Mathematics today has extensive history and mathematics is used to make objective evaluations and rational decisions. With each new phenomenon encountered by humankind, progress in mathematics has risen to the challenge. The logic of mathematics and the concept of numbers have been questioned time and time again throughout history. Mathematical innovations such as the establishment of coordinates, the discovery of complex numbers, and the founding of calculus are now taught in high school, but mathematical research continues to advance in the axioms of geometry or the concepts of continuity or transformation, etc.

While mathematics has developed creatively and innovatively seeking consistency within itself, it has also made leaps and bounds in relation to society. Twentieth-century mathematics made huge progress through abstraction and formalization, enabling its application across a broader range of issues. Mathematics has also had an enormous impact as the scientific basis to how phenomena are perceived.

In its interaction with society, mathematical innovation has progressed in the form of sublating existing research results. Mathematical research has even on occasion prepared in advance for its necessary application, and society has come to expect mathematics to be applied to real problems. The concepts of probability and statistics in modern society now play significant roles in making objective evaluations and rational decisions. As math-based information technology makes major advancements, it is increasingly necessary for people who observe, analyze, and integrate phenomena and those who develop mathematical tools to work together to build high quality mathematical models.

In these lectures, leading mathematical scientists will explore the history of mathematical innovation and how the power of accumulated tradition has been used to understand new phenomena and to address social problems, conveying the rich creativity that lies in mathematics today.

**The Shape of Mathematics**

#### Lecture 1 (Apr. 10) Mathematics:The Power of Tradition

#### Lecture 4 (May 8) Mathematics:The Power of Ideas

##### Kazuo Okamoto (National Institution for Academic Degrees and University Evaluation)

Overview: These lectures will review the process of the creation of mathematics in terms of the shape of mathematics. The material will be limited to content taught in high school based on the belief that a contemplation of how mathematical ideas have been created is significant and beneficial for students in both humanities and science courses. Lecture 1 will provide an overall introduction and consider the role of mathematics, and Lecture 2 will introduce the fun side of mathematics through actual examples of numbers and figures.

**Between Integers and Rationals**

#### Lecture 2 (Apr.17) Between Integers and Rationals: Part 1

#### Lecture 3 (Apr.24) Between Integers and Rationals: Part 2

##### Shihoko Ishii (Faculty of Science)

Overview: These lectures will focus on some problems of integers and rationals to clarify differences in the view points of contemporary mathematics and mathematics taught in high school.

**The Development of the Stochastic Process Model**

#### Lecture 5 (May 15) The Game of Sugoroku as a Stochastic Process Model

#### Lecture 6 (May 22) The Idea of Stochastic Differential Equations

##### Shigeo Kusuoka (Faculty of Science)

Overview: The stochastic process model is used in engineering, economics, finance, and other various fields as a tool to represent the indeterminate evolution of something over time. Lecture 1 explains the discrete-time Markov process model, which is a little complicated sugoroku game. The idea that time and space are continuous in this model may seem obvious, but this was not an easy task for mathematics. Lecture 2 will chronologically examine the ideas of mathematicians such as Paul Levy, Andrey Kolmogorov, and Kiyoshi Ito who created the continuous-time Markov process.

**The Past and Future of Statistics**

#### Lecture 7 (May 29) The History of Statistics

#### Lecture 8 (Jun. 5) The Future of Statistics

##### Akimichi Takemura (Faculty of Engineering)

Overview: The English word statistics is derived from the word ‘state,’ and thus statistics historically developed as a means to quantify the condition of a society or nation. This then became linked to the theory of probability, and by the early 20th century statistics was established as a general method to handle quantified data and data including uncertainties. As the world today has been described as the age of big data, with a sharp increase in the volume of data acquired from society and natural phenomena, statistics has been attracting more attention and the required methodologies are changing. These lectures will explain the history of statistics and its future role in society.

**Exponent Functions and Differential Equations**

**Lecture 9 (Jun. 12) Compound Interest and Exponential Functions**

**Lecture 10 (Jun. 19) Abstraction of the Equation Representing Residual Radioactivity**

**Lecture 11 (Jun. 26) How to Solve Differential Equations by Abstract Method**

##### Yoshikazu Giga (Faculty of Science)

Overview: Mathematics may often be described as abstract and unrealistic, but this abstract nature is what drives its widespread application. These lectures will explain how the abstract theory founded by Professor Kosaku Yosida, an alumnus of the University of Tokyo, has innovated the studies of differential equations.

**The Financial Crisis and Mathematical Sciences: Reflections of a Former Deputy Governor of the Bank of Japan**

**The Financial Crisis and Mathematical Sciences: Reflections of a Former Deputy Governor of the Bank of Japan**

**Lecture 12 (Jul. 3)The Global Financial Crisis and Mathematical Way of Thinking**

**Lecture 13 (Jul. 10)Economic Theory of Optimism and Pessimism**

##### Kiyohiko Nishimura (Faculty of Economics)

Overview: People tend to think that the succinct, abstract world of mathematical sciences lies opposite to the complex market economy of the real world. Predictions of complex market forces are made based on past data and the assumption of a stable structure. However, this structure underwent massive upheavals during the period surrounding the global financial crisis.Past experiences were rendered useless in making future predictions and the financial markets swayed between optimism and pessimism. It was then that the mathematical mind that considers how humans behave rationally in practically unknown situations became important in addressing the financial crisis. The first half of the lecture will explain how the unprecedented situation came to rise in the financial market, and the second half will provide a mathematical science perspective regarding the behavior of economic actors in a situation of fundamental uncertainty (often described as ‘unknown unknowns’).