The topology of two-dimensional real algebraic varieties

Sunto È noto che ogni spazio analitico reale è localmente omeomorfo al cono su un poliedro con caratteristica di Eulero-Poincaré pari. Si dimostra che questa condizione è anche sufficiente affinchè un poliedro (compatto) di dimensione due P sia omeomorfo ad una varietà algebrica reale affine P. Segue inoltre dalla costruzione che la P ottenuta ha, in un certo senso, un insieme di singolarità algebriche minimale, compatibilmente con la topologia di P.

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The authors are members of the G.N.S.A.G.A.     

The topology of parabolic character varieties of free groups

Let $G$ be a complex affine algebraic reductive group, and let $K\,\subset \, G$ be a maximal compact subgroup. Fix h $\,:=\,(h_{1}\,,\ldots \,,h_{m})\,\in \, K^{m}$ . For $n\, \ge \, 0$ , let $\mathsf X _{\mathbf{{h}},n}^{G}$ (respectively, $\mathsf X _{\mathbf{{h}},n}^{K}$ ) be the space of equivalence classes of representations of the free group on $m+n$ generators in $G$ (respectively, $K$ ) such that for each $1\le i\le m$ , the image of the $i$ -th free generator is conjugate to $h_{i}$ . These spaces are parabolic analogues of character varieties of free groups. We prove that $\mathsf X _{\mathbf{{h}},n}^{K}$ is a strong deformation retraction of $\mathsf X _{\mathbf{{h}},n}^{G}$ . In particular, $\mathsf X _{\mathbf{{h}},n}^{G}$ and $\mathsf X _{\mathbf{{h}},n}^{K}$ are homotopy equivalent. We also describe explicit examples relating $\mathsf X _{\mathbf{{h}},n}^{G}$ to relative character varieties.     

Dirichlet fundamental domains and topology of projective varieties

We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.     

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

Mathematische Zeitschrift - This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the...     

The Ehrhart Polynomial of the Birkhoff Polytope

The nth Birkhoff polytope is the set of all doubly stochastic n × n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these polytopes, which have been known for n $\leq$ 8. We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function counting the integer points in the dilated polytope. One reason to be interested in this counting function is that the leading term of the Ehrhart polynomial is—up to a trivial factor—the volume of the polytope. We implemented our methods in the form of a computer program, which yielded the Ehrhart polynomial (and hence the volume) of the ninth Birkhoff polytope, as well as the volume of the tenth Birkhoff polytope.     

The diameter of the acyclic Birkhoff polytope

In this work we give an interpretation of vertices and edges of the acyclic Birkhoff polytope, T_n=Ω_n(T), where T is a tree with n vertices, in terms of graph theory. We generalize a recent result relatively to the diameter of the graph G(T_n).     

The Birkhoff Variety Theorem for continuous algebras

Varieties of continuous algebras, i.e., classes presented by (in)equalities for terms, are characterized as precisely the HSP classes. Terms here are the usual algebraic terms enriched by iterated join symbols. A discussion of varieties which require only finitely many variables, or only the usual terms, is also presented.Presented by W. Taylor.     

The Birkhoff integral and the property of Bourgain

In this paper we study the Birkhoff integral of functions f:X defined on a complete probability space (,,) with values in a Banach space X. We prove that if f is bounded then its Birkhoff integrability is equivalent to the fact that the set of compositions of f with elements of the dual unit ball Z_f={x*,f:x*B_X*} has the Bourgain property. A non necessarily bounded function f is shown to be Birkhoff integrable if, and only if, Z_f is uniformly integrable and has the Bourgain property. As a consequence it turns out that the range of the indefinite integral of a Birkhoff integrable function is relatively norm compact. We characterize the weak Radon-Nikodým property in dual Banach spaces via Birkhoff integrable Radon-Nikodým derivatives. We also point out that a recently introduced notion of unconditional Riemann-Lebesgue integrability coincides with the notion of Birkhoff integrability. Some other applications are given.Mathematics Subject Classification (2000): 28B05, 46B22, 46G10Partially supported by the research grant BFM2002-01719 of MCyT (Spain). The second author was supported by a FPU grant of MECD (Spain).Acknowledgement We gratefully thank Gabriel Vera for lively discussions about some material considered in this paper. We also thank the referee for helpful suggestions that have improved the exposition of this work.     

The Birkhoff decomposition in groups of formal diffeomorphisms

Let $G_{\infty }$ be the group of one parameter identity-tangent diffeomorphisms on the line whose coefficients are formal Laurent series in the parameter ε with a pole of finite order at 0. It is well-known that the Birkhoff decomposition can be defined in such a group. We investigate the stability of the Birkhoff decomposition in subgroups of $G_{\infty }$ and give a formula for this decomposition. As proven by A. