Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co--cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply--connected manifold. In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are that there is a cell model operad for the moduli space of genus $g$ curves with $n$ punctures and a tangent vector at each of these punctures and that there exists a CW complex whose chains are isomorphic to a certain type of Sullivan Chord diagrams and that they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.Comment: 50 pages, 7 figures. Newer version has minor changes. Some material shifted. Typos and small things correcte

Chains

Chains is a poem that was inspired by the events surrounding the Steubenville Rape Case, and it is my interpretation of what the victim could have been feeling. The poem was written as a way for me to try to understand how something like this could have happened

Falling chains

The one-dimensional fall of a folded chain with one end suspended from a rigid support and a chain falling from a resting heap on a table is studied. Because their Lagrangians contain no explicit time dependence, the falling chains are conservative systems. Their equations of motion are shown to contain a term that enforces energy conservation when masses are transferred between subchains. We show that Cayley's 1857 energy nonconserving solution for a chain falling from a resting heap is incorrect because it neglects the energy gained when a transferred link leaves a subchain. The maximum chain tension measured by Calkin and March for the falling folded chain is given a simple if rough interpretation. Other aspects of this falling folded chain are briefly discussed.Comment: 9 pages, 1 figure; the Abstract has been shortened, three paragraphs have been re-written for greater clarity, and textual improvements have been made throughout the paper; to be published by the Am. J. Physic

Loewner Chains

These lecture notes on 2D growth processes are divided in two parts. The first part is a non-technical introduction to stochastic Loewner evolutions (SLEs). Their relationship with 2D critical interfaces is illustrated using numerical simulations. Schramm's argument mapping conformally invariant interfaces to SLEs is explained. The second part is a more detailed introduction to the mathematically challenging problems of 2D growth processes such as Laplacian growth, diffusion limited aggregation (DLA), etc. Their description in terms of dynamical conformal maps, with discrete or continuous time evolution, is recalled. We end with a conjecture based on possible dendritic anomalies which, if true, would imply that the Hele-Shaw problem and DLA are in different universality classes.Comment: 46 pages, 21 figure

Form factors of integrable Heisenberg (higher) spin chains

We present determinant formulae for the form factors of spin operators of general integrable XXX Heisenberg spin chains for arbitrary (finite dimensional) spin representations. The results apply to any "mixed" spin chains, such as alternating spin chains, or to spin chains with magnetic impurities.Comment: 24 page