Rebecca Goldin

Resolving Singularities of Plane Analytic Branches with one Toric Morphism

Let (C, 0) be an irreducible germ of complex plane curve. Let Γ ⊂ ℕ be the semigroup associated to it and C Γ ⊂ ℂ g+1 the corresponding monomial curve, where g is the number of Puiseux exponents of (C, 0). We show, using the specialization of (C 0) to (C Γ, 0), that the same toric morphisms ZΣ→ℂ g+1 which induce an embedded resolution of singularities of (C Γ, 0) also resolve the singularities of (C, 0) ⊂ (⊂ ℂ g+1, 0), the embedding being defined by elements of the analytic algebra O C, 0 whose valuations generate the semigroup Γ.

Orbifold cohomology of torus quotients

We introduce the_inertial cohomology ring_ NH^*_T(Y) of a stably almost complex manifold carrying an action of a torus T. We show that in the case that Y has a locally free action by T, the inertial cohomology ring is isomorphic to the Chen-Ruan orbifold cohomology ring H_{CR}^*(Y/T) of the quotient orbifold Y/T. For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two techniques that are standard in ordinary cohomology. We show that NH^*_T(Y) has a natural ring surjection onto H_{CR}^*(Y//T), where Y//T is the symplectic reduction of Y by T at a regular value of the moment map. We extend to NH^*_T(Y) the graphical GKM calculus (as detailed in e.g. [Harada-Henriques-Holm]), and the kernel computations of [Tolman-Weitsman, Goldin]. We detail this technology in two examples: toric orbifolds and weight varieties, which are symplectic reductions of flag manifolds. The Chen-Ruan ring has been computed for toric orbifolds, with \Q coefficients, in [Borisov-Chen-Smith]); symplectic toric orbifolds obtained by reduction by a connected torus (though with different computational methods), and extend them to \Z coefficients in certain cases, including weighted projective spaces.

Real loci of symplectic reductions

Let M be a compact, connected symplectic manifold with a Hamiltonian action of a compact n-dimensional torus T. Suppose that M is equipped with an anti-symplectic involution a compatible with the T-action. The real locus of M is the fixed point set M σ of σ. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction M//T has the same ordinary cohomology as its real locus (M//T) σred , with degrees halved. This extends Duistermaats original result on real loci to a case in which there is not a natural Hamiltonian torus action.

ORBIFOLD COHOMOLOGY OF HYPERTORIC VARIETIES

Hypertoric varieties are hyperkahler analogues of toric varieties, and are constructed as abelian hyperkahler quotients T*ℂn//// T of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrangements. By developing hyperkahler analogues of symplectic techniques developed by Goldin, Holm, and Knutson, we give an explicit combinatorial description of the Chen–Ruan orbifold cohomology of an orbifold hypertoric variety in terms of the combinatorial data of a rational cooriented weighted hyperplane arrangement . We detail several explicit examples, including some computations of orbifold Betti numbers (and Euler characteristics).

Cohomology of symplectic reductions of generic coadjoint orbits

Let O λ be a generic coadjoint orbit of a compact semi-simple Lie group K. Weight varieties are the symplectic reductions of O λ by the maximal torus T in K. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of O λ , and it makes explicit the dependence on A and a parameter in Lie(T)* =: t*.

Coordinate systems for dendritic spines: a somatocentric approach

Of fundamental importance in describing a neuron’s activity and constructing biologically plausible neural networks is the unambiguous description of its smallest element of input in the integration process. Among neuronal input units are the synaptic spines: highly regulated and coordinated elements on the dendrites, exchanging both electrical signals and molecules with the soma along the dendritic branches. Mapping the physiological parameters of dendritic branches and their spines in anatomically compatible coordinates is important because of the interactions between “close” spines and between spines and the soma. We present a simple method for quantitatively locating dendritic spines by separating their coordinates into two components. The first takes into account the position of the dendritic branch on which the spine lies. In this component, the distance between a branch and the soma is given by the number of bifurcations along the dendrite (“level”). We have formulaically described the difference in this parameter between any two spines (“distance”) in terms of the level of the common bifurcation farthest from the soma (“generator”). The second component of a spine’s location is its position on the dendritic branch. Our system is fully analytical and easily implementable. It also defines a biologically plausible distance between any two spines, and between a spine and the soma. Based on this labeling method, we present a coordinate system in which a spine is described by a matrix encoding physiological parameters of the generating branches. A second set of coordinates is introduced to describe a neural state with a matrix of spine parameters. Finally, a third matrix notation is proposed to take into account interactions between spines. This treatment leads to some interesting speculations, such as the possibility of describing input dynamics of a neuron in terms of operators on vector spaces.

Cohomology Pairings on the Symplectic Reduction of Products

Let M be the product of two compact Hamiltonian T-spaces X and Y. We present a formula for evaluating integrals on the symplectic reduction of M by the diagonal T action. At every regular value of the moment map for X × Y, the integral is the convolution of two distributions associated to the symplectic reductions of X by T and of Y by T. Several examples illustrate the computational strength of this relationship. We also prove a linear analogue which can be used to find cohomology pairings on toric orbifolds.

Inertia groups of a toric Deligne-Mumford stack, fake weighted projective stacks, and labeled sheared simplices

This paper determines the inertia groups (isotropy groups) of the points of a toric Deligne-Mumford stack [Z/G] (considered over the category of smooth manifolds) that is realized from a quotient construction using a stacky fan or stacky polytope. The computation provides an explicit correspondence between certain geometric and combinatorial data. In particular, we obtain a computation of the connected component of the identity element G0 ⊂ G and the component group G/G0 in terms of the underlying stacky fan, enabling us to characterize the toric DM stacks which are global quotients. As another application, we obtain a characterization of those stacky polytopes that yield stacks equivalent to weighted projective stacks and, more generally, to ‘fake’ weighted projective stacks. Finally, we illustrate our results in detail in the special case of labeled sheared simplices, where explicit computations can be made in terms of the facet labels. Introduction. Toric varieties have been studied for over 35 years. They provide an elementary but illustrative class of examples in algebraic geometry, while also offering insight into related fields such as integrable systems and combinatorics, where the corresponding combinatorial object is a fan. In their foundational paper [5], Borisov, Chen and Smith introduce the notion of a stacky fan, the combinatorial data from which one constructs toric Deligne-Mumford (DM) stacks, which are the stack-theoretic analogues of classical toric vari2010 AMS Mathematics subject classification. Primary 14M25, 57R18, Secondary 14L24.