March 21, 2016 Room K 9509 (map) The room is equipped with six blackboards, a data projector covering the middle two blackboards and an overhead/transparency projector if needed. Tentative schedule The miniconference will consist of talks and open problem session. If you would like to connect remotely by videoconferencing please follow these instructions (you do not need to contact Ross but you do need to install the software in advance).
Titles and abstracts: Generating functions' asymptotics' generating functions Michael Borinsky Sequences which admit some factorial growth are common in combinatorics and omnipresent in physics. I will talk about the algebraic properties of power series of this type. These power series form a subring of the ring of power series which is closed under composition. This subring can be equipped with a derivation which maps a power series to its asymptotic expansion. Leibniz and chain rules for this derivation can be deduced. In many cases full asymptotic expansions can be obtained easily in this formalism. Terminal chords in connected diagrams chords Julien Courtiel The topic of this work, resulting from a collaboration with Karen Yeats, is the enumerative study of connected chord diagrams, within the specific framework of quantum field. In fact, the solutions to certain Dyson-Schwinger equations can be defined in terms of connected chord diagrams with a particular parameter: the terminal chords. We study some statistics about these terminal chords: their asymptotic number, the position of the first terminal chord, their distribution with respect to the leading-log expansions, etc. We establish the means, the variances and the limit laws of some of these variables, and show the physics applications. Renormalizations, Motives, Determinant Hypersurfaces, and Kausz Compactifications Matilde Marcoli I will discuss a renormalization procedure (different from the physical one) for Feynman integrals, based on mapping the integral to the determinant hypersurface complement, and to a Kausz compactification. The talk is based on joint work with Paolo Aluffi (arXiv:0901.2107) and with Xiang Ni (arXiv:1408.3754) Combinatorics of the Legendre and Fourier transforms in PQFT Alejandro Morales The partition function, Z, the Fourier and Legendre transforms, are fundamental to quantum field theory. When Z is regarded as a function analytic obstacles are encountered, and heuristic arguments are introduced to go around them. Since Z may be formulated in terms of Feynman diagrams, it is reasonable to ask whether combinatorial methods may be used to determine which of its structures are independent of analytic assumptions. We demonstrate that this is so to an extent by constructing both algebraic and combinatorial analogues of the Legendre and Fourier transforms. Joint work with Achim Kempf and David M. Jackson. Differential Equations for Feynman Amplitudes Emad Nasrollahpoursamami A Feynman diagrams in scalar quantum field theory in dimension D, is a graph and a collection of external data which are vectors in R^D corresponding to the vertices. The amplitude of the diagram is a function of the external data which is defined by an integral which does not necessary converges. One can construct a maximal system of algebraic differential equations satisfied by the integral. It can be shown that in the case of divergent diagrams, the analytic continuation of the integral with respect to D is still a solution to these differential equation. The structure of these equations can be described by combinatorics of polytope which is constructed from the graph. The set of differential equations is "regular holonomic", which in particular implies that the space of solutions is finite dimensional. One can construct a basis for the space of solutions as both integrals and gamma series. In this talk I will talk about the construction of the polytope and differential equations, and show that certain gamma series are solutions to it. Hopf algebras of infrared divergences Erik Panzer Feynman graphs and their ultraviolet subdivergences form a Hopf algebra which was defined by Connes and Kreimer. Recently, Brown generalized this Hopf algebra by taking into account also the infrared divergences. I will define this Hopf algebra and demonstrate the corresponding new factorization of the second Symanzik polynomial. The signed permutation group on Feynman graphs Julian Purkart The Feynman rules assign to every graph an integral which can be written as a function of a scaling parameter L. Assuming L for the process under consideration is very small, so that contributions to the renormalizaton group are small, we can expand the integral and only consider the lowest orders in the scaling. The aim is to determine specific combinations of graphs in a scalar quantum field theory that lead to a remarkable simplification of the first non-trivial term in the perturbation series. It will be seen that the result is independent of the renormalization scheme and the scattering angles. To achieve that goal we will utilize the parametric representation of scalar Feynman integrals as well as the Hopf algebraic structure of the Feynman graphs under consideration. Moreover, we will present a formula which reduces the effort of determining the first-order term in the perturbation series for the specific combination of graphs to a minimum. Field diffeomorphisms and Bell polynomials Karen Yeats Kreimer and Velenich observed that if a diffeomorphism is applied to a free field theory then the terms in the perturbative expansion must cancel. I'll prove this at tree level using Bell polynomials. |