Number Theory

1. Geometric Number Theory
   1. Lattice Points and Geometry of Numbers
      1. Basics of Lattices
         1. Definition and examples of lattices in Euclidean space
         2. Lattice basis and fundamental domain
         3. Determinants of lattices
      2. Counting Lattice Points
         1. Gauss's circle problem and asymptotic formulas
         2. Lattice point enumeration techniques
         3. Rational points on curves and surfaces
      3. Applications in Cryptography
         1. Lattice-based cryptography
         2. Learning with errors (LWE) problem
   2. Minkowski's Theorem
      1. Statement of Minkowski's Theorem
      2. Applications in Geometry of Numbers
         1. Convex bodies and lattice point inclusion
         2. Minkowski's bound
      3. Generalizations and extensions
         1. Successive minima and related theorems
         2. Applications in algebraic number theory
   3. Fermat's Last Theorem
      1. Historical Overview
         1. Origins and statement of the theorem
         2. Contributions by mathematicians such as Euler and Sophie Germain
      2. Proof by Andrew Wiles
         1. Overview of the strategy using modular forms and elliptic curves
         2. Role of the Taniyama-Shimura-Weil Conjecture
      3. Consequences and implications
         1. Impact on number theory and algebraic geometry
         2. Recent advances and related problems
   4. Pell's Equation
      1. Statement and History
         1. Origin and historical significance
         2. Connection to continued fractions
      2. General Solutions
         1. Method of solving Pell's equation
         2. Chakravala method
      3. Applications
         1. Relationship with quadratic fields
         2. Role in cryptography
   5. Geometry and Algebraic Integers
      1. Units and Norms
         1. Connection between geometry of numbers and algebraic integers
         2. Dirichlet's unit theorem and geometric interpretation
      2. Relation with Diophantine equations
         1. Techniques for solving Diophantine equations using geometry of numbers
         2. Examples and historical problems
   6. Integer Programming and Optimization
      1. Formulation of Integer Programming Problems
         1. Linear vs. nonlinear integer programming
         2. Geometric approach to integer solutions
      2. Lattice-Based Methods
         1. Reduction algorithms and applications in optimization
         2. Branch and bound methods in integer programming
   7. Higher Dimensional Geometries
      1. Study of higher dimensional convex bodies
         1. Volume approximation and lattice point enumeration
         2. Role of lattices in multi-dimensional spaces
      2. Applications in Digital Communications
         1. Coding theory and error correction
         2. Lattice codes for communication channels