Geometry of Numbers

Abstract

Each participant will present one of the fifteen lectures in Siegel's book "Lectures on the geometry of numbers".

Content

## Minkowski's Two Theorems ## ### Lecture I ### 1. Convex sets 2. Convex bodies 3. Gauge function of a convex body 4. Convex bodies with a centre ### Lecture II ### 1. Minkowski's First Theorem 2. Lemma on bounded open sets in ℝ^n 3. Proof of Minkowski's First Theorem 4. Minkowski's theorem for the gauge function 5. The minimum of the gauge function for an arbitrary lattice in ℝ^n 6. Examples ### Lecture III ### 1. Evaluation of a volume integral 2. Discriminant of an irreducible polynomial 3. Successive minima 4. Minkowski's Second Theorem ### Lecture IV ### 1. A possible method of proof 2. A simple example 3. A complicated transformation 4. Volume of the transformed body 5. Proof of Minkowski's Second Theorem ## Linear Inequalities ## ### Lecture V ### 1. Vector groups 2. Construction of a basis 3. Relation between different bases for a lattice 4. Sub-lattices 5. Congruences relative to a sub-lattice 6. The number of sub-lattices with given index ### Lecture VI ### 1. Local rank of a vector group 2. Decomposition of a general vector group 3. Characters of vector groups 4. Conditions on characters 5. Duality theorem for character groups 6. Kronecker's approximation theorem ### Lecture VII ### 1. Periods of real functions 2. Periods of analytic functions 3. Periods of entire functions 4. Minkowski's theorem on linear forms ### Lecture VIII ### 1. Completing a given set of vectors to form a basis for a lattice 2. Completing a matrix to a unimodular matrix 3. A slight extension of Minkowski's theorem on linear forms 4. A limiting case 5. A theorem about parquets 6. Parquets formed by parallelepipeds ### Lecture IX ### 1. Products of linear forms 2. Product of two linear forms 3. Approximation of irrationals 4. Product of three linear forms 5. Minimum of positive-definite quadratic forms ## Theory of Reduction ## ### Lecture X ### 1. The problern of reduction 2. Space of all matrices 3. Minimizing vectors 4. Primitive sets 5. Construction of a reduced basis 6. The First Finiteness Theorem 7. Criteria for reduction 8. Use of a quadratic gauge function 9. Reduction of positive-definite quadratic forms ### Lecture XI ### 1. Space of symmetric matrices 2. Reduction of positive-definite quadratic forms 3. Consequences of the reduction conditions 4. The case n = 2 5. Reduction of lattices of rank two 6. The case n = 3 ### Lecture XII ### 1. Extrema of positive-definite quadratic forms 2. Closest packing of (solid) spheres 3. Closest packing in two, three, or four dimensions 4. Blichfeldt's method ### Lecture XIII ### 1. The Second Finiteness Theorem 2. An inequality for positive-definite symmetric matrices 3. The space P_K 4. Images of R ### Lecture XIV ### 1. Boundary points 2. Non-overlapping of images 3. Space defined by a finite number of conditions 4. The Second Finiteness Theorem 5. Fundamental region of the space of all matrices ### Lecture XV ### 1. Volume of a fundamental region 2. Outline of the proof 3. Change of variable 4. A new fundamental region 5. Integrals over fundamental regions are equal 6. Evaluation of the integral 7. Generalizations of Minkowski's First Theorem 8. A lower bound for the packing of spheres

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