{A}_Q(\mathbb{R}^{n})\to\mathbb{R}^{N(Q,n)}$; improve upon the estimate of the singular set of planar $\mathrm{D}$-minimizing functions by showing that it consists of isolated points.
Harmonic Analysis and Applications
The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.
Conversations on Optimal Transport
This work is closely tied to the renowned mathematics textbook series known as UNITEXT, tailored for university students pursuing bachelors or masters degrees. What sets this particular book apart in the Springer collection is its unique origin: it has been crafted through a meticulous process involving interviews handled with and by world-class mathematicians. The content featured in this book revolve around a highly relevant and engaging topic: Optimal Transport. These conversations involve not only authors from the UNITEXT series, but also members of the series Editorial Board. Additionally, they feature prominent figures in the field, including a Field Medalist. This work provides reader...
Transport Equations and Multi-D Hyperbolic Conservation Laws
The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements. This volume provides an up-to-date overview of the status and perspectives of two areas of research in PDEs, related to hyperbolic conservation laws. The captivating volume contains surveys of recent deep results and provides an overview of further developments and related open problems. Readers should have basic knowledge of PDE and measure theory.
Emancipatory Narratives & Enslaved Motherhood
Emancipatory Narratives & Enslaved Motherhood examines three major currents in the historiography of Brazilian slavery: manumission, miscegenation, and creolisation. It revisits themes central to the history of slavery and race relations in Brazil, updates the research about them, and revises interpretations of the role of gender and reproduction within them. First, about the preponderance of women and children in manumission; second, about the association of black female mobility with intimate inter-racial relations; third, about the racialised and gendered routes to freed status; and fourth, about the legacies of West African female socio-economic behaviours for modalities of family and fr...
The Healing Arts
"The book will appeal to students, teachers, health workers and general readers who wish to develop a critical awareness of medicine in the past. The essays are complemented by a selection of primary and secondary readings in the companion volume, Health, Disease and Society in Europe, 1500-1800: A Source Book."--BOOK JACKET.
Italy’s Eighteenth Century
In the age of the Grand Tour, foreigners flocked to Italy to gawk at its ruins and paintings, enjoy its salons and cafés, attend the opera, and revel in their own discovery of its past. But they also marveled at the people they saw, both male and female. In an era in which castrati were "rock stars," men served women as cicisbei, and dandified Englishmen became macaroni, Italy was perceived to be a place where men became women. The great publicity surrounding female poets, journalists, artists, anatomists, and scientists, and the visible roles for such women in salons, academies, and universities in many Italian cities also made visitors wonder whether women had become men. Such images, of course, were stereotypes, but they were nonetheless grounded in a reality that was unique to the Italian peninsula. This volume illuminates the social and cultural landscape of eighteenth-century Italy by exploring how questions of gender in music, art, literature, science, and medicine shaped perceptions of Italy in the age of the Grand Tour.
