In this talk we will introduce the notions of join, slice, (co)limits in the context of $\infty$-categories. We will also discuss some basic properties relevant to these constructions.

This talk is based on my joint paper with Masoud Khalkhali and Ali Moatadelro (arXiv:1610.04740). First I will recall Gilkey’s theorem on asymptotic expansion of heat kernels for the special case of Laplacians. I will also introduce the noncommutatvie 3-torus (NCT3) and then I will conformally perturb the standard volume form on it. Then the corresponding perturbed Laplacian will be discussed, and using Connes’ pseudodifferential calculus, I will define the scalar curvature of NCT3. Finally, introducing a rearrangement lemma I will compute an explicit formula for the scalar curvature of the curved noncommutative 3-torus.