#### Degree

Doctor of Philosophy

Mathematics

#### Supervisor

Professor Masoud Khalkhali

#### Abstract

In the first part of this thesis, a noncommutative analogue of Gross' logarithmic Sobolev inequality for the noncommutative 2-torus is investigated. More precisely, Weissler's result on the logarithmic Sobolev inequality for the unit circle is used to propose that the logarithmic Sobolev inequality for a positive element $a= \sum a_{m,n} U^{m} V^{n}$ of the noncommutative 2-torus should be of the form $$\tau(a^{2} \log a)\leqslant \underset{(m,n)\in \mathbb{Z}^{2}}{\sum} (\vert m\vert + \vert n\vert) \vert a_{m,n} \vert ^{2} + \tau (a^{2})\log ( \tau (a^2))^{1/2},$$ where $\tau$ is the normalized positive faithful trace of the noncommutative 2-torus. A possible approach to prove this inequality for arbitrary positive elements will involve a noncommutative multinomial expansion and seems to be exceedingly complicated. In this thesis the above inequality is proved for a particular class of elements of the noncommutative 2-torus. In the second part of this thesis, the scalar curvature of the curved noncommutative 3-torus is studied. In fact, the standard volume form on the noncommutative 3-torus is conformally perturbed and the corresponding perturbed Laplacian is analyzed. Then using Connes' pseudodifferential calculus for the noncommutative 3-torus, the first three terms of the small time heat kernel expansion for the perturbed Laplacian are derived. Moreover, by using the third term of this expansion and the Cauchy integral formula, the scalar curvature of the curved noncommutative 3-torus is defined. Finally, proving a rearrangement lemma, the scalar curvature is computed and an explicit local formula that describes the curvature in terms of the conformal factor is given.

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