Mathematics Faculty Scholarship

A Comparison of Continuously Controlled and Controlled K-theory

Douglas R. Anderson et al. — Wed, 19 Sep 2012 08:46:28 PDT

We define an unreduced version of the ǫ-controlled lower K -theoretic groups of Ranicki and Yamasaki, [?], and Quinn, [?]. We show that the reduced versions of our groups coincide (in the inverse limit and its first derived, lim1) with those of [?]. We also relate the controlled groups to the continuously controlled groups of [?], and to the Quinn homology groups of [?].

Stable Generalized Finite Element Method (SGFEM)

I. Babuska et al. — Wed, 19 Sep 2012 08:46:27 PDT

The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution of the associated linear system. In this paper, we address this issue and propose a modification of the GFEM, referred to as the Stable GFEM (SGFEM). We show that the conditioning of the stiffness matrix of SGFEM is not worse than that of the standard FEM. Moreover, SGFEM is very robust with respect to the parameters of the enrichments. We show these features of SGFEM on several examples.

Resolutions of Subsets of Finite Sets of Points in Projective Space

Steven P. Diaz et al. — Wed, 19 Sep 2012 08:46:26 PDT

Given a finite set, X, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is "as big as possible" inside X. Given a finite set of points in projective space for which the minimal free resolution of its homogeneous ideal is known, what can be said about possible resolutions of ideals of subsets of this finite set? We first give a maximal rank type description of the most generic possible resolution of a subset. Then we show that this generic resolution is not always achieved, by incorporating an example of Eisenbud and Popescu. However, we show that it is achieved for sets of points in projective two space: given any finite set of points in projective two space for which the minimal free resolution is known, there must exist a subset having the predicted resolution.

The Mixed Problem for Harmonic Functions in Polyhedra

Moises Venouziou et al. — Wed, 19 Sep 2012 08:46:25 PDT

R. M. Brown's theorem on mixed Dirichlet and Neumann boundary conditions is extended in two ways for the special case of polyhedral domains. A (1) more general partition of the boundary into Dirichlet and Neumann sets is used on (2) manifold boundaries that are not locally given as the graphs of functions. Examples are constructed to illustrate necessity and other implications of the geometric hypotheses.

Existence of Positive Definite Noncoercive Sums of Squares

Gregory C. Verchota — Wed, 19 Sep 2012 08:46:23 PDT

Positive definite forms

f 2 R[x₁, . . . , x_n] which are sums of squares of forms of R[x₁, . . . , x_n] are constructed to have the additional property that the members of any collection of forms whose squares sum to f must share a nontrivial complex root in Cⁿ.

Optimal Solvability for the Dirichlet and Neumann Problem in Dimension Two

Atanas Stefanov et al. — Wed, 19 Sep 2012 08:46:22 PDT

We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p₁,(partial D)(H^p (partial D)), p>2/3-E, where D C R² and E is a (small) number depending on the Lipschitz nature of D. This in turn implies that solutions to the Dirichlet problem with data in the Holder class C^1/2+E(partial D) are themselves in C^1/2+E(D). Both of these results are sharp. In fact, we prove a more general statement regarding the H^p solvability for divergence form elliptic equations with bounded measurable coefficients.
We also prove H^2/3-E and C^1/2+E solvability result for the regularity and Dirichlet problem for the biharmonic equation on Lipschitz domains.

Resolutions of Subsets of Finite Sets of Points in Projective Space

Steven P. Diaz et al. — Wed, 19 Sep 2012 08:46:21 PDT

Given a finite set, X, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of this finite set whose Hilbert function is "as big as possible'' inside X. Given a finite set of points in projective space for which the minimal free resolution of its homogeneous ideal is known, what can be said about possible resolutions of ideals of subsets of this finite set? We first give a maximal rank type description of the most generic possible resolution of a subset. Then we show that this generic resolution is not always achieved, by incorporating an example of Eisenbud and Popescu. However, we show that it is achieved for sets of points in projective two space: given any finite set of points in projective two space for which the minimal free resolution is known, there must exist a subset having the predicted resolution.

