Abstract: SPECIAL FIELDS POSTDOCTORAL FELLOW TALK

I will first give a brief introduction to the metric aspects of noncommutative geometry and ideas from spectral geometry that have played an important role in their development. Noncommutative tori $\mathbb{T}_\theta^n$ are important $C^*$-algebras that have been studied vastly in noncommutative geometry due to their importance, among which is their role in the study of foliated manifolds. In a recent seminal paper, A. Connes and P. Tretkoff proved the Gauss-Bonnet theorem for the noncommutative two torus $\mathbb{T}_\theta^2$ equipped with its canonical conformal structure. In a series of joint works with M. Khalkhali, we extended this result to general translation invariant conformal structures, computed the scalar curvature, and proved the analog of Weyl's law and Connes' trace theorem for $\mathbb{T}_\theta^2$. Our final formula for the curvature of $\mathbb{T}_\theta^2$ precisely matches with the one computed independently by A. Connes and H. Moscovici. A purely noncommutative feature is the appearance of the modular automorphism from Tomita-Takesaki theory in the computations and the final formula for the curvature. In this talk I will review these results and will then turn to part of our recent work on the curved geometry of noncommutative four tori $\mathbb{T}_\theta^4$. That is, I will explain the computation of scalar curvature and the analog of the Einstein-Hilbert action for $\mathbb{T}_\theta^4$, and show that metrics with constant curvature are critical points of this action.