Thomas H. Parker

On Witten's proof of the positive energy theorem

This paper gives a mathematically rigorous proof of the positive energy theorem using spinors. This completes and simplifies the original argument presented by Edward Witten. We clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant Dirac operator.

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem.The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S2 V S2 V ... →N. The main result is that the images of these renormalized maps converge in L1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class.While the main focus is on holomorphic maps, the bubble tree construction applies to other conformally invariant problems, including minimal surfaces and Yang-Mills fields.

Gauge theories on four-dimensional Riemannian manifolds

This paper develops the Riemannian geometry of classical gauge theories — Yang-Mills fields coupled with scalar and spinor fields — on compact four-dimensional manifolds. Some important properties of these fields are derived from elliptic theory: regularity, an “energy gap theorem”, the manifold structure of the configuration space, and a bound for the supremum of the field in terms of the energy. It is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities (this extends a theorem of K. Uhlenbeck).

The Riemannian geometry of the Yang-Mills moduli space

The moduli space ℳ of self-dual connections over a Riemannian 4-manifold has a natural Riemannian metric, inherited from theL2 metric on the space of connections. We give a formula for the curvature of this metric in terms of the relevant Green operators. We then examine in great detail the moduli space ℳ1 ofk=1 instantons on the 4-sphere, and obtain an explicit formula for the metric in this case. In particular, we prove that ℳ1 is rotationally symmetric and has “finite geometry:” it is an incomplete 5-manifold with finite diameter and finite volume.

A Study of Core-Plus Students Attending Michigan State University

1. INTRODUCTION. One important measure of the effectiveness of a high school mathematics program is the success students have in subsequent university mathematics courses. Yet this measure is seldom considered in studies of high school curricula. This article describes a study that we undertook to quantify this measure for one particular curriculum. Readers of this MONTHLY may be aware of the general context leading to the need for such evaluative studies. Over the past two decades there has been a growing awareness of the inadequacy of the mathematical skills of American high school graduates. That was the assessment of the 1983 report A Nation at Risk [14] and confirmed by many subsequent studies. A recent National Assessment of Educational Progress (NAEP) Report [3] concluded that only 17 percent of U.S. twelfth graders were “proficient” at mathematics. 1 International comparisons also indicate a relatively low level of mathematics achievement by U.S. high school students. The Third International Mathematics and Science Study (TIMSS) assessed the “Mathematics Literacy” of end-ofsecondary students in twenty-two countries and found that U.S. students outperformed only two countries, Cyprus and South Africa [18]. Related studies suggest that the mathematics courses taken by American high school students are often at a lower level than those taken by their international peers and that U.S. high schools are offering a wide assortment of courses that lack the focus and coherence found in many foreign curricula [19]. This situation has been of particular concern on college and university campuses, where large numbers of entering students require remedial courses to bring their mathematical knowledge and skills up to what is required for a wide variety of college courses. One effort to improve school mathematics began in 1989 with the publication of Curriculum and Evaluation Standards for School Mathematics (generally known as “The NCTM Standards”) by the National Council of Teachers of Mathematics [5]. The National Science Foundation (NSF) subsequently funded the development and implementation of thirteen elementary, middle, and high school mathematics curricula based on theseStandards, programs variously referred to as “reform,” “Standardsbased,” or “NSF-sponsored” by their publishers and others. Many of these programs have been controversial. In particular, there has been concern that the NSF-sponsored curricula moved from pilot testing to large-scale implementation without sufficient independent evaluations of their efficacy in preparing students for college mathematics and science courses. Although studies of the effectiveness of these curricula have yielded promising conclusions (see, for example, the collection of such evaluations in [17]), most have been conducted by persons associated with their writing or implementation. A recent National Research Council (NRC) report, On Evaluating Curricular Effectiveness [8], also expresses concerns about the methodological adequacy of these studies. This NRC 1 The notion of “proficient” used in this statistic is identified by the National Assessment Governing Board

Gauge choice in Witten's energy expression

AbstractWittens equation

The yamabe problem

\not D\psi = 0

The symplectic sum formula for Gromov-Witten invariants

can be interpreted as a gauge fixing condition for classical supergravity. We rigorously prove the existence of asymptotically constant solutions of the more general gauge condition