We present an axisymmetric, equilibrium model for late-type galaxies which consists of an exponential disk, a Sersic bulge, and a cuspy dark halo. The model is specified by a phase space distribution function which, in turn, depends on the integrals of motion. Bayesian statistics and the Markov Chain Monte Carlo method are used to tailor the model to satisfy observational data and theoretical constraints. By way of example, we construct a chain of 10^5 models for the Milky Way designed to fit a wide range of photometric and kinematic observations. From this chain, we calculate the probability distribution function of important Galactic parameters such as the Sersic index of the bulge, the disk scale length, and the disk, bulge, and halo masses. We also calculate the probability distribution function of the local dark matter velocity dispersion and density, two quantities of paramount significance for terrestrial dark matter detection experiments. Though the Milky Way models in our chain all satisfy the prescribed observational constraints, they vary considerably in key structural parameters and therefore respond differently to non-axisymmetric perturbations. We simulate the evolution of twenty-five models which have different Toomre Q and Goldreich-Tremaine X parameters. Virtually all of these models form a bar, though some, more quickly than others. The bar pattern speeds are ~ 40 - 50 km/s/kpc at the time when they form and then decrease, presumably due to coupling of the bar with the halo. Since the Galactic bar has a pattern speed ~50 km/s/kpc we conclude that it must have formed recently.Comment: 54 pages, 20 figure

In this paper, we study the interplay between modules and sub‐objects in holomorphic Poisson geometry. In particular, we define a new notion of ‘residue’ for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. We then explore the interaction between the residues of the canonical line bundle of a Poisson manifold and its degeneracy loci, where the rank of the Poisson structure drops. As an application, we provide new evidence in favor of Bondal's conjecture that the rank at most 2k locus of a Fano Poisson manifold always has dimension at least 2k + 1. In particular, we show that the conjecture holds for Fano 4‐folds. We also apply our techniques to a family of Poisson structures defined by Feĭgin and Odesskiĭ, where the degeneracy loci are given by the secant varieties of elliptic normal curves.

Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula. Contents

This paper is motivated by the question of how motivic Donaldson-Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi-Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal An-singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, as plethystic exponentials of simple rational functions. While our results are limited by the standard dimensional reduction techniques that we employ, they nevertheless allow us to conjecture formulae for more interesting cases, such as the elliptic Sklyanin algebras.