Geometry of numbers

Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by . The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity. Minkowski's theorem Suppose that is a lattice in -dimensional Euclidean space and is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if , then contains a nonzero vector in . Minkowski's second theorem The successive minimum is defined to be the inf of the numbers such that contains linearly independent vectors of . Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that In 1930-1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies. Subspace theorem Siegel's lemmavolume (mathematics)determinant and Parallelepiped In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972. It states that if n is a positive integer, and L1,...,Ln are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x in n coordinates with lie in a finite number of proper subspaces of Qn. normed vector space Banach space and F-space Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces.

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Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

Göttingen

Göttingen (ˈɡɜːtɪŋən, USalsoˈɡɛt-, ˈɡœtɪŋən; Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. The origins of Göttingen lay in a village called Gutingi, first mentioned in a document in 953 AD. The city was founded northwest of this village, between 1150 and 1200 AD, and adopted its name. In medieval times the city was a member of the Hanseatic League and hence a wealthy town.

Vital role of magnetocrystalline anisotropy in cubic chiral skyrmion hosts

Henrik Moodysson Rønnow, Jonathan White

Magnetic anisotropy is anticipated to govern the formation of exotic spin textures reported recently in cubic chiral magnets, like low-temperature tilted conical and skyrmion lattice (SkL) states and metastable SkLs with various lattice geometry. Motivated ...

NATURE RESEARCH2021

The present paper is a summary and overview of results obtained in the author's thesis, "L-2-Betti Numbers of Locally Compact Groups", wherein the definition of L-2-Betti numbers for countable groups is extended to locally compact unimodular groups. In man ...

Elsevier2013

Algebraic methods for channel coding

Frédérique Oggier

This work is dedicated to developing algebraic methods for channel coding. Its goal is to show that in different contexts, namely single-antenna Rayleigh fading channels, coherent and non-coherent MIMO channels, algebraic techniques can provide useful tool ...

EPFL2005

Lattices, Minkowski's Theorem MATH-504: Integer optimisation

Explores lattices, Minkowski's Theorem, and properties of symmetric convex sets.

Lattices: More on Hermite's Theorem MATH-489: Number theory II.c - Cryptography

Explores Hermite's Theorem in lattices, including Minkowski's first theorem and properties of centrally symmetric lattices.

Defects in Crystals: Point Defects and Dislocations

Explores point defects and dislocations in crystals, highlighting their impact on material properties and the importance of energy relaxation.