Mathematics

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.

This course provides research training for students through the experience of belonging to a specific laboratory at the University of Tokyo. Students carry out an original research project under the guidance of assigned faculty members. Through a full-time commitment, students will be able to improve their research skills by applying the basic principles and knowledge from the literature related to the research questions, and by developing the skills to collect, interpret, and critique data in order to resolve a research question or evaluate a design for a research project. At the conclusion of the program, students submit their final work (paper, presentation, report etc.) as instructed by their lab supervisors

The course provides theoretical foundations for key concepts appearing in analysis: open sets, closed sets, compact sets, connected sets, continuous maps. This is done in the context of metric and topological spaces.  

Recently, deep leaning has been the successful tool for various tasks of data analysis. Also, the theoretical structure of deep neural network (DNN) has been clarified gradually. On the other hand, such theoretical structure is crucially based on elementary linear algebra. Thus it is worth studying machine learning from scratch, that is, elementary linear algebra. Upon completion of the course, students will be able to understand topics on machine learning including elementary deep neural network and reservoir computing.

The course covers a broad range of optimization algorithms and models. Topics include linear programming, simplex algorithm, duality, sensitivity analysis, two player zero-sum games, network optimization, minimum cost flow, network simplex algorithm, integer programming, branch and bound methods, cutting plane methods, and dynamic programming.

This course includes knowledge of common methods in asymmetric encryption, as well as possible attacks in faulty implementations of these methods: RSA, El-Gamal, Diffie-Hellman-Key-Exchange, elliptic curves, and selected methods of Post-Quantum-Cryptography. Students who completed this course possess profound knowledge of cryptographic methods. They are able to correctly and securely use cryptographic protocols. They are proficient in verifying the security of One-Way-Functions and (Pseudo-)Random-Number-Generators. Furthermore, they are able to recognize and avoid typical mistakes in asymmetric encryption.

The course covers some of the most prominent tools in modelling and simulation. Both deterministic and stochastic models are covered. These include mathematical optimization, the application of sophisticated mathematical methods to make optimal decisions, and simulation, the playing-out of real-life scenarios in a (computer-based) modelling environment. Topics may include formulation of management problems using linear/nonlinear and network models (these could include binary, integer, convex, and stochastic programming models) as well as solving these problems and analyzing the solutions; generating random variables using Monte Carlo simulation; discrete event simulation; variance reduction techniques; Markov Chain Monte Carlo methods. The course teaches students to use modelling and simulation computer packages.

This course introduces the mathematical formalism of quantum information theory. Topics include a review of probability theory and classical information theory (random variables, Shannon entropy, coding); formalism of quantum information theory (quantum states, density matrices, quantum channels, measurement); quantum versus classical correlations (entanglement, Bell inequalities, Tsirelson's bound); basic tools (distance measures, fidelity, quantum entropy); basic results (quantum teleportation, quantum error correction, Schumacher data compression); and quantum resource theory (quantum coding theory, entanglement theory, application: quantum cryptography). 

In this course, students are taught the foundational concepts of major stochastic fields and associated topics, including Statistics, probability, and combinatorics. The course is presented in “flipped-classroom” format, such that students are expected to learn concepts on their own, and then practice application in the classroom.