Mathematics

Topics in this Mathematical Methods I course include: differential equations; systems of ordinary differential equations; functions of complex variables. 

This course is an introduction to finance. It starts by introducing the value of money, interest rates, and financial contracts, in particular, what are fair prices for contracts and why no one uses fair prices in real life. Then, there is a review of probability theory followed by an introduction to financial markets in discrete time. In discrete time, students learn how the ideas of fair pricing apply to price contracts commonly found in stock exchanges. The next block focuses on continuous time finance and contains an introduction to the basic ideas of Stochastic calculus.

This course provides individual research training for students in the Junior Year Engineering Program through the experience of belonging to a specific laboratory at Tohoku University. Students are assigned to a laboratory with the consent of the faculty member in charge. They participate in various group activities, including seminars, for the purposes 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.

This course examines core skills in the manipulation, statistical analysis, and communication of data. Using examples from the biological, earth, and environmental sciences and using the R programming language, students will examine the role of statistics in addressing scientific questions with different goals, including determining causes, describing variation, and predicting outcomes. 

This course offers an introduction to cryptography. Topics include: mathematical foundations of cryptography; classic cryptography; symmetric encryption; key distribution and asymmetric encryption; hash functions, MAC, and authenticated encryption; digital signatures schemes; public key infrastructure; user authentication.

This calculus course covers the following topics: real numbers; sequences and series of real numbers; continuous functions; derivative; theorems about differentiable functions; Taylor expansions; applications of the derivative; Riemann integral and techniques of integration; improper integrals; applications of integration.

This course covers algebraic number fields and their rings of integers; trace, norm, and discriminants; prime decomposition in Dedekind domains and rings of integers; prime decomposition in quadratic and cyclotomic number fields; decomposition theory in Galois extensions; decomposition- and inertia groups and fields; quadratic reciprocity via decomposition theory; Frobenius automorphisms; the prime divisors of the discriminant and ramification; finiteness of class numbers; Dirichlet's unit theorem; the first case of Fermat's last theorem for regular primes.

Linear algebra is an essential tool to handle multi-component quantities. It is used not only in mathematics but also in natural and social sciences. This course provides basic notions and understanding of linear algebra such as vectors and matrices. 

Stochastic processes find applications in a wide variety of fields and offer a refined and powerful framework to examine and analyze time series. This course presents the basics for the treatment of stochastic signals and time series. Topics covered include models for stochastic dependence; concepts of description of stationary stochastic processes in the time domain including expectation, covariance, and cross-covariance functions; concepts of description of stationary stochastic processes in the frequency domain including effect spectrum and cross-spectrum; Gaussian process, Wiener process, white noise, and Gaussian fields in time and space; Stochastic processes in linear filters including relationships between in- and out-signals, autoregression and moving average (AR, MA, ARMA), and derivation and integration of stochastic processes; the basics in statistical signal processing, estimation of expectations, covariance function, and spectrum; and application of linear filters: frequency analysis and optimal filters.