Chord diagrams and BPHZ subtractions

The combinatorics of the BPHZ subtraction scheme for a class of ladder graphs for the three point vertex in $\phi^3$ theory is transcribed into certain connectivity relations for marked chord diagrams (knots with transversal intersections). The resolution of the singular crossings using the equivalence relations in these examples provides confirmation of a proposed fundamental relationship between knot theory and renormalization in perturbative quantum field theory.Comment: 12 pages, 5 Postscript figures, LaTex 2

Analysis of existing mathematics textbooks for use in secondary schools.

Thesis (Ed.M.)--Boston University Thesis (M.A.)--Boston Universit

Counting cusped hyperbolic 3-manifolds that bound geometrically

We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.Comment: 17 pages, 7 figures; to appear in Transactions AM

Hyperbolic low-dimensional invariant tori and summations of divergent series

We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coefficients that grow at most as a power of a factorial and a remainder that to any order N is bounded by the (N+1)-st power of the argument times a power of $N!$. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theoryComment: 32 pages, 5 figure