Yonsei University Logic Seminar Series (hosted by Byunghan Kim)

The Connes Embedding Problem, MIP*=RE, and Model Theory

Lectures given by Isaac Goldbring

Meeting time: Tuesdays at 6pm PST, January 4-February 22, 2022

Email Isaac Goldbring at isaac@math.uci.edu if you want access to the Zoom link.

#### Abstract:

The Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras.
Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras.
In early 2020, a group of computer scientists proved a landmark result in quantum complexity theory called MIP*=RE and, as a corollary,
gave a negative solution to the CEP. However, the derivation of the negative solution of the CEP from MIP*=RE involves several very
complicated detours through C*-algebra theory and quantum information theory.

In this series of lectures, I will present the "standard" derivation of the failure of CEP from MIP*=RE. In addition, I will present joint work with Bradd Hart
where we show how some relatively simple model-theoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while
simultaneously yielding a stronger, Gödelian-style refutation of CEP as well as the existence of "many" counterexamples to CEP.

I will assume no prior knowledge of operator algebras, quantum complexity theory, or model theory and will try to develop as much of the story from scratch as possible.

#### Lectures notes and videos

- Lecture 1 (January 4)
- Lecture 2 (January 11)
- Lecture 3 (January 18)
- Lecture 4 (January 25)
- Lecture 5 (February 1)
- Lecture 6 (February 8)
- Lecture 7 (February 15)
- Lecture 8 (February 22)

#### References