Combinatorial Quantum Field Theory and Gluing Formula for Determinants

We define the combinatorial Dirichlet-to-Neumann operator and establish a gluing formula for determinants of discrete Laplacians using a combinatorial Gaussian quantum field theory. In case of a diagonal inner product on cochains we provide an explicit local expression for the discrete Dirichlet-to-Neumann operator. We relate the gluing formula to the corresponding Mayer-Vietoris formula by Burghelea, Friedlander and Kappeler for zeta-determinants of analytic Laplacians, using the approximation theory of Dodziuk. Our argument motivates existence of gluing formulas as a consequence of a gluing principle on the discrete level.Comment: 26 pages, accepted for publication at Letters in Math. Physic

Heights and measures on analytic spaces. A survey of recent results, and some remarks

This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related results.Comment: 41 pages, final version. To appear in: Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry, edited by Raf Cluckers, Johannes Nicaise, Julien Seba

The dynamical Manin-Mumford problem for plane polynomial automorphisms

Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a time-reversal symmetry; in particular the Jacobian $\mathrm{Jac}(f)$ must be a root of unity. As a step towards this conjecture, we prove that the Jacobian of $f$ and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed $\mathrm{Jac}(f)$ is a root of unity. We use these results to show in various cases that any two automorphisms sharing an infinite set of periodic points must have a common iterate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.Comment: 45 pages. Theorems A and B are now extended to automorphisms defined over any field of characteristic zer

Finite type coarse expanding conformal dynamics

We continue the study of non-invertible topological dynamical systems with expanding behavior. We introduce the class of {\em finite type} systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, level-transitive group action (in the sense of V. Nekrashevych) is of finite type. The proof makes essential use of an analog of the finiteness of cone types property enjoyed by hyperbolic groups.Comment: Updated versio

Three projective problems on Finsler surfaces

Two Finsler metrics on the same manifold are called projectively equivalent, if they have the same unparametrized, oriented geodesics. A vector field on the manifold is called projective for a Finsler metric, if its flow takes geodesics to geodesics as unparametrized curves. In this dissertation, after an introduction to the general theory of Finsler metrics and its projective aspects, results to three projective problems on Finsler metrics on surfaces are presented: Firstly. Inspired by a problem posed by Sophus Lie, it is proven that every Finsler metric, admitting three independent projective vector fields, is projectively equivalent to a Randers metric. An explicit list of such metrics is given, complete up to isometry and projective equivalence. Secondly. The problem of local, fiber-global projective metrization asks whether a given system of unparametrized, oriented curves describes the geodesics of some fiber-globally defined Finsler metric - and if yes, how unique this metric is. It is shown that the set of such metrizations is, up to the trivial freedom, in 1-to-1 correspondence with measures on the space of prescribed curves, satisfying a certain equilibrium property. Thirdly. It is proven that on surfaces of negative Euler characteristic, two real-analytic Finsler metrics can only be trivially projectively related: they are projectively equivalent, if and only if they differ by multiplication with a positive number and addition of a closed 1-form

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Path Integration in Two-Dimensional Toplogical Quantum Field Theory

A positive, diffeomorphism-invariant generalized measure on the space of metrics of a two-dimensional smooth manifold is constructed. We use the term generalized measure analogously with the generalized measures of Ashtekar and Lewandowski and of Baez. A family of actions is presented which, when integrated against this measure, give the two-dimensional axiomatic topological quantum field theories, or TQFT's, in terms of which Durhuus and Jonsson decompose every two-dimensional unitary TQFT as a direct sum.Comment: 13 pages, LaTeX with amsfonts and psfig, 3 postscript figures and .bbl file tar'd and compressed by uufile