A joint CIMPA-ICTP research school on Lattices and applications to cryptography and coding theory Saigon University, Ho Chi Minh City, Vietnam. August 1st - 12th, 2016 dfasjk asdgsiji

Introductory Classes

 Introduction to lattices, René Schoof Scalar product, Euclidean vector space, Gram-Schmidt orthonormalization; Lattices, Gram matrix, isometries, sphere packing, Kissing number, Hermite constant. Introduction to complex analysis, Francesco Pappalardi Complex numbers,series, exponential, sinus and cosinus of comple numbers; Complex differentiation, holomorphic functions and the Cauchy Riemann Equations; Complex integration, Cauchy theorem and Cauchy integral formula; Taylor series, Laurent series and residue theorem. Introduction to algebraic number theory, Peter Stevenhagen Unique factorization in $\mathbb{Z}$ and $\mathbb{Z}[i]$; Uniquue ideal factorization in rings of integers: units, class groups; Finiteness results from embeddings in Euclidean spaces and Minkowski's theorem; Explicit computations.

Courses

 Lattices and geometry of numbers, Michel Waldschmidt Lecture 1 : Subgroups of $\mathbb{R}^n$: discrete, closed, dense; Topological groups; Lattices; Fundamental parallelepiped, volume; Packing, covering, tiling; Subgroup of $\operatorname{Hom}_{\mathbb{R}}(\mathbb{R}^n, \mathbb{R})$ associated with a subgroup of $\mathbb{R}^n$. Lecture 2: Convex Sets, Star Bodies and Distance Functions; Minkowski's convex body theorem; Minkowski's theorems on linear forms; Gauge functions; Minkowski's theorem on successive minima. Lecture 3 Minima of quadratic forms; Sum of two squares; Sum of four squares; Primes of the form $x^2 + ny^2$; Discriminant of a number field; Units of a number field: Dirchlet's s theorem; Geometry of numbers and transcendence. Exercises Coding theory, Dung Duong Hamming distance; Linear codes; Cyclic codes; Quadratic residue codes; dual codes; MacWilliams identities; Perfect codes. Lattices and cryptography, Phong Q. Nguyen 1. Basic Algorithms: Gram-Schmidt Orthogonalization and Computing a Basis. Notes. 2. Finding Short Lattice Vectors: Hermite's inequality and the LLL algorithm; Mordell's inequality and Blockwise algorithms; Minkowski's inequality, worst-case to average-case reductions and sieve algorithms; Lattice enumeration. 3. Cryptography from lattices: SIS and LWE; One-way Functions from Lattices; Lattice-based Key Exchange and Public-Key Encryption; Lattices in number theory, Ha Tran 1. The Arakelov class group: Notes Arakelov divisors (degree, covolume,...); Ideal lattices; The Arakelov class group and its structure; Some examples. 2. Reduced Arakelov divisors: Notes Metric on the Arakelov class group; Definition and examples (in quadratic fields) of Reduced Arakelov divisors; Properties of Reduced Arakelov divisors. 3. Computations with reduced Arakelov divisors:Notes Reduction algorithm; Composition algorithm and inversion algorithm; Scan algorithm.
 Lattices and Mordell-Weil groups, Francesco Pappalardi and Peter Stevenhagen 1. Elliptic curves over number fields: Projective coordinates and projective space, homogeneous equations, singular points; Weierstrass equation, sum of points, duplication of points, points at infinity, explicit formulas for operations; The group of rational points over a field, points of order two and points of order three; Points of finite order and endomorphisms of elliptic curves. 2. Three proofs of the associativity of the group law: Computer assisted proof; Proof via the Pappus theorem Proof via Picard group 3. Mordell-Weil lattices The weak Mordell-Weil theorem $+$ descent $\Rightarrow$ the Mordell-Weil theorem heights on projective spaces Neron-Tate height Mordell-Weil lattices Lattices and Lie algebras, Laura Geatti Abstract root systems; Existence of a base; The lattice associated to an abstract root system (root lattice); The classification of root systems; Examples of root lattices. Lecture 2. Complex semisimple Lie algebras; Cartan subalgebras and root decomposition; The Lie algebraa $\mathfrak{sl}_2$ and its representations. Lecture 3. The root system of a semisimple Lie algebra Cartan matrix and Dynkin diagram. The weight lattice. Notes References J. E. Humphreys, Introduction to Lie algebras and Representation theory, Graduate Texts in Mathematics, Springer Verlag, New York, 1972 J.-P. Serre, Complex semisimple Lie algebras, Springer Verlag, New York, 1966Course notes. Lattices and modular forms, Valerio Talamanca Lecture 1 Lattices in the complex plane and elliptic curves; Action of $\operatorname{SL}_2(\mathbb{Z})$ on the set of lattices in the complex plane; Moduli space for complex elliptic curves. Lecture 2 Classsical theta functions and their properties; Modular forms for $\operatorname{SL}_2(\mathbb{Z})$, cusp forms. The space $M_k$ of weight 2k modular forms; Invariants of elliptic curves as modular functions; Eisenstein series and the explicit determination of a basis of $M_k$. Lecture 3 Theta series of lattices; Even Unimodular lattices; The theta series of even unimodular lattices is a modular form; Explicit examples. References J.-P. Serre, A Course in Arithmetic, Springer Verlag, New York, 1973 (Chapter VII)

Lectures

 Poisson summation, René Schoof Dual lattices; Fourier analysis; Poisson summation formula. Elliptic curves over ${\mathbb C}$, Peter Stevenhagen calculus for algebraic integrals: Riemann surfaces; elliptic functions, Weierstrass parametrization of elliptic curves; Eisenstein series, modular functions; Endomorphism rings, CM elliptic curves. Notes on complex elliptic curves The Leech lattice, René Schoof Construction and properties of the Leech lattice.

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