Pure Mathematics is the core of mathematical sciences, which focuses on the development of mathematical concepts, theorems, and methods in an abstract and theoretical manner. It explores the fundamental structures, patterns, and properties of pure mathematics, without considering practical applications at first. The goal is to build a solid foundation and deepen understanding of the underlying mathematical structures. In Pure Mathematics, mathematicians delve into fundamental concepts such as number theory, abstract algebra, geometry, topology, and mathematical logic. They build complex theories, formulate formal definitions, prove theorems, and explore relationships between concepts. This allows them to deeply understand mathematical properties and apply rigorous logic in mathematical reasoning. One of the main goals of Pure Mathematics is to discover and prove theorems that are universal, identifying patterns and structures that cover the entire domain of mathematics. The interrelationships between theorems and concepts come into focus, and mathematicians seek to understand the fundamentals of mathematics more deeply. The discovery of pure theorems is often the basis for further development of mathematics.

Statistics is a subfield of Mathematical Sciences that plays an important role in collecting, analyzing, and interpreting data. Its main focus is to understand phenomena or events in the real world through data obtained from various sources such as surveys, experiments, or observations. The main goal of Statistics is to provide accurate, reliable, and meaningful information from the data at hand, and develop robust statistical methods to analyze the data appropriately. One important aspect of Statistics is data collection. This involves designing appropriate experiments or surveys to obtain relevant and representative data. Data collection methods must be valid, measurable, and appropriate to the purpose of the research or analysis to be conducted. Once the data is collected, statisticians organize, analyze, and interpret this data. Data processing is the next step in the statistical process. It includes techniques to summarize data in the form of descriptive statistics, such as mean, median, mode, and so on. In addition, statisticians use various inferential methods to make inferences or generalizations about the population based on the sample taken. These techniques include hypothesis testing, parameter estimation, and regression analysis. Statistics also plays an important role in decision-making. The results of statistical analysis help individuals, organizations, or institutions to make better and more informed decisions. For example, in the business field, statistics are used to analyze market behavior, consumer trends, and investment decisions. In the healthcare field, statistics help in disease research, drug development, and evaluation of treatment effectiveness.

Operations research is a subfield of mathematics, focusing on the development and application of operations techniques on various mathematical objects, including but not limited to numbers, functions, and geometric structures. These mathematical objects are the main focus in the analysis, modeling, and solving of complex mathematical problems. The main goal of this group is to gain a deeper understanding of the fundamental properties of these mathematical objects and develop effective algorithms and computational methods to manipulate and understand them better. Operations research has a wide scope and can be applied in various fields, including cryptography, dynamical systems modeling, number theory, and more. In cryptography, operations research helps develop secure and efficient encryption and decryption algorithms to protect sensitive information. In the field of dynamical systems modeling, operations research helps in formulating mathematical models to understand the behavior of complex systems. Also, in number theory, operations research is used to analyze the unique properties of numbers. Operations research experts use various techniques, algorithms, and mathematical approaches to solve problems and explore mathematical structures. They also utilize computation to facilitate complex analysis and solve difficult problems. These skills are essential in developing solutions to modern mathematical challenges, as well as in contributing to the development of technology, information security, and science in general.

Mathematical Computing is a subfield that focuses on the development and application of computer technology to solve various mathematical problems. It covers a wide range of areas, including numerical modeling, simulation, data analysis, optimization, image processing and more. The goal is to develop effective computational methods to tackle complex and deep mathematical problems, while gaining a deeper understanding of the underlying mathematical properties. The application of computer technology in Mathematical Computing enables faster and more accurate mathematical problem solving than traditional manual methods. Through techniques such as algorithms, programming, data analysis and mathematical modeling, Mathematical Computing experts can solve problems involving high complexity and large volumes of data. In the field of numerical modeling, they develop mathematical models and use computation to simulate complex physical, biological, or economic phenomena. The applications of Mathematical Computing are vast and impactful in many fields, including physics, biology, economics, and engineering. In physics, Mathematical Computing is used to understand and predict the behavior of natural phenomena, such as fluid dynamics or particle interactions. In biology, computing helps in modeling and simulating evolution, protein structure, and other biological processes. In economics, data analysis and optimization are used to understand market behavior and economic decision-making. In engineering, computing plays a key role in designing and optimizing structures and systems.