The Space of Virtual Solutions to the Warped Product Einstein Equation

Chenxu He et al. — Wed, 19 Sep 2012 08:46:20 PDT

In this paper we introduce a vector space of virtual warping functions that yield Einstein metrics over a fixed base. There is a natural quadratic form on this space and we study how this form interacts with the geometry. We use this structure along with the results in our earlier paper "Warped product rigidity" to show that essentially every warped product Einstein manifold admits a particularly nice warped product structure that we call basic. As applications we give a sharp characterization of when a homogeneous Einstein metric can be a warped product and also generalize a construction of Lauret showing that any algebraic soliton on a general Lie group can be extended to a left invariant Einstein metric.

Warped Product Rigidity

Chenxu He et al. — Wed, 25 Jul 2012 06:41:33 PDT

In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. This warped product structure will be used to study warped product Einstein structures in our paper "The space of virtual solutions to the warped product Einstein equation".

Warped Product Einstein Metrics Over Spaces with Constant Scalar Curvature

Chenxu He et al. — Wed, 25 Jul 2012 06:41:32 PDT

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.

On the Classification of Warped Product Einstein Metrics

Chenxu He et al. — Wed, 25 Jul 2012 06:41:31 PDT

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the m-Quasi Einstein equation, but we will also call it the (lambda,n+m)-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some interesting new canonical examples. We also derive some new formulas involving curvatures which are analogous to those for the gradient Ricci solitons. As an application, we characterize warped product Einstein metrics when the base is locally conformally flat.

On the Classification of Gradient Ricci Solitons

Peter Petersen et al. — Wed, 25 Jul 2012 06:41:29 PDT

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with constant scalar curvature and suitably decaying Weyl tensor.

On Gradient Ricci Solitons with Symmetry

Peter Petersen et al. — Wed, 25 Jul 2012 06:41:28 PDT

We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in our paper "Rigidity of gradient Ricci solitons" to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature.

Rigidity of Gradient Ricci Solitons

Peter Petersen et al. — Wed, 25 Jul 2012 06:41:27 PDT

We define a gradient Ricci soliton to be rigid if it is a flat bundle N*_GammaR^kwhere N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

Comparison Geometry for the Bakry-Emery Ricci Tensor

Guofang Wei et al. — Wed, 25 Jul 2012 06:41:25 PDT

For Riemannian manifolds with a measure (M, g, e^−fdvol_g) we prove mean curvature and volume comparison results when the 1-Bakry-Emery Ricci tensor is bounded from below and f is bounded or Theta_rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f

is necessary for these results.

Complete Shrinking Ricci Solitons have Finite Fundamental Group

William Wylie — Wed, 25 Jul 2012 06:41:24 PDT

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case.

Noncompact Manifolds with Nonnegative Ricci Curvature

William Wylie — Wed, 25 Jul 2012 06:41:22 PDT

Let (M,d) be a metric space. For 0

Refinement of Operator-Valued Reproducing Kernels

Yuesheng Xu et al. — Wed, 25 Jul 2012 06:39:32 PDT

This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representations of operator-valued reproducing kernels. Concrete examples of refining translation invariant and finite Hilbert-Schmidt operator-valued reproducing kernels are provided. Other examples include refinement of Hessian of scalar-valued translation-invariant kernels and transformation kernels. Existence and properties of operator-valued reproducing kernels preserved during the refinement process are also investigated.

A Remark on the Topology of (n,n) Springer Varieties

Stephan M. Wehrli — Thu, 26 Jan 2012 07:59:17 PST

We prove a conjecture of Khovanov [Kho04] which identifies the topological space underlying the Springer variety of complete flags in C²ⁿ stabilized by a fixed nilpotent operator with two Jordan blocks of size n.

Khovanov Homology, Sutured Floer Homology, and Annular Links

J. Elisenda Grigsby et al. — Thu, 26 Jan 2012 07:49:06 PST

Lawrence Roberts, extending the work of Ozsvath-Szabo, showed how to associate to a link, L, in the complement of a fixed unknot, B in S³, a spectral sequence from the Khovanov homology of a link in a thickened annulus to the knot Floer homology of the preimage of B inside the double-branched cover of L. In a previous paper, we extended Ozsvath-Szabo's spectral sequence in a different direction, constructing for each knot K in S³ and each positive integer n, a spectral sequence from Khovanov's categorification of the reduced, n-colored Jones polynomial to the sutured Floer homology of a reduced n-cable of K. In the present work, we reinterpret Roberts' result in the language of Juhasz's sutured Floer homology and show that our spectral sequence is a direct summand of Roberts'.