Mathematics Commons^™

The Expected Total Curvature Of Random Polygons, Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, Clayton Shonkwiler

Robert Kusner

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution.

We then consider the symmetric measure on closed polygons of fixed total length constructed by …

A Table Of Elliptic Curves Over The Cubic Field Of Discriminant −23, Steve Donnelly, Paul Gunnells, Ariah Klages-Mundt, Dan Yasaki

Paul Gunnells

Abstract. Let F be the cubic field of discriminant −23 and OF its ring of integers. Let 􀀀 be the arithmetic group GL2(OF ), and for any ideal n ⊂ OF let 􀀀0(n) be the congruence subgroup of level n. In [16], two of us (PG and DY) computed the cohomology of various 􀀀0(n), along with the action of the Hecke operators. The goal of [16] was to test the modularity of elliptic curves over F. In the present paper, we complement and extend the results of [16] in two ways. First, we tabulate more elliptic curves than were found …

Metaplectic Demazure Operators And Whittaker Functions, Gautam Chinta, Paul Gunnells, Anna Pusk´As

Paul Gunnells

Abstract. In [CG10] the first two named authors defined an action of a Weyl group on rational functions and used it to construct multiple Dirichlet series. These series are related to Whittaker functions on an n-fold metaplectic cover of a reductive group. In this paper, we define metaplectic analogues of the Demazure and Demazure-Lusztig operators. We show how these operators can be used to recover the formulas from [CG10], and how, together with results of McNamara [McN], they can be used to compute Whittaker functions on metaplectic groups over p-adic fields.

Parametric Sensitivity Analysis For Biochemical Reaction Networks Based On Pathwise Information Theory, Yannis Pantazis, Markos Katsoulakis, Dionisios G. Vlachos

Markos Katsoulakis

Background: Stochastic modeling and simulation provide powerful predictive methods for the intrinsic understanding of fundamental mechanisms in complex biochemical networks. Typically, such mathematical models involve networks of coupled jump stochastic processes with a large number of parameters that need to be suitably calibrated against experimental data. In this direction, the parameter sensitivity analysis of reaction networks is an essential mathematical and computational tool, yielding information regarding the robustness and the identifiability of model parameters. However, existing sensitivity analysis approaches such as variants of the finite difference method can have an overwhelming computational cost in models with a high-dimensional parameter space. …

Torus Orbits On Homogeneous Varieties And Kac Polynomials Of Quivers, Paul Gunnells, Emmanuel Letellier, Fernando Rodriguez Villegas

Paul Gunnells

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.

On The Cohomology Of Linear Groups Over Imaginary Quadratic Fields, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schurmann, Mathieu Dutour Sikiric, Dan Yasaki

Paul Gunnells

Let 􀀀 be the group GLN(OD), where OD is the ring of integers in the imaginary quadratic field with discriminant D < 0. In this paper we investigate the cohomology of 􀀀 for N = 3, 4 and for a selection of discriminants: D −24 when N = 3, and D = −3,−4 when N = 4. In particular we compute the integral cohomology of 􀀀 up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for 􀀀 developed by Ash [4, Ch. II] and Koecher [18]. Our results extend work of Staffeldt [29], who treated the case n = 3, D = −4. In a sequel [11] to this paper, we will apply some of these results to the computations with the K-groups K4(OD), when D = −3,−4.

On K4 Of The Gaussian And Eisenstein Integers, Mathieu Dutour Sikiric, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki

Paul Gunnells

Abstract. In this paper we investigate the structure of the algebraic K-groups K4(Z[i]) and K4(Z[ρ]), where i := √ −1 and ρ := (1 + √ −3)/2. We exploit the close connection between homology groups of GLn(R) for n 6 5 and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GLn(R) acts. Our main results are (i) K4(Z[i]) is a finite abelian 3-group, and (ii) K4(Z[ρ]) is trivial.

On Thickness And Packing Density For Knots And Links, Robert Kusner

Robert Kusner

We describe some problems, observations, and conjectures concerning density of the hexagonal packing of unit disks in R2.thickness and packing density of knots and links in S3 and R3. We prove the thickness of a nontrivial knot or link in S3 is no more than 4 , the thickness of a Hopf link. We also give arguments and evidence supporting the conjecture that the packing density of thick links in R3 or S3 is generally less than √12 , the density of the hexagonal packing of unit disks in R2.

On Toric Varieties And Modular Forms, Paul Gunnells

Paul Gunnells

No abstract provided.

Wonderful Blowups Associated To Group Actions, Lev A. Borisov, Paul Gunnells

Paul Gunnells

A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary sub-groups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev.

Triunduloids: Embedded Constant Mean Curvature Surfaces With Three Ends And Genus Zero, Karsten Grosse-Brauckmann, Robert Kusner, John M. Sullivan

Robert Kusner

We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.

The Spinor Representation Of Surfaces In Space, Robert Kusner, Nick Schmitt

Robert Kusner

The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan [32], which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the canonical line bundle K = T(M). Given a conformal immersion of M into R3, the unique spin strucure on S2 pulls back via the Gauss map to a spin structure S on M, and gives rise to a pair of smooth sections (s1, s2) of S. Conversely, any pair of sections of S generates a (possibly periodic) conformal immersion …

Moduli Spaces Of Embedded Constant Mean Curvature Surfaces With Few Ends And Special Symmetry, Karsten Grosse-Brauckmann, Robert Kusner

Robert Kusner

We give necessary conditions on complete embedded cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are twodimensional varieties in the moduli spaces of general cmc surfaces. We characterize fundamental domains of our cmc surfaces by associated great circle polygons in the three-sphere.