- Professor Takeshi Kawazoe
Introduction of the discussion
In general, a set is a “collection of things”. However, whether or not the “thing” given here belongs to a certain set is determined by an objective condition independent of the determiner. A set is just a “gather” and no mathematical order is required.
On the other hand, space is almost a synonym for “set”. However, in mathematics, “space” implies a special property. It is not just a “collection of things”, but a “collection of things” with a “structure” as a whole. Examples are “vector space” and “Euclidean space”.
In particular, the topological space means a “set” in which a structure called topology is incorporated. The phase in mathematics is not a sense of distance such as “closeness” or “remoteness”, but a “grouping” based on mathematical properties.
Finding the problem
Then, what is the concept of the “manifold” that is the main subject of modern geometry?
Argument
To imagine a manifold, it can be recognized locally as an expansive space, but it is too large to understand the whole picture. In other words, It is easy to understand if you imagine something like the universe for modern people.
Here, let us consider a means of grasping the whole manifold. Similar to creating a map (atlas) that covers the entire earth by connecting segmented images transmitted from satellites, a “map book” that aggregates local maps of all segments over all segments is created. If it can be created, it will be the “world map”.
A phase space provided with such a “map book” is called a manifold. Now consider how to analyze functions on manifolds. A method is used in which manifolds are segmented and each segment is once projected onto a phase space (Euclidean space) via a map book.
Conclusion
Therefore, I would like to study the mathematical properties of Lie groups, which are differentiable manifolds with group structure.
Examining the conclusion
By studying with Professor Takeshi Kawazoe, who specializes in the mathematical properties of Lie groups, I am sure that I could obtain various perspectives regarding my research and it would open up new horizons for me.
Therefore, I believe that Keio University’s Faculty of Policy Management is the most suitable place for pursuing my research and social contributions, and I am aspiring to enter your school and study in your laboratory.
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