REU Meeting - 2026-06-22

This following is a brief summary of our research meeting on 2026-06-22.

Meeting summary

We outlined our basic plan for the summer, which roughly breaks down into three phases:

• Phase I: Get familiar with classical Lie theory, by learning about matrix Lie groups and their associated Lie algebras
• Phase II: Learn the more general definitions of Lie groups and algebras, and general facts about the categories of each
• Phase III: Learn and investigate the structures of the categories of Lie groups and Lie algebras in relation to other categories; e.g., what is special about the category of Lie groups? how is it intrinsically defined as a category in its own right?

Tasks for next meeting

• Learn about classical Lie groups by reading Chapter 1 ("Matrix Lie Groups") in Hall
  • Check out the following exercises in Section 1.6: 1, 2, 3, 7, 11, 16, 17
• Learn about the classical matrix exponential function by reading Chapter 2 ("The Matrix Exponential") in Hall
  • Check out the following exercises in Section 2.6: 2, 4, 8, 9

References

Brian C. Hall, Lie groups, Lie algebras, and representations (Chapters 1 and 2)