Mathematics

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.

This introductory course covers mathematical topics closely related to computer science. Topics include: logic, sets, functions, relations, countability, combinatorics, proof techniques, mathematical induction, recursion, recurrence relations, graph theory, and number theory. The course emphasizes the context and applications of these concepts within computer science. Prerequisites: No prior programming experience is assumed. 

This course provides a basic introduction to mathematical theory and methods in biology, with enough scope to enable the student to handle biologically phrased problems. Topics covered include population models with discrete or continuous time, pharmacokinetics and -dynamics, qualitative analysis of systems of differential equations, modelling of the spread of infectious diseases, bifurcations, limit cycles, and excitable media with applications to, e.g., predator-prey models, spatial methods with application to diffusion, and nerve conduction.

The course covers properties of the real numbers R: completeness axiom, Cauchy sequences, cardinality of rational, and irrational numbers; Topology in Rn: open and closed sets, p-norms, convergence, compactness, the Bolzano-Weierstrass theorem, and connected sets; Continuous functions in Rn: intermediate value theorem, min-max theorem, uniform continuity, continuity of inverse functions, implicit function theorem; Convergence of sequences and series of functions: pointwise, absolute, and uniform convergence, term wise differentiation and integration, power series;  and examples of applications to selected topics relevant to mathematical research at the center for mathematical sciences. Admission to the course requires at least 30 credits in mathematics including knowledge corresponding to MATA31 Analysis in One Variable, 15 credits, MATA32 Algebra and Vector Geometry, 7.5 credits and NUMA01 Computational Programming with Python, 7.5 credits.

In this course, students learn about vectors spaces, subspaces, bases, inner products, linear transformations, rank/nullity, matrices of linear maps, change of basis, eigenvalues/eigenvectors, Jordan normal form, diagonalization, and special classes of linear transformations and their matrices.

The course covers the translation between biology and mathematics; population models and spatial models, simulations: Deterministic versus stochastic simulations of mathematical models; weaknesses, strengths, and applicability; the Gillespie algorithm for stochastic simulations: Naive implementation and possible optimizations for large systems; cost functions; optimization methods including local optimization, thermodynamic methods, particle-swarm optimization, and genetic algorithms; and sensitivity analysis: Estimation of the uncertainty of determined parameter values. Strategies to achieve robustness. Admission to the course requires 90 credits Science studies, including knowledge equivalent to BERN01 Modelling in Computational Science, 7.5 credits or FYTN03 Computational physics, 7.5 credits and English 6/B. Admission to the course also requires knowledge in programming in Python equivalent to NUMA01, 7.5 credits or similar knowledge in Matlab, C++ or the like programming language.

The course is an introduction to vector calculus and a specialization of differential and integral calculus of functions of several variables. The course covers line and surface integrals; Green's formula, Gauss divergence theorem, and Stokes theorem; Basic potential theory. To be eligible for the course, 45 credits in courses in mathematics equivalent to MATA21 Analysis in One Variable (15 credits), MATA22 Linear Algebra 1 (7.5 credits), MATA21 Analysis in Several Variables 1 (7.5 credits), MATB22 Linear Algebra 2 (7.5 credits) and one of the courses NUMA01 Computational Programming with Python (7.5 credits) and MATA23 Foundations of Algebra, (7.5 credits) are required.