Transformation and Individuation in Giordano Bruno's Monadology

The essay explores the systematic relationship in the work of Giordano Bruno (1548-1600) between his monadology, his metaphysics as presented in works such as De la causa, principio et uno, the mythopoeic cosmology of Lo spaccio de la bestia trionfante, and practical works like De vinculis in genere. Bruno subverts the conceptual regime of the Aristotelian substantial forms and its accompanying cosmology with a metaphysics of individuality that privileges individual unity (singularity) over formal unity and particulars over substantial forms without sacrificing a metaphysical perspective on the cosmos. The particular is individuated as a unique site of desire, continually transforming but able to entrain itself and others through phantasmatic ‘bonding’, the new source of regularity in Bruno’s polycentric universe. Bruno thus tries to do justice to the demands of intelligibility as well as transformative eros. The essay concludes with a note on Bruno’s geometry as it relates to his general conception of form

Krichever Maps, Faa' di Bruno Polynomials, and Cohomology in KP Theory

We study the geometrical meaning of the Faa' di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faa' di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.Comment: 16 pages, LaTex using amssymb.sty. To be published in Lett. Math. Phy

Fa\`a di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B^+(P(X)) in the non-commutative Connes-KreimerHopf algebra of planar rooted trees H, where B^+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A_P of\H. We describe all the formal series P such that A_P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H, organized into three isomorphism classes: a first one, restricted to a olynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Fa\`a di Bruno Hopf algebra. By taking the quotient, the last classe gives an infinite set of embeddings of the Fa\`a di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Fa\`a di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, togetherwith a non commutative version of this embedding.Comment: 23 pages, final version, to appear in Advances in Mathematic

The 1/2--Complex Bruno function and the Yoccoz function. A numerical study of the Marmi--Moussa--Yoccoz Conjecture

We study the 1/2--Complex Bruno function and we produce an algorithm to evaluate it numerically, giving a characterization of the monoid $\hat{\mathcal{M}}=\mathcal{M}_T\cup \mathcal{M}_S$. We use this algorithm to test the Marmi--Moussa--Yoccoz Conjecture about the H\"older continuity of the function $z\mapsto -i\mathbf{B}(z)+ \log U(e^{2\pi i z})$ on $\{z\in \mathbb{C}: \Im z \geq 0 \}$, where $\mathbf{B}$ is the 1/2--complex Bruno function and $U$ is the Yoccoz function. We give a positive answer to an explicit question of S. Marmi et al [MMY2001].Comment: 21 pages, 11 figures, 2 table

Unsolvability of the isomorphism problem for [free abelian]-by-free groups

The isomorphism problem for [free abelian]-by-free groups is unsolvable.Comment: added reference to a paper by Bruno Zimmermann containing a similar result for (free abelian)-by-surface group