Lecture Notes on Geometry of Numbers

This book serves as an illuminating introduction to the intricacies of the geometry of numbers. It commences by exploring basic concepts of convex sets and lattices in Euclidean space and goes on to delve into Minkowski's fundamental theorem for convex bodies and its applications. It discusses critical determinants and successive minima before explaining the core results of packings and coverings. The text goes on to delve into the significance of renowned conjectures such as Minkowski's conjecture regarding the product of linear forms, Watson's conjecture, and the conjecture of Bambah, Dumir, and Hans-Gill concerning non-homogeneous minima of indefinite quadratic forms.

Dedicated to Prof. R.P. Bambah on his 98th birthday, a living legend of number theory in India, this comprehensive book addresses both homogeneous and non-homogeneous problems, while sprinkling in historical insights and highlighting unresolved questions in the field. It is ideally suited for beginnersembarking on self-study as well as for use as a text for a one- or two-semester introductory course.

Presents as an introductory one- or two-semester course in geometry of numbers Introduces Minkowski's conjecture, Watson's conjecture and the conjecture of Bambah, Dumir, and Hans-Gill Honours Prof. R.P. Bambah on his 98th birthday, a living legend of number theory in India

Auteur

R. J. Hans-Gill is Emeritus Professor at Panjab University, Chandigarh, India, since 2007. She was NBHM Visiting Professor (200508) and INSA Senior Scientist (200914) at the same university. She also held the position of Dean of University Instruction at Panjab University (200204). A recipient of the Narasinga Rao Gold Medal from the Indian Mathematical Society (1971), she is Fellow of the Indian Academy of Sciences, Bangalore (1982); Indian National Science Academy, Delhi (1992); National Academy of Sciences India, Allahabad (1998); and the Academy of Sciences for the Developing World (TWAS) (2006). She was Member of the INSA Council (200304) and its Vice-President (200405). She was awarded the Srinivasa Ramanujan Birth Centenary Gold Medal by the Indian Science Congress Association (ISCA), in 2010. She earned her Ph.D. (1965) from Ohio State University and then joined the faculty there, after that she moved to the University of Wisconsin, Madison, before joining the Centre of Advanced Study in Mathematics at Panjab University, Chandigarh. With 52 publications to her credit, her research interests are in geometry of numbers and discrete geometry.

Madhu Raka is the National Academy of Science India (NASI) Senior Scientist at Panjab University, Chandigarh, India. Earlier, she worked as CSIR Emeritus Scientist at the same institute for three years, after retiring from Panjab University as professor in 2018. She also held the positions of Chairperson of the Department of Mathematics and Dean Research and Dean University Instruction at Panjab University. She was honoured with the Young Scientist Award by the Indian National Science Academy (INSA) in 1985 and was elected as Fellow of the NASI in 2008. With 58 publications to her credit, her research interests are in number theory and algebraic coding theory.

Ranjeet Sehmi retired as Professor from Punjab Engineering College (deemed to be university), Chandigarh in 2022. She has topped in the university throughout her career. She has worked in various capacities in the administration and management of the Institution, in particular as Head of the Department of Applied Science. With 20 publications to her credit, her research interests are in number theory and algebraic coding theory.

Contenu

1. Preliminaries.- 2. Minkowski's Fundamental Theorem and its Applications.- 3. Lattices.- 4. Minima of Positive De nite Quadratic Forms.- 5. Critical Determinant.- 6. Successive Minima.- 7. Packings Density.- 8. Coverings.- 9. Homogeneous Minimum.- 10. Inhomogeneous Problems.