Double bubbles in the 3-torus

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3.Comment: 13 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200

Constructing Buildings and Harmonic Maps

In a continuation of our previous work, we outline a theory which should lead to the construction of a universal pre-building and versal building with a $\phi$-harmonic map from a Riemann surface, in the case of two-dimensional buildings for the group $SL_3$. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for $SL_2$. Our conjectural construction would determine the exponents for $SL_3$ WKB problems, and it can be put into practice on examples.Comment: 61 pages, 24 figures. Comments are welcom

Isoperimetric Regions in Spaces

We study the isoperimetric problem, the least-perimeter way to enclose given area, in various surfaces. For example, in two-dimensional Twisted Chimney space, a two-dimensional analog of one of the ten flat, orientable models for the universe, we prove that isoperimetric regions are round discs or strips. In the Gauss plane, defined as the Euclidean plane with Gaussian density, we prove that in halfspaces y ≥ a vertical rays minimize perimeter. In Rn with radial density and in certain products we provide partial results and conjectures

Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York

Proof of the bounded conformal conjecture

Given any asymptotically flat 3-manifold $(M,g)$ with smooth, non-empty, compact boundary $\Sigma$, the conformal conjecture states that for every $\delta>0$, there exists a metric $g' = u^4 g$, with $u$ a harmonic function, such that the area of outermost minimal area enclosure $\overline{\Sigma}_{g'}$ of $\Sigma$ with respect to $g'$: $|\Sigma - \overline{\Sigma}_{g'}|_g' < \delta$. Recently, the conjecture was used to prove the Riemannian Penrose inequality for black holes with zero horizon area, and was proven to be true under the assumption of the existence of a finite number of minimal area enclosures of the boundary $\Sigma$, and boundedness of the harmonic function $u$. We prove the conjecture assuming only the boundedness of $u$

The Demarcation of Land and the Role of Coordinating Institutions

This paper examines the economic effects of the two dominant land demarcation systems, metes and bounds (MB) and the rectangular system (RS). Under MB property is demarcated by its perimeter as indicated by natural features and human structures and linked to surveys within local political jurisdictions. Under RS land demarcation is governed by a common grid with uniform square shapes, sizes, alignment, and geographically-based addresses. In the U.S. MB is used principally in the original 13 states, Kentucky, and Tennessee. The RS is found elsewhere under the Land Ordinance of 1785 that divided federal lands into square-mile sections. We develop an economic framework for examining land demarcation systems and draw predictions. Our empirical analysis focuses on a 39-county area of Ohio where both MB and RS were used in adjacent areas as a result of exogenous historical factors. The results indicate that topography influences parcel shape and size under a MB system; that parcel shapes are aligned under the RS; and that the RS is associated with higher land values, more roads, more land transactions, and fewer legal disputes than MB, all else equal. The comparative limitations of MB appear to have had negative long-term effects on land values and economic activity in the sample area.

Double bubbles in the three-torus

13 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200113 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200113 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200113 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200