2024 Spring TP403: Introduction to Geometry
Differential equations are mathematical tools for modeling natural and social phenomena. The solution to the differential equation is a differentiable function, which gives simulated results as well as estimated results. We can find such solutions both in mathematical formulas and numerical data. To interpret the solution, we need to understand the definition of properties of derivatives and integrals. In this course, we learn basic concepts of differentiation and interaction on single variable functions and various techniques for solving differential equations. We also emphasize modeling real-world problems using differential equations and how to formulate proper research questions and answer those questions mathematically. We will learn how to use Matlab to solve those problems.
Geometry, the oldest subject of study in mathematics, is not only widely studied by modern mathematicians, but also applies to numerous scientific areas. It is particularly important in Physics, Mechanical Engineering, Computer Science, Bioinformatics, Economy, Numerical Analysis, and much more. Geometry consists of various research areas of mathematics, such as Differential Geometry, Differential Topology, Lie groups, and Lie algebras. In this course, we will study the essential results of Differential Geometry and Differential Topology. Meanwhile, we will cover some basic properties of Topology which are necessary to understand the context of these subjects.
Differential Geometry is a study of differential structures of manifolds and their properties. It lays out the foundation of many scientific areas such as Newtonian mechanics and general relativity in Physics. The subjects in Differential Geometry are also used in motion control in Mechanics and Robotics, and visual analysis in Computer Science.
Differential Topology is a study of differential manifolds using their topological properties. While Differential Geometry focuses on manifolds as rigid space, Differential Topology allows topological elasticity on manifolds because it studies differential functions defined on manifolds and the deformation of manifolds.
Lie groups are manifolds with group structures. It is a vital subject to understand contemporary mathematics. One of the most famous examples is the matrix group, and it is important in research areas such as optimization, molecular symmetry, rigid motion, etc.
This course aims at students who are willing to study mathematics at a deeper level. The prerequisite is Vector Calculus. I will deliver mathematical concepts and their properties as thoroughly as possible. I will require students to actively think about how such concepts can develop to research questions of their own. After completing this course, a successful student can
This class is not a serious math-major class, but my lecture style will reveal how we should learn mathematics. I strongly believe that all students who orient themselves toward scientific areas must be well-trained mathematically. The class will be perfect for students who seek more than Calculus but have no idea where to start.
There are some difficulties that rises from the nature of this class. First, there are so many subjects in Geometry and it is impossible to go through all the details in a single semester. Second, almost all content in Geometry requires some background in Linear Algebra, Analysis, and Topology. So I will introduce the basic concepts in Topology and Analysis before we cover Geometry. I will put a few questions every week and we will investigate mathematical ideas related to those questions. I will cover one or two theorems each week and prove them as rigorously as possible. You should write out what you think about the question and subject and derive your own answers. I expect that at the end of the semester, you will get an idea of how to write mathematical arguments correctly.
In 1872, Felix Klein (1849-1925) proposed a new observation of geometry, known as the Erlangen program. What he proposed is the following. Geometry should be understood by the symmetry of space under the group action. (The full text of Klein about the Erlangen program can be found here.) For example, in order to understand the structure of the torus, one should look at the $\mathbf Z^2$-action on the two-dimensional space $\mathbb E^2$. This idea was truly revolutionary. The main goal of this course is to share the same spirit in the Erlangen program. Once we cover the basic developments of classical results, I will introduce examples of symmetric space through examples of matrix groups.
The class is run in a PBL (project-based learning) style. There will be five project topics. You will be assigned to a group for each topic. Each group consists of 4-5 members. There will be a worksheet in each class. You should seat near your team members and work together to answer the questions in the worksheet. The questions in the worksheets ask for careful reading and thoughtful thinking. Although I will try to give you enough time to answer the question, it is highly recommended to read the worksheet for the next week ahead and prepare it before the class begins.
During the class, you should write the answers on the worksheet by hand-writing. After each class, you should submit your worksheets which will be returned back to you once I record your activities. Your participation in the class and the writing on worksheets will cover 70% of your final grade. Do not worry too much about minor mistakes or leaving some blank answers, because I will not be harsh on those issues.
There will be reading material for each class which is crucial for the worksheets. I will print and hand out them to you at the beginning of the class, so you do not have to print them out for yourself. However, it is highly advised to read the material beforehand. Reading materials are collected from books listed in the reference section of each project.
If you have an official situation to miss a class, you should report it beforehand. Otherwise, I will not make up for your lost grade. Any work done on the worksheet outside the class does not count for the grade. If you have a valid reason for missing the class, I will make up the missing grade proportionally.
At the end of the semester, you should write an essay individually based on what you have learned in class. The report should be written within 3 pages on A4-sized sheets. The main component of grading for the essay is how much you condensed your ideas into the report. That means you should use every margin of pages efficiently so that you can deliver as much content as possible.
Academic integrity is fundamental to this course. It requires all students to maintain honesty, fairness, and responsibility in their academic work. Upholding academic integrity entails submitting original work, properly crediting sources, and adhering to ethical standards in all assignments and assessments. Any form of academic dishonesty, including but not limited to plagiarism and cheating, is strictly prohibited and will result in appropriate disciplinary action.
You are required to attend every class and participate in discussions actively. You should write your answers to your own worksheets and participate in group discussions as well. At the end of the semester, you should write the final reports based on topics covered in the class.
In-class activities are graded by groups. Worksheets and final reports are graded as a group. The percentages of each component to the final grade are the following.
If you miss more than 7 classes, you will fail the class.
If you don’t submit the final report, you will fail the class.
| Score | Tags |
|---|---|
| 90 or above | A+ |
| 80 or above | A0 |
| 75 or above | A- |
| 65 or above | B+ |
| 55 or above | B0 |
| 50 or above | B- |
| 40 or above | C+ |
| 30 or above | C0 |
| 25 or above | C- |
| 15 or above | D+ |
| 5 or above | D0 |
| below 5 | D- |
| 0 | F |
Munkres, Analysis on Manifold
We will consult the following texts. Students are not required to have all of those.
Pressley, Elementary Differential GeometrySpivak, Comprehensive Introduction to Differential Geometrydo Carmo, Riemannian GeometrySpivak, Calculus on ManifoldGuillemin and Pollack, Differential TopologyMunkres, TopologyBaker, Matrix GroupsCurtis, Matrix Groups| --- | --- | --- |