Some basic problems and results from Invariant Theory

Invariant theory deals with properties of symmetries - actions of groups on sets of objects.

It has been slower to join its sibling mathematical theories in the computer science party, but we now see it more and more in studies of computational complexity, pseudorandomness, and the analysis of algorithms.

I will discuss some basic questions and results of this field, and how they lead to fertile interaction with CS (in which symmetries occur perhaps more often than you think). In particular, I will give the necessary background on invariant rings of linear actions that is used in the recent works on Operator Scaling and on Brascamp Lieb inequalities from my joint works wih Garg, Gurvits and Olivera).

No special background will be assumed.

PS If you need more motivation - Invariant Theory is beautiful! Here is Hermann Weyl's poetic quote about its origins:
"The theory of invariants came into existence in the middle of the nineteenth century somewhat like Minerva:
a grown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head."