Welcome to Mathematical Sciences
History and Evolution
The Department of Mathematics was established in 1946 with the founding of Seoul National University’s College of Liberal Arts and Sciences. Following the relocation to the Gwanak Campus in 1975, the Department of Mathematics (College of Liberal Arts and Sciences) and the Department of Applied Mathematics (College of Engineering) were integrated into the Department of Mathematics under the College of Natural Sciences, while the Department of Computer Science and Statistics was separated. In 1999, the department was officially renamed the Department of Mathematical Sciences.
Academic Excellence and Research
The Department of Mathematical Sciences is dedicated to world-class education and research in core areas of modern mathematics, including Algebra, Analysis, Geometry/Topology, and Applied Mathematics. We provide specialized major programs for undergraduate and graduate students, as well as foundational and general education for students in other disciplines. Our research activities are vibrant, centered around the BK21 Four Mathematical Sciences Division, Leading Research Centers, and the Research Institute of Mathematics. Currently, our department is home to 37 full-time faculty members, over 200 graduate students, 200 undergraduate students, and approximately 40 researchers.
Future and Career Pathways
To support academic and research excellence, we strive to provide a wide range of scholarships for both undergraduate and graduate students. Our graduates pursue diverse career paths, including advanced studies in prestigious domestic and international graduate schools, academia, research institutes, corporate sectors, and educational institutions. As the demand for creative talent equipped with mathematical logic and fundamental expertise grows to support rapid technological advancements, the career prospects for our graduates are expected to become even more diverse and promising.
Algebra
Algebra is a branch of mathematics that studies fundamental structures and operations, having developed around the concepts of symmetry and classification. Group theory is a prominent example of a theory that investigates symmetry, which plays a crucial role throughout mathematics by defining invariants and systematically classifying various mathematical objects.
Algebra is divided into several subfields based on its objects of study and methods of approach. Combinatorics explores discrete structures and properties, while number theory investigates the arithmetic properties of numbers. Algebraic geometry is concerned with cases where symmetry transformations can be expressed as rational functions, studying geometric structures defined by polynomial equations. As such, it is a key bridge between number theory and geometry. Representation theory examines groups and algebras by expressing them as linear transformations on vector spaces, providing a powerful tool for analyzing symmetry and group actions. It is widely applied in various fields, including physics, combinatorics, and number theory, to understand underlying symmetry structures. Ring theory and field theory focus on algebraic operations and solutions to equations, with commutative algebra being closely linked to algebraic geometry and number theory. In field theory, Galois theory allows for the analysis of solutions to equations and their symmetry structures.
Algebra has also had a profound impact on modern science and technology. In cryptography, number theory has led to the development of secure systems, while in computer science, algebraic techniques are employed in algorithm design and complexity theory. Furthermore, in quantum information science, group theory provides the theoretical foundation for quantum operations and computing, while algebraic geometry and representation theory have become essential concepts in theoretical physics.
Thus, algebra continues to evolve not only as a field of purely theoretical research but also through its intersections with various branches of mathematics and science. The exploration of symmetry and algebraic structures has given rise to numerous fundamental concepts, which play an indispensable role across mathematics and the sciences.
Research Areas
Algebraic Geometry
Number Theory
Combinatorics
Representation Theory
Analysis
Analysis is a branch of mathematics that studies mathematical structures based on concepts such as limits, continuity, sequences, series, functions, and calculus. It provides the foundation for precisely describing natural phenomena and plays an essential role throughout mathematics. Depending on the characteristics of the functions studied and the research methods employed, analysis is divided into various subfields, including harmonic analysis, functional analysis, complex analysis, partial differential equations, and probability theory.
