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.

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.

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) {x_n} produced the recursive formula x_n+1=ax_n(1?x_n). Even a relatively straightforward recursive formula that is defined by a quadratic equation produces a sequence {x_n} 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 {x_n}, 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 {x_n} 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.

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.

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.

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.

Stochastic processes investigate patterns which expand or diverge over time. It is a central part of the rationalization of randomness. Subjects of study include Brownian motion, stochastic integrals, infinite order analysis, and Markov processes. Their findings are applied in a variety of fields, such as communication, stochastic control, finance, quantum statistics, and mathematical biology.

The "representation" of group G are its elements arranged in a matrix. Suppose G is a finite group. In such a case, representational investigations into the construction of G would principally rely on the tools of algebra. Now, if we suppose G is a Lie group (a group of matrices), we would use a Hilbert space composed by functions V. Investigation of the functional space in which V operates on an already well understood Lie group G, would fall into the purview of mathematical analysis. If we consider V as a manifold of all functions, and wish to investigate the foundation of V, we examine the geometric construction of the manifold. Hence, we apply geometry.
Mathematics is often subdivided into the fields of algebra, geometry, and analysis. However, this distinction is ultimately artificial. Some aspects of mathematics simply cannot be so easily categorized. Representation theory is a unified methodology for the whole of mathematics, and how it is reckoned varies from researcher to researcher. Furthermore, the combinatorial phenomena and structures of representation theory itself are extraordinarily interesting and actively researched by modern mathematicians.