Some examples of mathematical theories are number theory, graph theory, game theory, and set theory.
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Mathematics is a vast subject that deals with logical reasoning, calculations, and measurements. There are numerous mathematical theories, each with its own set of rules and principles. Some of the most important mathematical theories are:

Number Theory: Number theory is a branch of mathematics that deals with the properties of numbers, primarily integers. It is one of the oldest branches of mathematics and has fascinated mathematicians since antiquity. Number theory is concerned with questions that are easy to ask but difficult to answer, such as the distribution of prime numbers, properties of perfect numbers, and the like.

Graph Theory: Graph theory is the study of graphs, which are mathematical structures that represent sets of objects connected by links. Graphs have numerous applications in computer science, engineering, and other fields. The most wellknown result in graph theory is the FourColor Theorem, which states that any map can be colored using only four colors such that adjacent regions have different colors.

Game Theory: Game theory is the study of decisionmaking in situations where two or more parties have conflicting interests. It is used to analyze economic, political, and social situations. Game theory is concerned with finding optimal strategies for players in games such as chess, poker, and the like. Game theory was first introduced by John von Neumann and Oskar Morgenstern in their book, Theory of Games and Economic Behavior.

Set Theory: Set theory is a branch of mathematics that deals with sets, which are collections of objects. Set theory is the foundation of mathematics and provides the language and rules for dealing with mathematical structures such as groups, rings, fields, and the like. Set theory was developed by Georg Cantor in the late nineteenth century.
In the words of the famous mathematician, Bertrand Russell, “Mathematics, rightly viewed, possesses not only truth but supreme beauty—a beauty cold and austere, without the gorgeous trappings of painting or music.” Mathematics is a beautiful subject that has fascinated humans for centuries. Here are some interesting facts about mathematical theories:
 The famous P versus NP problem, which asks whether every problem whose solution can be verified by a computer can also be solved by a computer in polynomial time, is one of the most important open problems in computer science and mathematics.
 The Golden Ratio, which is approximately equal to 1.618, is a mathematical concept that appears in art, architecture, and nature. It is used to create aesthetically pleasing designs and is often referred to as the “divine proportion.”
 The Riemann Hypothesis, which deals with the distribution of prime numbers, is one of the most important unsolved problems in mathematics. It was first introduced by Bernhard Riemann in 1859 and has numerous applications in computer science, physics, and engineering.
 Chaos Theory, which deals with the behavior of deterministic systems that are highly sensitive to initial conditions, is a fascinating branch of mathematics that has applications in a wide range of fields, from meteorology to economics.
Table: Mathematical Theories and their Applications
Mathematical Theory  Applications 

Number Theory  Cryptography, computer science, physics 
Graph Theory  Computer networks, social networks, electrical networks 
Game Theory  Economics, political science, sociology 
Set Theory  Mathematics, computer science, philosophy 
See the answer to “What are some examples of mathematical theories?” in this video
This video covers the history of mathematics and its applications, discussing topics such as set theory, logic, the Euclidean algorithm, and calculus. It also covers group theory and its applications in physics and chemistry, and mentions some of the most famous unsolved mathematical problems.
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List of mathematical theories
 Algebraic Ktheory.
 Almgren–Pitts minmax theory.
 Approximation theory.
 Asymptotic theory.
 Automata theory.
 Bifurcation theory.
 Braid theory.
 Brill–Noether theory.
 BBD decomposition theorem ( algebraic geometry)
 BEST theorem ( graph theory)
 Babuška–Lax–Milgram theorem ( partial differential equations)
 Baily–Borel theorem ( algebraic geometry)
A few important theorems are:
 Theorem 1: Equal chords of a circle subtend equal angles, at the centre of the circle.
 Converse of Theorem 1: If two angles subtended at the centre, by two chords are equal, then the chords are of equal length.
Mathematics does not have theories in the same way that Physics or Chemistry does. In Physics/Chemistry one has theories which are supported by empirical experiments but may be proven false by later experiments.
In Mathematics we start with a set of axioms/postulates and prove theorems using them. These can’t be proven false, since they’ve already been proven true. The proof does not depend on empirical evidence found in the physical world, but on nothing more than logic.
That all said there have been Mathematical conjectures which were widely believed to be true which were later proven false. Below is one example.
In 1838 Peter Gustav Lejeune Dirichlet [ https://en.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet ] came up with an approximation to the Primecounting function [ https://en.wikipedia.org/wiki/Primecounting_function ] (denoted [math]\pi(x)[/math]known as the Logarithmic integral function [ https://en.wikipedia.org/wiki/Logarithmic_integral_function ] (denoted [math]\…
This is a list of mathematical theories . Algebraic Ktheory. Almgren–Pitts minmax theory. Approximation theory. Asymptotic theory. Automata theory. Bifurcation theory. Braid theory. Brill–Noether theory.
Surely you will be interested in these topics
For example, if one tosses a coin and the question of interest is how many times the coin needs to be tossed before the first tail occurs, this number does not have an upper bound in the sense that no larger value may be observed than the bound.
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