By transforming to momentum space, the integrals used to compute the Feynman weight of a graph can be simplified. After carrying out this process, I will compute the partition function of two simple systems, quantum mechanics on a circle and circle-valued quantum mechanics. Finally I will discuss how these methods easily generalize to quantum field theory in the case of a free scalar bosonic field.

A quasi-shift endomorphism is a unital normal *-endomorphism acting on a von Neumann algebra, of which tail and fixed point algebras coincide. Our purpose, in this presentation, is to discuss several asymptotic characterizations of quasi-shifts associated with representations of Cuntz algebras. Joint work with T. Wood.

I shall first derive the semi-classical approximation of Feynman path integrals and give several examples. This involves regularized determinants of differential operators in terms of spectral zeta functions. This idea will then be applied to a supersymmetric quantum system defined by the Dirac operator of a closed spin manifold and after some non-trivial algebraic manipulations will lead to a proof the Atiyah-Singer index theorem.

Failure of some (important) properties of a holomorphic mapping manifest themselves as degeneracies in the family of fibres of the mapping. Among these properties, openness and flatness are our main object of interest. The first goal in my thesis is to develop criteria such that first, they effectively (i.e., computationally) detect such degeneracies in the family of fibres, and second, they are applicable to the case of mappings with singular targets. For flatness, no such algorithms that work in the general setting of a singular target were known before. We prove that a mapping (with locally irreducible target) is flat (open) if and only if the fibre above the origin of the pullback by the blowing-up is not an (isolated) irreducible component. Algebraically, this flatness criterion reduces to the following straightforward prescription: compute the the local ring of the pullback, and check to see if the exceptional divisor is not a zero divisor. The second goal is to characterize different modes of such degeneracies. We take an invariant, called verticality index, as a gauge which measures the level of non-openness for mappings. We obtain some results about verticality index, specially on its behaviour and computation over singular targets.

If $k$ is a number field, $G_k=\mathrm{Gal}(\overline{k}|k)$ its absolute Galois group and $p$ a prime number, $p$-adic Galois representations $\rho: G_k\to GL_n(\mathbb{Q}_p)$ naturally arise in algebraic geometry, coming from the action of $G_k$ on etale cohomology groups of varieties defined over $k$. Fontaine and Mazur make a fundamental conjecture giving a precise characterization of those $p$-adic representations 'coming from algebraic geometry' in the above sense. In this talk we discuss consequences of the Fontaine-Mazur Conjecture for representations which are unramified at primes above $p$. After recalling results due to N. Boston and K. Wingberg providing evidence for the conjecture in the unramified case, we report on recent work on tamely ramified pro-$p$-extensions of $\mathbb{Q}$ by J. Labute.