#### Thesis Format

Integrated Article

#### Degree

Doctor of Philosophy

Mathematics

#### Supervisor

Khalkhali, Masoud

#### Abstract

In noncommutative geometry, the metric information of a noncommutative space is encoded in the data of a spectral triple $(\mathcal{A}, \mathcal{H},D)$, where $D$ plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace $\textrm{Tr}(e^{-tD^2})$. In Chapter 2, we will compute the Ricci curvature of a curved noncommutative three torus. The computation is done for both conformal and a non-conformal perturbation of the flat metric. By applying Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second density of the heat trace expansion for the perturbed Laplacians on both functions and $1-$forms. On the other hand, in noncommutative geometry one also wants to get a good notion of an action functional which depends only on the spectrum of $D$, called spectral action functional. It is known that such a functional can be expressed as $\textrm{Tr(f(D))}$ for some function $f$. In chapter 3, we show that the von Neumann entropy, average energy, and negative free energy of the Gibbs state of the second quantized Dirac operator $d\Gamma D$ has a spectral action functional interpretation of the original Dirac operator $D$. To be able to carry on the computations, we have to incorporate the chemical potential $\mu$. All those spectral action coefficients can be given in terms of the modified Bessel functions.

#### Summary for Lay Audience

In mathematics one can describe the topological properties of a compact Hausdorff space $M$ via the algebra $C(M)$ of all continuous complex-valued functions over $M$, which is a commutative $C^*-$algebra. By analogy, a noncommutative $C^*-$algebra encodes all the topological information of a noncommutative space.

In Section 1.1 and 1.2, we shall briefly review the definition of a spectral triple which encodes the geometric information of a noncommutative space. Then in Chapter 2, we shall recall the definition of the noncommutative three tori. Then we will compute the Ricci density of curved noncommutative three tori under the conformally flat metric and a specific non-conformal metric by analyzing the spectral properties of the Laplace operators.

In Section 1.3 and 1.4, we shall give a brief introduction to the spectral action principle and the second quantization. A spectral action functional is an additive functional with respect to direct sum of the spectral triples and the second quantization is one method to describe a multi-particle system in the quantum statistical mechanics. We will show, in Chapter 3, that the entropy and energy of the Gibbs state of the second quantized Dirac operator can be interpreted as spectral action functionals of the original Dirac operator. Moreover, we will explicitly compute all the spectral action coefficients for the above quantities in both Bosonic and Fermionic cases.