Abstract: We will write down a complete classification of all Clifford algebras of the form $Cl(n,0)$ and $Cl(0,n)$. From here, we will look at a theorem that relates periodicity of Clifford algebras with Bott periodicity in K-theory. Next, we will consider the Pauli-spin and Dirac matrices and see how they generate important algebras such as the real quaternion algebra, the complex quaternion algebra, the real Dirac algebra and higher order Clifford algebras. From here, we will start a discussion about the Dirac operator and Dirac equation, and subsequently introduce important examples such as the Pauli-Dirac and Dirac-Yukawa operators.

Abstract: We establish connections between various algebras and module categories that can be associated to a finite directed graph. These connections involve the singularity category for certain finite dimensional algebras, von Neumann regular algebras, Leavitt path algebras, $C^*$-graph algebras, AF algebras, some ideas from symbolic dynamics, and the space of Penrose tilings as a special case.