#  # ###   Research Research Centers

#### Dynamical system LabProfessors: Kyewon Koh Park, Jungseob Lee, Younghwa Ha

The dynamical system theory started as an effort to describe and predict natural and social phenomena which evolve as time goes. The theory includes ergodic theory, topological dynamics, chaos and fractal theory as subfields. Moreover, it is related to various areas of mathematics, and has applications to natural sciences, engineering and social sciences. Recently it extends to data compression, storage and bioinformatics.

#### Applied Mathematics LabProfessors: Hyungchun Lee, Youngmok Jeon, Youngwoo Choi

In Applied Mathematics Lab, mathematical modelling, analysis, and numerical solution of partial differential equations are studied. With the aid of computer programming and simulation, we attempt to explain and predict various phenomena of nature. Partial differential equations derived from engineering problems related to elasticity, fluid dynamics, and electro-magnetics, are typical subjects of research. Methods from harmonic analysis such as Fourier transform and wavelet transform are used as important tools. Solutions are approximated and simulated using finite element, boundary element, and spectral methods.

#### Probability/Statistics LabProfessors: Seungho Lee, Kijung Lee

• - Application of statistical theories and methods to natural science and social science
• - Multiple Decision
• - Statistical Calibration

We deal with the uncertainty in natural or social phenomena. We find the information of probability distributions of the random quantity(random variables, vectors) we are interested in. We use this information to compute probability of important events related to the random quantity. Probability and Statistics are complementary each other.
The important information of a given distribution is included in several forms of deterministic quantities. For instance, the moments of a random variable contain most valuable probabilistic information; we obtain the mean and the variance from the first two moments. Given samplings, Statistics provide us with the methods of approximating these quantities and giving us the confidence on the size of samplings and the approximation. Based on the information of the distribution we get through Statistics, we use probability theory to approximate the probability of events of interest. Many probability theories of attacking complex and abstruse probabilistic event has been developed.

#### Topology/Geometry Lab Professors: Jaeseok Jeon, Seungjin Bang, Suyoung Choi

• - Differential topology
• - Differential geometry, Riemannian geometry
• - Toric topology
A torus is a classical compact abelian Lie group, a product of ﬁnitely many circles. Toric topology is the study of topological spaces having very nice symmetries (i.e., torus actions) and topological aspects of torus actions. A study of torus actions has been studied for over a century as an important subbranch of equivariant topology. Recently, the study of them has become more important in various areas of mathematics such as algebraic geometry, combinatorial and convex geometry, commutative and homological algebra, diﬀerential topology, and homotopy theory.
We work on toric topology with the viewpoint of algebraic topology, commutative and homological algebra, and polytope theory.
• - Transformation group theory

#### Algebra LabProfessors: Soojin Cho, Dongseon Hwang

Algebra is one of the most fundamental fields in mathematics, whose objectives are the sets with binary operations. Algebraists consider the natural structures or the symmetries of algebraic objects, make descriptions of rules in nature using mathematical languages. The theories have wide applications in natural sciences, and cryptography as well as in engineering.
Our research interests in the `Algebra Lab’ include;

• - Algebraic Geometry
Investigating the classification problems in algebraic geometry with a special focus on singular surfaces and higher dimensional algebraic varieties. Applications of the research to other fields such as topology, differential geometry, symplectic geometry, mirror symmetry, and coding theory are also being conducted.
• - Algebraic Combinatorics
We try to understand the algebraic objects in combinatorial ways; we obtain new algebraic results through combinatorial methods and vice versa. The most important example in algebraic combinatorics is the group of permutations, and we use two tableaux to represent a permutation. By observing the properties of tableaux we have better understanding of the representations of the symmetric groups.
• - Enumerative Combinatorics
We closely look at the combinatorial objects which has symmetries; by classifying and counting them due to their natural properties. We also try to find natural correspondence between our objects and other well known ones. This enables us to have better understanding on the objects of interest.   