This volume includes articles spanning several research areas in number theory, such as arithmetic geometry, algebraic number theory, analytic number theory, and applications in cryptography and coding theory. Most of the articles are the results of collaborations started at the 3rd edition of the Women in Numbers Europe (WINE) conference between senior and mid-level faculty, junior faculty, postdocs, and graduate students. The contents of this book should be of interest to graduate students and researchers in number theory.

Inspired by the September 2016 conference of the same name, this second volume highlights recent research in a wide range of topics in contemporary number theory and arithmetic geometry. Research reports from projects started at the conference, expository papers describing ongoing research, and contributed papers from women number theorists outside the conference make up this diverse volume. Topics cover a broad range of topics such as arithmetic dynamics, failure of local-global principles, geometry in positive characteristics, and heights of algebraic integers. The use of tools from algebra, analysis and geometry, as well as computational methods exemplifies the wealth of techniques available to modern researchers in number theory. Exploring connections between different branches of mathematics and combining different points of view, these papers continue the tradition of supporting and highlighting the contributions of women number theorists at a variety of career stages. Perfect for students and researchers interested in the field, this volume provides an easily accessible introduction and has the potential to inspire future work.

This groundbreaking work explores the powerful role of communities in mathematics. It introduces readers to twenty-six different mathematical communities and addresses important questions about how they form, how they thrive, and how they advance individuals and the group as a whole. The chapters celebrate how diversity and sameness bind colleagues together, showing how geography, gender, or graph theory can create spaces for colleagues to establish connections in the discipline. They celebrate outcomes measured by mathematical results and by increased interest in studying mathematics. They highlight the value of relationships with peers and colleagues at various stages of their careers. Together, these stories offer a guide—rather than a template—for building and sustaining a mathematical community. They call attention to critical strategies of rotating leadership and regular assessment and evaluation of goals and programs, and promote an ongoing awareness of the responsibilities of life that impinge on mathematical creativity and contributions. Whether you are giving thought to starting a group, joining one already in existence, or encouraging a colleague to participate in the broader mathematical community, this book will meet you where you are—and move you beyond. It contains a plethora of ideas to foster a sense of belonging in the exciting discipline of mathematics.

Covering topics in graph theory, L-functions, p-adic geometry, Galois representations, elliptic fibrations, genus 3 curves and bad reduction, harmonic analysis, symplectic groups and mould combinatorics, this volume presents a collection of papers covering a wide swath of number theory emerging from the third iteration of the international Women in Numbers conference, “Women in Numbers - Europe” (WINE), held on October 14–18, 2013 at the CIRM-Luminy mathematical conference center in France. While containing contributions covering a wide range of cutting-edge topics in number theory, the volume emphasizes those concrete approaches that make it possible for graduate students and postdocs to begin work immediately on research problems even in highly complex subjects.

In the early Middle Ages, the conversion of the early English kingdoms acted as a catalyst for significant social and cultural change. One of the most visible of these changes was the introduction of a new type of household: the monastic household. These reproduced through education and training, rather than biological means; their inhabitants practised celibacy as a lifelong state, rather than as a stage in the life course. Because monastic households depended on secular households to produce the next generation of recruits, previous studies have tended to view them as more mutable than their secular counterparts, which are implicitly regarded as natural and ahistorical. Katharine Sykes charts some of the significant changes to the structure of households between the seventh to eleventh centuries, as ideas of spiritual, non-biological reproduction first fostered in monastic households were adopted in royal households in the tenth and eleventh centuries, and as ideas about kinship that were generated in secular households, such as the relationship between genealogy and inheritance, were picked up and applied by their monastic counterparts. In place of binary divisions between secular and monastic, biological and spiritual, real and imagined, Sykes demonstrates that different forms of kinship and reproduction in this period were intimately linked.