REU Meeting - 2026-07-29 1
This following is a brief summary of our research meeting on 2026-08-05.
Meeting summary
Today's meeting was very brief. With only a week or so left in the summer, it's time to focus on last odds-and-ends, writing up what we've done and thinking about the poster.
Tasks for next meeting
For completeness, we should at least find answers to the following:
- What are Lie’s three theorems?
- Can a given group G always be equipped with a Lie group structure? Can it be equipped with different Lie group structures?
- Which topological spaces can be equipped with a Lie group structure?
- Can non-isomorphic Lie groups have isomorphic Lie algebras?
- What is Lie integration?
- Is there a classification of all Lie groups?
And then the next level of abstraction (which is actually needed for some of the higher-end physics applications and even some automatic-proof-checking type shenanigans) is to tiptoe into one (or more) of following topics:
- Lie algebra objects (analogous to group objects internal to a Cartesian closed category)
- Lie groupoids and Lie algebroids (Maybe even just starting with groupoids and the idea of “oidification”)
- Higher “dimensional” versions of Lie’s three theorems, e.g., analogues for Lie groupoids, Lie 2-groups, etc.
- The Lie operad (which also involves learning what an operad is)
For the immediate future, let’s focus on the first list of questions. Then we can investigate some of the second batch and see what seems most exciting.
Also continue working on writing up what we've learned, and start thinking about how you'd like to arrange that info on a poster.