Chapter Ⅰ 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 IRn
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 IRn
6.Examples
Lecture Ⅲ
1.Evaluation of a volume integral
2.Discriminant of an irreducible polynomial
3.Successive miruma
4.Minkowski's Second Theorem （Theorem 16）
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 Theorem 16 （Minkowski's Second Theorem）
Chapter Ⅱ 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 quadrat，ic forms
Chapter Ⅲ Theory of Reduction
Lecture Ⅹ
1.The problem 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 packinng 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 PK
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
References
