Coordinator: Takashi Tsuboi, Faculty of Science Navigator: Yoshiko Ogata, Faculty of Science 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.