The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [Seg04] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW02]. Contents 1. Introduction and results 2. The cobordism category and its sheaves 2.1. The cobordism category 2.2. Recollection from [MW02] on sheaves 2.3. A sheaf model for the cobordism category 2.4. Cocycle sheaves 3. The Thom spectra and their sheaves 3.1. The spectrum MT (d) and its infinite loop space 3.2. Using Phillips' submersion theorem 4. Proof of the main theorem 5. Tangential structures 6. Connectedness issues 6.1. Discussion 6.2. Surgery 6.3. Connectivity 6.4. Parametrized surgery 7. Harer type stability and C 2 References

Wiskott-Aldrich syndrome protein (WASp) is a hematopoietic-specific, multidomain protein whose mutation is responsible for the immunodeficiency disorder Wiskott-Aldrich syndrome. WASp contains a binding motif for the Rho GTPase CDC42Hs as well as verprolin/cofilin-like actinregulatory domains, but no specific actin structure regulated by CDC42Hs-WASp has been identified. We found that WASp colocalizes with CDC42Hs and actin in the core of podosomes, a highly dynamic adhesion structure of human blood-derived macrophages. Microinjection of constitutively active V12CDC42Hs or a constitutively active WASp fragment consisting of the verprolin/cofilin-like domains led to the disassemly of podosomes. Conversely, macrophages from patients expressing truncated forms of WASp completely lacked podosomes. These findings indicate that WASp controls podosome assembly and, in cooperation with CDC42Hs, podosome disassembly in primary human macrophages.

D. Mumford conjectured in [33] that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes κ i of dimension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by BΓ ∞ , where Γ ∞ is the group of isotopy classes of automorphisms of a smooth oriented connected surface of "large" genus. Tillmann's theorem [44]

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Let M and N be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings emb(M, N ) should come from an analysis of the cofunctor V → emb(V, N ) from the poset O of open subsets of M to spaces. We therefore abstract some of the properties of this cofunctor, and develop a suitable calculus of such cofunctors, Goodwillie style, with Taylor series and so on. The terms of the Taylor series for the cofunctor V → emb(V, N ) are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from O to spaces, and show that the Taylor series of an analytic cofunctor F converges to F . Deep excision theorems due to Goodwillie and Goodwillie-Klein imply that the cofunctor V → emb(V, N ) is analytic when dim(N ) − dim(M ) ≥ 3.

Objective: To review the current evidence and make practice recommendations regarding the diagnosis and treatment of limb-girdle muscular dystrophies (LGMDs).Methods: Systematic review and practice recommendation development using the American Academy of Neurology guideline development process.

Abstract.Orthogonal calculus is a calculus of functors, similar to Goodwillie's calculus. The functors in question take finite dimensional real vector spaces (with an inner product) to pointed spaces. Prime example: F(V) = BO(V), where O(V) is the orthogonal group of V . In this example, and in general, first derivatives in the orthogonal calculus reproduce and generalize much of the theory of Stiefel-Whitney classes. Similarly, second derivatives in the orthogonal calculus reproduce and generalize much of the theory of Pontryagin classes.

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