# The Theory of Causal Fermion Systems

## Overview of Mathematical Aspects

### Basic Definitions

The general definitions of a causal fermion system and the causal structure are given.

### Generalizations and Special Cases

An overview of different generalizations and specializations is given.

## Inherent Structures I

A causal fermion system involves structures which are *inherent* in the sense that they do not give additional input but merely give information already encoded in the causal fermion system a useful name. Here is an overview of the most important inherent structures:

### Spin Spaces and Physical Wave Functions

These structures bear similarity with a topological vector bundle, with the physical wave functions as sections thereof.

### The Fermionic Projector

The kernel of the fermionic projector induces relations between space-time points. It is also the kernel of an integral operator.

## Analytic Structures

### The Euler-Lagrange Equations

Critical points of the causal action principle satisfy the Euler-Lagrange equations.

### The Linearized Field Equations

Linearizing families of solutions of the Euler-Lagrange equations gives rise to the linearized field equations.

### Existence Theory for the Linearized Field Equations

The existence theory for solutions of the linearized field equations are outlined. The methods are based on energy estimates.