Orientations, modular properties and the Witten genus

This thesis is the recollement of the three research papers I wrote together with my advisor, Prof. Domenico Fiorenza and my Ph.D. student colleague Eugenio Landi, during my three year Ph.D. fellowship at the University of Rome Tor Vergata. Thought their subjects are actually interrelated, the three articles do not fall under a single mathematical topic, so what follows should be read as a personal academic research path to understand phenomena and prove results that occur in the various mathematical areas that mostly caught my interest over these three years. The first paper, An exposition of the topological half of the Grothendieck-HirzebruchRiemann-Roch theorem in the fancy language of spectra, aims to explain (or demystify, to use the words from the referee) pushforwards and orientations in general cohomology from the point of view of spectra in order to derive a purely topological/spectral version of the famous Grothendieck-Hirzebuch-Riemann-Roch theorem from the properties of the ∞-category of spectra. This article is in press for Expositiones Mathematicae. The second paper, A very short note on the (rational) graded Hori map, contains a description of the graded Hori map in the rational homotopy theory approximation of T-duality. This article is in press for Communications in Algebra. Finally, in the most recent paper, The (anti-)holomorphic sector in C/Λ-equivariant cohomology, and the Witten class, we introduce a subcomplex of the C/Λ-equivariant Cartan complex of a space, called the antiholomorphic sector, which allow us to prove that the Witten class Wit(X) appears in a formal localization formula for the conformal double loop space Maps(C/Λ, X) of a rationally string manifold X. This article has received a first positive report from Journal of Geometry and Physics and its revised version is currently under review.