Mathematics Department Faculty Selected Works pagesCopyright (c) 2016 University of Massachusetts - Amherst All rights reserved.
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Recent documents in Mathematics Department Faculty Selected Works pagesen-usThu, 04 Feb 2016 01:06:46 PST3600The Spinor Representation of Surfaces in Space
http://works.bepress.com/robert_kusner/18
http://works.bepress.com/robert_kusner/18Thu, 18 Jun 2015 08:08:47 PDT
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 of M under a suitable integrability condition, which for a minimal surface is simply that the spinor sections are meromorphic. A spin structure S also determines (and is determined by) the regular homotopy class of the immersion by way of a Z2-quadratic form qS. We present an analytic expression for the Arf invariant of qS, which decides whether or not the correponding immersion can be deformed to an embedding. The Arf invariant also turns out to be an obstruction, for example, to the existence of certain complete minimal immersions. The later parts of this paper use the spinor representation to investigate minimal surfaces with embedded planar ends. In general, we show for a spin structure S on a compact Riemann surface M with punctures at P that the space of all such (possibly periodic) minimal immersions of M\P into R3 (upto homothety) is the the product of S1×H3 with the Grassmanian of 2-planes in a complex vector space K of meromorphic sections of S. An important tool – a skew-symmetric form defined by residues of a certain meromorphic quadratic differential on M – lets us compute how K varies as M and P are varied. Then we apply this to determine the moduli spaces of planar-ended minimal spheres and real projective planes, and also to construct a new family of minimal tori and a minimal Klein bottle with 4 ends. These surfaces compactify in S3 to yield surfaces critical for the M¨obius invariant squared mean curvature functional W. On the other hand, Robert Bryant [5] has shown all W-critical spheres and real projective planes arise this way. Thus we find at the same time the moduli spaces of W-critical spheres and real projective planes via the spinor representation.
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Robert Kusner et al.On Thickness and Packing Density for Knots and Links
http://works.bepress.com/robert_kusner/17
http://works.bepress.com/robert_kusner/17Wed, 17 Jun 2015 13:11:51 PDT
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.
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Robert KusnerModuli Spaces Of Embedded Constant Mean Curvature Surfaces With Few Ends And Special Symmetry
http://works.bepress.com/robert_kusner/16
http://works.bepress.com/robert_kusner/16Wed, 17 Jun 2015 12:58:00 PDT
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.
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Karsten Grosse-Brauckmann et al.Triunduloids: Embedded Constant Mean Curvature Surfaces With Three Ends and Genus Zero
http://works.bepress.com/robert_kusner/15
http://works.bepress.com/robert_kusner/15Wed, 17 Jun 2015 12:52:35 PDT
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.
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Karsten Grosse-Brauckmann et al.Wonderful Blowups Associated To Group Actions
http://works.bepress.com/paul_gunnells/48
http://works.bepress.com/paul_gunnells/48Wed, 17 Jun 2015 12:38:36 PDT
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.
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Lev A. Borisov et al.Torus Orbits On Homogeneous Varieties And Kac Polynomials Of Quivers
http://works.bepress.com/paul_gunnells/47
http://works.bepress.com/paul_gunnells/47Wed, 17 Jun 2015 11:58:18 PDT
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.
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Paul Gunnells et al.On Toric Varieties And Modular Forms
http://works.bepress.com/paul_gunnells/46
http://works.bepress.com/paul_gunnells/46Wed, 17 Jun 2015 11:51:28 PDTPaul GunnellsOn The Cohomology Of Linear Groups Over Imaginary Quadratic Fields
http://works.bepress.com/paul_gunnells/45
http://works.bepress.com/paul_gunnells/45Wed, 17 Jun 2015 11:45:25 PDT
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.
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Herbert Gangl et al.On K4 Of The Gaussian And Eisenstein Integers
http://works.bepress.com/paul_gunnells/44
http://works.bepress.com/paul_gunnells/44Wed, 17 Jun 2015 11:34:40 PDT
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.
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Mathieu Dutour Sikiric et al.Metaplectic Demazure Operators And Whittaker Functions
http://works.bepress.com/paul_gunnells/43
http://works.bepress.com/paul_gunnells/43Wed, 17 Jun 2015 11:23:26 PDT
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.
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Gautam Chinta et al.