# Mathematical Foundations

1. **Discrete Mathematics:**

– Introduction to Discrete Mathematics

– Sets, Relations, and Functions

– Graph Theory and Trees

– Combinatorics and Discrete Probability

– Boolean Algebra and Logic Gates

2. **Algorithms and Complexity:**

– Fundamentals of Algorithms

– Big O Notation and Complexity Analysis

– Sorting and Searching Algorithms

– Graph Algorithms

– NP-Completeness and Computational Complexity

3. **Number Theory and Cryptography:**

– Basic Concepts in Number Theory

– Prime Numbers and Factoring

– Modular Arithmetic and Cryptographic Algorithms

– Public Key Cryptography

– Hash Functions and Digital Signatures

4. **Linear Algebra and Matrices:**

– Matrix Operations and Transformations

– Systems of Linear Equations

– Eigenvalues and Eigenvectors

– Applications of Linear Algebra in Computer Science

– Advanced Topics: Singular Value Decomposition

5. **Calculus for Computer Science:**

– Limits, Continuity, and Differentiation

– Integration and its Applications

– Multivariable Calculus Concepts

– Differential Equations

– Discrete vs. Continuous Mathematics

6. **Probability and Statistics:**

– Probability Fundamentals

– Random Variables and Probability Distributions

– Statistical Inference and Hypothesis Testing

– Regression Analysis and Correlation

– Applications of Statistics in Computer Science

7. **Formal Languages and Automata Theory:**

– Introduction to Formal Languages

– Finite Automata and Regular Expressions

– Context-Free Grammars

– Turing Machines and Computability

– Complexity Classes and Language Hierarchies

8. **Information Theory and Coding:**

– Introduction to Information Theory

– Entropy and Information Content

– Coding Theorems and Data Compression

– Error Correction and Coding

– Applications in Data Transmission and Storage

9. **Mathematical Logic and Proof Techniques:**

– Propositional and Predicate Logic

– Proof Strategies and Techniques

– Recursive Definitions and Structural Induction

– Program Verification and Logic in Computer Science

– Gödel’s Incompleteness Theorems

10. **Optimization and Operations Research:**

– Linear Programming and Optimization

– Integer Programming and Combinatorial Optimization

– Decision Analysis and Game Theory

– Network Flow and Transportation Problems

– Heuristics and Approximation Algorithms

Each of these categories can be a mini-series within your larger series, providing a comprehensive foundation in the mathematical concepts crucial to computer science.