Abstract Algebra

1. Applications of Abstract Algebra
   1. Cryptography
      1. Fundamental Concepts
         1. Encryption and Decryption
         2. Key Management and Distribution
      2. Public Key Cryptosystems
         1. RSA Algorithm
            1. Key Generation
            2. Encryption and Decryption Process
            3. Security Basis on Number Theory
         2. Diffie-Hellman Key Exchange
            1. Protocol Description
            2. Security via Discrete Logarithms
         3. Comparison with Symmetric Key Algorithms
      3. Symmetric Key Cryptography
         1. Group-based Symmetric Algorithms
      4. Modern Cryptographic Techniques
         1. Applications of Finite Fields
         2. Use of Elliptic Curves
            1. Definition and Properties of Elliptic Curves
            2. Elliptic Curve Cryptography (ECC)
               1. Key Generation
               2. Digital Signatures
               3. Encrypting Data with Elliptic Curves
   2. Coding Theory
      1. Error-Detecting and Error-Correcting Codes
         1. Basic Concepts and Objectives
         2. Application in Data Transmission
      2. Linear Codes
         1. Definition and Examples
         2. Coding Theorems
            1. Hamming Codes
            2. Reed-Solomon Codes
         3. Bounds and Distance Measures
      3. Algebraic Code Construction
         1. Use of Polynomials in Code Construction
         2. Cyclic Codes
            1. Generator and Parity-check Polynomials
      4. Advanced Coding Techniques
         1. Turbo Codes and LDPC Codes
         2. Applications of Group Theory in Encoding and Decoding
   3. Algebraic Geometry
      1. Introduction to Algebraic Geometry
         1. Basic Definitions and Concepts
         2. Role of Polynomials and Rings
      2. Varieties and Schemes
         1. Classes of Varieties
            1. Affine and Projective Varieties
         2. Schemes as Generalization of Varieties
      3. Intersection Theory
         1. Intersections in Algebraic Varieties
         2. Role of Homological Algebra
      4. Applications in Number Theory and Cryptography
         1. Elimination Theory
         2. Use in Cryptographic Protocols
   4. Mathematical Physics
      1. Use of Algebra in Physics
      2. Symmetries and Conservation Laws
         1. Group Theory and Physical Symmetries
         2. Noether's Theorem and Conservation Laws
      3. Quantum Groups
         1. Definition and Concept
         2. Applications in Quantum Mechanics
            1. Quantum Algebras
            2. Lie Algebras and their Representations
         3. Role in Particle Physics
            1. Symmetric Spaces
            2. Representation Theory in Quantum Physics
   5. Other Applications
      1. Robotics and Motion Planning
         1. Use of Group Theory in Path Planning
         2. Algebraic Kinematics
      2. Chemistry and Crystallography
         1. Symmetry in Molecular Structures
         2. Group Theory in Crystallography
      3. Economics and Game Theory
         1. Application of Algebraic Structures in Auction Theory
         2. Coalition Game Models using Algebraic Techniques