Derived Smooth Manifolds

We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^\infty$-rings that is obtained by patching together homotopy zero-sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection $A\cap B$ of submanifolds $A,B\subset X$ exists on the categorical level in our theory, and a cup product formula $$[A]\smile[B]=[A\cap B]$$ holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a {\em categorification} of intersection theory.Comment: 57 pages. Reformulation of author's PhD thesis. To appear in Duke Math J

Hydrology

When applied to wetlands, the science of hydrology is concerned with how the storage and movement of water into and out of a wetland affects the plants and animals, and the soils on which they grow. Most wetland scientists agree that the single most important factor determining both wetland type and function is hydrology. Consequently, changes in hydrology are the leading causes of wetland degradation or destruction. The two case studies in this chapter illustrate how water was returned to a previously drained lowland swamp and a peat bog and the effects on the vegetation communities. Both sites had been drained as further dry land was desired for farming and urban development, a common scenario throughout New Zealand

On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilisers

Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \g. We also establish when the centralisers G_v of vectors v\in V_min and stabilisers \Stab_G of 1-spaces \subset V_min are smooth; that is, when \dim G_v=\dim\g_v or \dim \Stab_G=\dim\Stab_\g.Comment: This contains corrections and should be used instead of the published versio

Class Problem!: Why the Inconsistent Application of Rule 23\u27s Class Certification Requirements During Overbreadth Analysis is a Threat to Litigant Certainty

Rule 23 of the Federal Rules of Civil Procedure is home to the class action device. It is well-documented that this rule significantly impacts our legal system. As a result, the need for its effective utilization has been apparent since its introduction. Despite this, federal courts have inconsistently applied the rule during their analyses of overbroad class definitions at the class certification stage. Consequently, parties involved in such litigation have been exposed to unnecessary costs and the potential for forum shopping. Nonetheless, this judicial inconsistency has gone largely unrecognized because it does not implicate the results of class certification. Hence, courts here must first recognize the general need for uniformity before a precise standard for overbreadth analysis may be chosen. Only then, this Note argues, may the aforementioned detrimental consequences be avoided