A Wams school on
Introductory topics in Number Theory and Differential Geometry
with a course on Dynamical Systems

King Khalid University, Abha. Saudi Arabia
June 16th- 23rd, 2019

Differential Geometry of Curves and surfaces:
Ali Alkhaldi and Mohammad Hasan Shahid
  1. Curves in 3-dimensional Euclidean space E3
  2. Curves in 3-dimensional Minkowski spaceE31
  3. Curves in 4-dimensional Minkowski space E41
  4. Surfaces in 3-dimension Euclidean space E3
  5. Shape operator, curvature and new type of surfaces.
Introduction to Diophantine Equations:
Florian Luca & Michel Waldschmidt
  1. Classical Diophantine equations in two variables, their theory, relations with Diophantine Approximation and with Algebraic Number Theory
    • linear equation
    • Pell equation
    • Thue equation
    • elliptic, hyperelliptic, superelliptic equations
    • general equation, Siegel's finiteness theorem.
  2. Baker's method, effective results
  3. From effective to explicit: the Las-Vegas Principle
  4. Continued Fractions and Baker-Davenport Lemma
  5. Explicit solution of simultaneous Pell equations
  6. Explicit solutions of Thue equations.
  7. Elliptic curves, Mordell-Weil Theorem. Solving elliptic equations using elliptic logarithms.
Introduction to Dynamical Systems:
Gamal Mokhtar Mahmoud
  1. Introduction to Differential Equations and Dynamical Systems
  2. Introduction to Chaos
  3. Local Bifurcations
  4. Averaging and other Perturbation methods
  5. Fixed points and their stability (e.g. real and complex Lorenz systems)
Introduction to Enumerative Combinatorics:
Abdulaziz Alanazi
  1. Permutations and binomial coefficients
  2. Inclusion and exclusion formula.
  3. Linear recurrences. The Fibonacci sequence
  4. Catalan numbers
  5. Generating functions
  6. Partitions
  7. Euler's generating function for partitions
  8. pentagonal formula.
Introduction to Elliptic Curve over finite fields:
Francesco Pappalardi
  1. Weierstrass Equations, The Group Law,
  2. The j-Invariant,
  3. Elliptic Curves in Characteristic 2,
  4. Endomorphisms, Singular Curves, Elliptic Curves mod n.
  5. Torsion points: Division Polynomials,
  6. The Weil Pairing.
  7. The Frobenius Endomorphism,
  8. Determining the Group Order, Schoof's Algorithm, Supersingular Curves.
Exercise Sessions

There will be exercise sessions where the participants will solve problems assigned by the lecturers.