Introduction to Mathematics

• Mathematics is the abstract study of quantity, structure, space, and change.
• The scope and definition of mathematics are subjects of debate among different schools of thought.
• Mathematicians identify patterns and develop new conjectures.
• Mathematical proofs are used to determine the truth or falsity of conjectures.
• Mathematical problems arise in natural sciences, engineering, medicine, finance, computer science, and the social sciences.
• Applied mathematics uses mathematical tools to address problems in various fields, including natural sciences, engineering, medicine, finance, computer science, and the social sciences.
• Mathematical proof uses logic to demonstrate the general truth of a statement.
• Mathematical research often focuses on identifying essential characteristics of mathematical objects.
• The aim is to establish concise and widely applicable principles.
• Principles with broad application are preferred over those specific to particular cases.
• Mathematics is an essential tool in natural science, engineering, medicine, finance, and social sciences.
• Applied mathematics has led to new mathematical disciplines like statistics and game theory.
• Pure mathematics is pursued by mathematicians without immediate concern for application.
• Practical uses for pure mathematics are often discovered later.
• Mathematics is critical to the advancement of science and technology.
• Elementary mathematics includes arithmetic, algebra, geometry, and trigonometry.
• Arithmetic involves numerical operations, recording, and retrieving numbers.
• Algebra deals with equations, formulas, and mathematical structures.
• Geometry focuses on shapes and their properties.
• Trigonometry is the study of triangles and trigonometric functions.
• Calculus, typically taught at the undergraduate level, involves functions, limits, derivatives, integrals, and infinite series.
• Mathematics enables the creation of abstract representations of reality.
• Constraints can be added to these abstractions to modify them.
• Mathematical objects are defined axiomatically.
• Proof serves as sufficient justification in mathematics.
• Mathematical structures are sets with one or more operations.

Set Theory

• Set theory studies sets, which are collections of objects, including numbers, functions, and other sets.
• It provides a foundation for modern mathematics.

Logic

• Logic is the study of valid reasoning and argumentation.
• It proves theorems and constructs mathematical arguments in mathematics.
• Mathematical logic applies formal logic techniques to mathematical problems.

Category Theory

• Category theory is a general theory dealing with mathematical structures and their relationships.
• It is used to study abstract mathematical concepts and develop new mathematical tools.

Number Theory

• Number theory is a branch of pure mathematics focused on integers and integer-valued functions.
• Analytic number theory employs mathematical analysis to solve problems concerning integers.
• Diophantine geometry studies Diophantine equations using methods from algebraic geometry.
• Order theory studies the concept of order using binary relations.

Group Theory

• Group theory studies groups, which are algebraic structures with sets of elements and operations that satisfy specific axioms.

Representation Theory

• Representation theory studies algebraic structures by representing their elements as linear transformations of vector spaces.

Field Theory

• Field theory studies fields, which are algebraic structures with addition, subtraction, multiplication, and division operations that meet certain axioms.
• Commutative algebra studies commutative rings.

Algebraic Geometry

• Algebraic geometry combines abstract algebra techniques with geometry to solve geometric problems.

Topology

• Topology studies properties of spaces that remain unchanged under continuous deformations like stretching, twisting, crumpling, and bending.

Analysis

• Analysis studies continuous change and includes differentiation, integration, measurement, limits, infinite series, and analytic functions.
• Real analysis focuses on real numbers and real-valued functions of a real variable.
• Complex analysis investigates functions of complex numbers.
• Functional analysis studies vector spaces and operators acting on them.

Differential Equations & Dynamical systems

• Differential equations are mathematical equations relating functions and their derivatives.
• Dynamical systems evolve over time according to a fixed rule.
• Chaos theory describes the behavior of dynamical systems with sensitivity to initial conditions.
• Fractal geometry studies self-similar geometric shapes called fractals.
• Measure theory generalizes notions like length, area, and volume.
• Numerical analysis studies algorithms using numerical approximation for mathematical analysis problems.

Probability & Statistics

• Probability theory studies probability.
• Statistics involves the collection, analysis, interpretation, presentation, and organization of data.
• Game theory studies strategic interactions between rational agents.
• Optimization finds the best solution to a problem with constraints.
• Operations research uses mathematical models and algorithms to solve complex decision-making problems.
• Linear programming optimizes a linear objective function subject to linear constraints.

Discrete Mathematics

• Discrete mathematics studies discrete mathematical structures.
• Combinatorics studies arrangements of discrete objects.
• Graph theory studies graphs, which are mathematical structures modeling pairwise relations.
• Coding theory studies code properties and their suitability for applications.
• Cryptography studies techniques for secure communication.

Geometry

• Geometry studies shapes, sizes, positions, and properties of space.
• Euclidean geometry is based on Euclid's axioms.
• Differential geometry uses calculus to study geometric problems.
• Algebraic geometry uses abstract algebra to solve geometric problems.
• Trigonometry studies relationships between angles and sides of triangles.
• Convex geometry studies convex sets in Euclidean space.

An introduction to mathematics and its applications across different fields, including natural sciences, engineering, and computer science. Mathematics is the abstract study of quantity, structure, space, and change. Mathematicians seek patterns, formulate conjectures, and prove their validity using logic.