Here are some of the main areas of pure mathematics, along with brief definitions:

**Algebra:** Algebra is the study of mathematical structures and the properties of operations on them. It includes the study of equations, polynomials, groups, rings, and fields.

**Analysis:** Analysis is the study of functions and their properties. It includes the study of limits, continuity, differentiability, and integrability, as well as the study of metric spaces and functional analysis.

**Geometry:** Geometry is the study of shape and space. It includes the study of points, lines, angles, surfaces, and volumes, as well as the properties of geometric objects.

**Topology:** Topology is the study of the properties of space that are preserved under continuous transformations. It includes the study of spaces, continuity, convergence, and connectedness.

**Number theory:** Number theory is the study of the properties of integers and related structures. It includes the study of prime numbers, modular arithmetic, and Diophantine equations.

**Combinatorics**: Combinatorics is the study of counting and arrangements of objects. It includes the study of permutations, combinations, graph theory, and discrete mathematics.

**Logic:** Logic is the study of formal reasoning and the principles of valid reasoning. It includes the study of propositional logic, predicate logic, and formal systems.

**Lattice Theory:** Lattice Theory is a branch of mathematics that deals with the study of partially ordered sets in which any two elements have both a unique greatest lower bound (also called infimum) and a unique least upper bound (also called supremum). These partially ordered sets are called lattices.

Field theory is a branch of mathematics that studies the algebraic structures known as fields. A field is a set of elements, along with two binary operations (usually called addition and multiplication), that satisfy certain axioms.

**Field Theory:** The study of field theory includes topics such as field extensions, Galois theory, algebraic geometry, and algebraic number theory. Some of the key concepts in field theory include:

- Field extension: A field extension is a larger field that contains a smaller field as a subfield. The degree of a field extension is the dimension of the larger field over the smaller field.
- Galois theory: Galois theory is a branch of algebra that studies field extensions and their automorphisms. It provides a powerful tool for studying the solvability of polynomial equations.
- Algebraic geometry: Algebraic geometry is the study of algebraic varieties, which are geometric objects defined by polynomial equations. Fields play an important role in algebraic geometry, as they are used to define the coefficients of the polynomials.
- Algebraic number theory: Algebraic number theory is the study of number fields, which are finite extensions of the rational numbers. Fields play a central role in algebraic number theory, as they are used to study the properties of number fields and their rings of integers.

**Complex Analysis:** Complex analysis is a branch of mathematics that deals with the study of complex-valued functions of a complex variable. In other words, it is the study of functions that map complex numbers to complex numbers.

Here are some of the main interdisciplinary areas of mathematics, along with brief definitions:

**Mathematical biology:** Mathematical biology is the application of mathematical methods to problems in biology, including population dynamics, genetics, and ecology.

**Mathematical physics:** Mathematical physics is the application of mathematical methods to problems in physics, including quantum mechanics, relativity, and classical mechanics.

**Computational science:** Computational science is the application of mathematical methods and computer simulations to solve problems in science and engineering. It includes the study of algorithms for scientific computing, high-performance computing, and data visualization.

**Operations research:** Operations research is the application of mathematical methods to optimize complex systems, including supply chains, transportation networks, and manufacturing processes.

**Mathematical finance:** Mathematical finance is the application of mathematical methods to problems in finance, including risk management, portfolio optimization, and derivatives pricing.

**Cryptography:** Cryptography is the study of techniques for secure communication in the presence of third parties. It includes the study of encryption algorithms, digital signatures, and cryptographic protocols.

**Data science:** Data science is the study of methods for extracting insights from large and complex data sets. It includes the study of statistical modelling, machine learning, and data visualization.

**Numerical Analysis**: Numerical analysis is the study of algorithms for solving mathematical problems that cannot be solved analytically, while in engineering and physics, numerical methods are used to solve problems that arise in designing and analyzing physical systems.

**Differential Equations: **Differential Equations is a branch of mathematics that studies equations that involve derivatives of functions. These equations can describe many physical, biological, and social phenomena, such as the growth of populations, the spread of disease, and the motion of particles.

Overall, interdisciplinary areas of mathematics involve the application of mathematical methods to solve problems in other fields, and they often involve collaboration with experts in those fields. These interdisciplinary areas are important for solving real-world problems and have many applications in various fields, including science, engineering, finance, and business.