Mathematics

The course covers how to generate documents containing mathematical formula, elementary programming skills, and computer simulation methods for mathematical study

This course examines the nature of mathematical concepts and thinking. Students will reflect on their experience as a learner in the process of sense making, reasoning and co-constructing knowledge in mathematics. By examining origins and development of mathematical ideas in historical-cultural context and individual minds, students will understand means and issues in supporting mathematics learning

This course examines the theory and techniques of optimization. It covers unconstrained and constrained optimization; necessary conditions and sufficient conditions for optimality, convexity, duality; and algorithms and numerical examples. 

In this course, students learn about a number of key topics in the philosophy of mathematics. It ensure students are familiar with the main views such as Platonism, nominalism, logicism, formalism, intuitionism, and structuralism, as well as the main criticisms of each. Students learn about the philosophical significance of Russell’s paradox and Gödel’s incompleteness theorems. From here, they consider topics in the philosophy of mathematical practice, such as the nature of mathematical proofs, the use of diagrams in mathematical reasoning, explanation and understanding in mathematics, mathematical knowledge, and the ethics of mathematics.  

The primary focus of this course is on the core machine learning techniques in the context of high-dimensional or large datasets (i.e. big data). The first part of the course covers elementary and important statistical methods including nearest neighbors, linear regression, logistic regression, regularization, cross-validation, and variable selection. The second part of the course deals with more advanced machine learning methods including regression and classification trees, random forests, bagging, boosting, deep neural networks, k-means clustering and hierarchical clustering. The course will also introduce causal inference motivated by analogy between double machine learning and two-stage least squares. All the topics are delivered using illustrative real data examples. Students also gain hands-on experience using R or Python (programming languages and software environments for data analysis, computing and visualization).

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.

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 the program.

This course offers an introduction to the techniques of locating critical points in infinite-dimensional spaces in order to understand the variational formulations of mechanics in physics, including the principles of minimum action that give rise to the Euler-Lagrange equations and the Hamilton-Jacobi equation. Topics include: calculation in spaces of functions; necessary conditions; change of variables-- Hamilton-Jacobi; sufficient conditions.

This course is a continuation of MA1101 Linear Algebra I. The course presents more advanced topics and concepts in linear algebra. A key difference from MA1101 is that there is a greater emphasis on conceptual understanding and proof techniques than on computations. Major topics: matrices over a field; determinant; vector spaces; subspaces; linear independence; basis and dimension; linear transformations; range and kernel; isomorphism; coordinates; representation of linear transformations by matrices; change of basis; eigenvalues and eigenvectors; diagonalizable linear operators; Cayley-Hamilton Theorem; minimal polynomial; Jordan canonical form; inner product spaces; Cauchy-Schwartz inequality; orthonormal basis; Gram-Schmidt Process; orthogonal complement; orthogonal projections; best approximation; adjoint of a linear operator; normal and self-adjoint operators; orthogonal and unitary operators.

Topics include elementary properties of integers; functions and their behavior; an introduction to recursion; algorithms and complexity; graphs including Euler’s Theorem; shortest path algorithm and vertex coloring; trees - applications include problem solving and spanning trees; directed graphs including networks; dynamic programming; codes and cyphers - with Hamming codes and RSA.