Lecture Notes: The Diverse World of Mathematics

Introduction

• School math only shows a small part of the vast field.
• Aim: To explore the amazing diversity of mathematics.

Historical Overview

Origin of Mathematics

• Began with counting, evidence from prehistoric times.
• Non-human animals can count.

Key Historical Developments

• Egyptians: First equation.
• Ancient Greeks: Advances in geometry and numerology.
• China: Invention of negative numbers.
• India: First use of zero.
• Golden Age of Islam: Persian mathematicians wrote the first book on algebra.
• Renaissance: Boom in mathematics with science.

Modern Mathematics

Main Areas

• Pure Mathematics: Study for its own sake.
• Applied Mathematics: Developed to solve real-world problems.
• Crossover between pure and applied math: Discoveries in pure math can become useful in real-world applications.

Value of Pure Mathematics

• Has intrinsic beauty and elegance, similar to art.

Areas of Pure Mathematics

Study of Numbers

• Natural Numbers and arithmetic.
• Integers: Include negative numbers.
• Rational Numbers: Fractions.
• Real Numbers: Include irrationals like pi.
• Complex Numbers and others.
• Interesting properties: Prime numbers, pi, exponential functions.
• Infinite sets: More real numbers than integers.

Study of Structures

• Algebra: Manipulation of equations using variables.
• Linear Algebra: Vectors, matrices.
• Number Theory: Properties of prime numbers.
• Combinatorics: Structures like trees and graphs.
• Group Theory: Study of object relations, e.g., Rubik's cube.
• Order Theory: Arrangement of objects, e.g., natural numbers.

Study of Shapes

• Geometry: Includes Pythagorean theorem.
• Trigonometry: School-level familiarization.
• Fractal Geometry: Scale-invariant patterns.
• Topology: Properties of spaces, e.g., Möbius strip.
• Measure Theory: Assigning values to spaces.
• Differential Geometry: Shapes on curved surfaces.

Study of Changes

• Calculus: Integrals and differentials.
• Vector Calculus: For vectors.
• Dynamical Systems: Evolving systems.
• Chaos Theory: Systems sensitive to initial conditions.
• Complex Analysis: Functions with complex numbers.

Areas of Applied Mathematics

Physics

• Uses a wide range of pure math topics.
• Close relation with mathematical and theoretical physics.

Other Sciences

• Mathematical Chemistry: Molecular modeling.
• Biomathematics: Evolutionary biology.
• Engineering: Historically math-intensive.
• Control Theory: Used in complex systems, e.g., airplanes.
• Numerical Analysis: Approximations for complex problems.

Social Sciences

• Game Theory: Best choices in set scenarios, used in economics and psychology.
• Probability: Study of random events.
• Statistics: Analysis of data.
• Mathematical Finance: Financial modeling.
• Optimization: Best choice among options.

Computer Science

• Derives rules from pure math.
• Machine Learning: Uses linear algebra, optimization.
• Cryptography: Uses combinatorics, number theory.

Foundations of Mathematics

• Study of the properties of mathematics itself.
• Mathematical Logic, Set Theory, Category Theory: Explore fundamental rules, axioms.
• Gödel’s Incompleteness Theorems: Mathematics lacks a complete, consistent set of axioms.
• Theory of Computation and Complexity Theory: Study models of computing and problem-solving efficiency.

Conclusion

• Mathematics provides epiphany moments, revealing the universe's nature.
• Creating a map of mathematics was popular and well-received.
• Efforts to convey the subject's beauty and complexity.
• Anticipation of future videos.