Analysis is a discipline that conducts a detailed examination of the properties of functions and their variations, focusing on function spaces, transformation theory, and the generalization of differentiation and integration. Harmonic analysis, based on Fourier analysis, emphasizes understanding the relationships between functions and their transformations. Key concepts in this field include the expansion of orthogonal functions, singular integral theory, and spectral analysis. Functional analysis deals with operations in infinite-dimensional spaces, such as Banach spaces and Hilbert spaces, and explores advanced concepts such as the spectral theory of operators. Complex analysis studies the properties of holomorphic functions defined on the complex plane, where the notion of analytic continuation plays a crucial role. Partial differential equations (PDEs) investigate equations that describe the rates of change in functions, focusing on the existence and uniqueness of solutions. Probability theory is a significant applied field of analysis that provides a mathematical framework for studying uncertainty. In the 20th century, probability theory was rigorously axiomatized through measure theory, leading to the development of stochastic processes and stochastic calculus. Beyond describing randomness in physical systems, probability theory plays an essential role in finance, data science, and optimization, where analytical techniques have refined it into a more sophisticated theoretical framework.
Analysis is not only a field of pure theoretical research but also an indispensable component of mathematics and science. Furthermore, its influence continues to expand into modern technology, demonstrating its broad applicability across disciplines.
Research Areas
Complex Analysis
Harmonic Analysis
Partial Differential Equations
Functional Analysis
Probability Theory
Geometry and Topology
Geometry and Topology are two major fields of mathematics that stem from humanity's long-standing quest to understand the nature of space. These disciplines have evolved complementarily through different perspectives and tools. Geometry originated in ancient Egypt and Greece, driven by the practical need to measure length, area, and volume, and it has developed into a discipline that studies the shapes, properties, and transformations of space. In contrast, topology investigates the properties of space, such as connectivity and the number of holes, which remain unchanged under continuous deformations. In modern times, both fields focus on understanding higher-dimensional abstract spaces beyond two and three dimensions using the language of manifolds.
To conduct such research, a variety of mathematical tools and languages, including algebra and analysis, are utilized. For example, algebraic tools can be used to understand invariants of space, or symmetry of space can be used to study the spaces. Furthermore, the properties of space can be explored by studying dynamical systems based on functions defined on that space. Various geometric structures—such as the ones from Riemannian geometry, complex geometry, symplectic geometry, and hyperbolic geometry—and their collective study in terms of moduli spaces also provide ways to analyze spaces.
Geometry and topology, while employing distinct methodologies and perspectives, share the common goal of understanding the intrinsic nature of space. This research finds broad applications in modern science and technology, including physics, data science, computer graphics, machine learning, robotics, and medical imaging. For instance, general relativity is founded on the theoretical framework provided by manifolds and Riemannian geometry, while in data science, topological methods offer critical insights for analyzing and interpreting complex data structures.
Research Areas
Geometric Topology
Algebraic Topology
Dynamics
Physical Mathematics
Complex Geometry
Symplectic Geometry
Applied mathematics
Applied mathematics is a branch of mathematics that focuses on the practical application of mathematical techniques and theories to solve real-world problems. While pure mathematics explores theoretical and abstract concepts, applied mathematics emphasizes mathematical modeling, computational methods, and quantitative analysis. This field encompasses various areas, including numerical analysis, industrial mathematics, cryptography, financial mathematics, mathematical algorithms, and machine learning, playing a crucial role across multiple industries and academic disciplines.
Applied mathematics is widely utilized in engineering, physics, finance, biology, and computer science. In engineering, it is essential for structural design, mechanical analysis, and system optimization. In finance, it is used for risk assessment, portfolio optimization, and the development of algorithms for pricing complex financial products. In the natural sciences, applied mathematics contributes to the modeling of complex phenomena such as fluid dynamics, population growth, and chemical reactions. Additionally, probabilistic methods are integral to machine learning and artificial intelligence, enabling pattern extraction and predictive modeling.
Applied mathematics also supports the development of data analysis, machine learning, and artificial intelligence algorithms, becoming increasingly important in today's data-driven world. In particular, optimization techniques and probability theory are often employed together to make optimal decisions in uncertain environments. Many modern challenges are closely tied to understanding and optimizing complex systems, and applied mathematics provides essential tools for tackling these problems. Applied mathematics has become an indispensable discipline in contemporary society by leveraging mathematical techniques to analyze quantitative data and make informed decisions.
Research Areas
Financial Mathematics
Machine Learning
Industrial Mathematics
Mathematical Algorithm
Numerical Analysis
Mathematical Cryptology