## Educational Program

### Course for Mathematical Sciences

Theory of Numbers

Unlock the Secrets of Numbers
Number theory is the oldest subject in all of mathematics. Number theory, as Gauss once put it, is "the queen of mathematics." It is a study not only of integers themselves, but also of all mathematical operations "involving integers but also extending beyond them.” Our research seeks to continue the development of this ancient field. Riemann Hypothesis

Group Theory
The Beauty of Symmetry
Group theory studies the algebraic structures known as groups. This theory was defined by Augustin - Louis Cauchy and Leopold Kronecker in 19th century. The concept of group is applied to the development of other fields like geometry. Regular and Quasi-Regular Polyhedrons
Dynamical Systems

Tracing the Orbits of Moving Points

High school students learn about Newton's laws of dynamics that govern the movement of objects. Dynamics is the study of movements, called dynamical systems, which adhere to these established rules. For example, a recursive sequence can be represented as a series of points that follow a set rule, making its behavior a type of dynamical system. Specifically, suppose one was to examine the sequence (series of points) {xn} produced the recursive formula xn+1=axn(1−xn). Even a relatively straightforward recursive formula that is defined by a quadratic equation produces a sequence {xn} whose behavior is incredibly complex. Another mysterious and fascinating phenomenon is how the slightest adjustment to the fixed value a, or the initial value x0 can create entirely different behavior in the sequence. Figure 1 on the right is a computer rendering of the sequence when a=4 and x0=0.5. The intersections of the polygonal lines and diagonal line y = x correspond to the sequence {xn}, which meanders chaotically throughout the interval [0,1]. Meanwhile, the horizontal axis of Figure 2 shows values of a on the horizontal axis, and the values for {xn} on the vertical axis to illustrate the cluster points of the sequence. We can see that the cluster points become increasingly complex as the value of a increases from 0 to 4. Even high school-level mathematical concepts such as recursive formulas produce highly unpredictable behavior, making them useful subjects of study in the field of dynamical systems. figure 1 figure 2

Differential Geometry

Seeing Curves in Space
Euclidean geometry, the most well-known type of geometry, took shape over 2000 years ago. Based on Euclid's five exceedingly simple postulates, this geometry logically explains the concepts at work in the figures we encounter as junior high school students. In the 19th century, a new dimension of geometry, called hyperbolic geometry, sprang into existence, and denied the certainty of Euclid's fifth postulate regarding parallel lines. Riemannian geometry is based on surface theory, and can explain for the geometry of a variety of spaces, including both Euclidean and hyperbolic geometries. It has had a huge impact on all spatial principles and geometric scholarship to follow. Differential geometry is the study of Riemannian and various other geometries. Straight (geodesic) lines on spherical, flat, and curved (disc model) surfaces
Theory of Differential Equations
Studying Dynamics in Natural Phenomena
Differential equation is the fundamental method to describe natural phenomena which appears in physics, chemistry and biology.
We build on some mathematical models for phenomena by differential equations and study properties of their solutions to understand a kind of system sealed in nature.
We also derive some relationship between theory of numbers, geometry and differential equations, and find the mathematical structures there. the Beat of a Drum Represented by Bessel Functions
Algebraic Geometry
Higher Dimesional Figures Represented by Polynomials
Algebraic varieties, not only curves defined by quadratic polynomials such as parabola, ellipse and hyperbola but also higher dimensional varieties defined by polynomials of high degree.
It develops in connection with various branches of mathematics such as theory of numbers, differential equations, mathematical physics and cryptography.  