An algebraic Birkhoff decomposition for the continuous renormalization group

This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited

The Universal coefficient theorem for cohomology

This project is an expository survey of the Universal Coefficient Theorem for Cohomology. Algebraic preliminaries, homology, and cohomology are discussed prior to the proof of the theorem

Ore's theorem on subfactor planar algebras

This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics (2000). We deduce a link between combinatorics and representations in finite group theory.Comment: 14 pages. It reproduces some preliminaries of arXiv:1702.02124 and arXiv:1703.04486, for being self-containe

Scaled-free objects II

This work creates two categories of "array-weighted sets" for the purposes of constructing universal matrix-normed spaces and algebras. These universal objects have the analogous universal property to the free vector space, lifting maps completely bounded on a generation set to a completely bounded linear map of the matrix-normed space. Moreover, the universal matrix-normed algebra is used to prove the existence of a free product for matrix-normed algebras using algebraic methods.Comment: 46 pages. Version 4 fixed a few minor typos. Version 3 added matricial completion; fixed an arithmetic error in Example 3.5.10. Version 2 added a preliminaries section on weighted sets and matricial Banach spaces, incorporating much of "Matricial Banach spaces" in summary; fixed a domain issue in Lemma 3.3.2; simplified Examples 3.5.10 and 4.11; added more proofs to Sections 4 and