This is a course on Lie groups, Lie algebras and their representations, with an emphasis on the later two topics. In particular, we will mostly focus on algebraic and combinatorial aspects appearing in the structure and representation theory of semisimple Lie algebras and only briefly deal with Lie groups and their geometric properties.
Representation Theory is a broad subject with many connections/applications (to physics, number theory via Langlands program etc.). In this class we will study Lie algebras, with a focus on understanding the structure and representation theory of semi-simple Lie algebras, by understanding their root systems. This should be useful for students working many areas of algebras, including number theory (for example, understanding root systems should allow you to understand how to work not only with GL_n objects (like automorphic forms etc.), but objects on more general groups like Sp_{2n} etc. )
The focus of the class is on
0) brief introduction and motivation (including relation to Lie groups)
1) understanding the structure theory, classification and representation theory of semisimple Lie algebras over C. (approximately sections 2-4 of Humphreys)
topics including:
1.1) structure theory for semisimple Lie algebras over C
1.2) combinatorics of root systems
1.3) finite dimensional representation of ss Lie algebras and their characters.
2) basic facts about compact Lie groups and their representation theory (Brocker, tom Dieck )
3) advanced topics for Lie algebras (focus on BGG category O and the Kazhdan-Lusztig conjecture - Humphreys, BBG category O book)

Basic algebra and representation theory including Math 301, 302, 403 or equivalent.
If you haven't taken Math 403 and are interested in this course, consider looking at Chapters 2, 3 and 4 of Etingof's book:
https://math.mit.edu/~etingof/reprbook.pdf
Differential geometry is not needed; it can be useful to understand some of the motivation.
Math 501 and/or the maturity gained in that class is useful, but not needed.
Feel free to contact the instructor if you are unsure about your prerequisites.

There will be some homework (less than for the usual 300-400 level class) and we will decide between midterm, presentation and/or term paper. There will likely not be a final exam.

A standard reference for Lie algebras and representation theory is
Introduction to Lie Algebras and Representation Theory, James E. Humphreys, GTM9, third edition
This book does not deal with Lie groups. There are many other book dealing with both Lie algebras and Lie groups, including:
Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Brian C. Hall, GTM222, Springer
Representation theory: A First Course, William Fulton and Joe Harris, GTM
Groupes et algèbres de Lie, Bourbaki
Lie groups beyond an introduction, Anthony W. Knapp
Representations of Compact Lie Groups, Theodor Brocker, Tammo tom Dieck, GTM
Some related/advanced books:
(physics) Lie Algebras in Particle Physics, Howard Georgi
(quantum groups) A guide to quantum groups, Vyjayanthi Chari and Andrew Pressley, Cambridge University Press
(advanced) Representations of Semisimple Lie Algebras in the BGG Category O, James E. Humphreys, American mathematical Society

in person teaching