8
votes
2answers
320 views

What is the physical difference between states and unital completely positive maps?

Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into ...
13
votes
1answer
795 views

Lagrange's theorem for Hopf algebras

Under what conditions is a Hopf algebra free over any of its sub-Hopf algebras? I am reading "Hopf algebras and their actions on rings" by Susan Montgomery, specifically chapter 3. Lagrange's theorem ...
24
votes
4answers
1k views

Algebraic P vs. NP

I recently attended a lecture where the speaker mentioned that what he was talking about was connected to the algebraic version of the $P$ vs. $NP$ problem. Could someone explain what that means in a ...
13
votes
1answer
725 views

A synopsis of Adyan’s solution to the general Burnside problem?

Where can I find a high-level overview of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent? Additionally: If possible, would an expert ...
26
votes
6answers
3k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...