Mathematics

This course examines the methods frequently used to find numerical solutions to problems that arise in applied mathematics. The topics covered include methods for solving linear and nonlinear algebraic equations, interpolation, differentiation, integration and the numerical solution of ordinary differential equations.

The course is an introduction to three important tools of applied mathematics, namely ordinary differential equations, Fourier-series, and partial differential equations.  Some basic theoretical properties are proved and solution methods presented. Ordinary differential equations: linear differential equations of order n, the Cauchy problem, Picard's existence theorem, solution by power series and equations with singular points.  Fourier series: convergence point-wise, uniformly and in the mean-square, Parseval's equation. Partial differential equations: the heat equation and the wave equation solved on a finite interval by separation of variables and Fourier series and their solutions compared, the Dirichlet problem for the Laplace equation on the rectangle and the disc, the Poisson integral formula.

This course examines methods for understanding the behavior of solutions to ordinary differential equations. Qualitative and elementary numerical methods for obtaining information about solutions are discussed, as well as some analytical techniques for finding exact solutions in certain cases.

In this course, students gain in-depth knowledge of pricing and hedging of financial derivatives in equity markets, basic stochastic calculus, Ito’s formula, Black-Scholes models for European, American and path-dependent options such as Barrier, Asian and Lookback options. The course requires students to take prerequisites.

Students complete an internship with a local organization or company. Each placement includes oversight and regular check-ins with an internship supervisor from the company or organization. The Internship Methodology Seminar accompanies the internship placement and offers a platform for reflection, enhancement of skills, and development of cultural competence. It focuses on practical skill application, cultural understanding, and adaptability within professional environments to provide a bridge between academic learning and real-world experience.

This course provides research training for exchange students. Students work on a research project under the guidance of assigned faculty members. Through a full-time commitment, students improve their research skills by participating in the different phases of research, including development of research plans, proposals, data analysis, and presentation of research results. A pass/no pass grade is assigned based a progress report, self-evaluation, midterm report, presentation, and final report.

In this online course, learn how to construct graphs and visualizations according to the theory Grammar of Graphics. Learn how to create visualizations yourself using the software R and its package ggplot2. A central part of creating visualizations is making choices. Through the choices you make, your visualizations are more or less intelligible and also highlight different aspects of the data. An important element of the course is therefore to review visualizations by other course participants. Topics covered in the course include introduction to R and ggplot2; choice of color, symbols, scales, and perspective (2D, 3D); summation and abstraction; interactive visualizations; maps and spatial data; visualization of statistical models.

This course provides students with a further grounding in the important statistical and probabilistic techniques and models relevant to the non-life insurance industry.

This course introduces more advanced topics in calculus and ordinary differential equations. The course introduces students to multi-dimensional vector calculus and differential operators, and to the calculus of variations and the concept of variational problems. Differential equations play a key role in both pure and applied mathematics. The importance of these ideas is emphasized by the inclusion of a number of applications in physics, engineering, and biology. 

This six-week summer course provides individual research training through the experience of belonging to a specific laboratory at Tohoku University. Students are assigned to a laboratory research group with Japanese and international students under the supervision of Tohoku University faculty. They participate in various group activities, including seminars, for the purpose of training in research methods and developing teamwork skills. The specific topic studied depends on the instructor in charge of the laboratory to which each student is assigned. The methods of assessment vary with the student's project and laboratory instructor. Students submit an abstract concerning the results of their individual research each semester and present the results near the end of this program.