# Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods

## TL;DR

This paper introduces the Farrell-Jones Conjecture in algebraic K-theory, discusses its current status, and reviews the main tools—controlled algebra and trace methods—used to approach it.

## Contribution

It provides a concise overview of the conjecture, its significance, and the primary techniques employed in recent research.

## Key findings

- Survey of the current status of the Farrell-Jones Conjecture
- Explanation of controlled algebra and trace methods
- Illustration of applications in algebraic K-theory

## Abstract

We give a concise introduction to the Farrell-Jones Conjecture in algebraic $K$-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods.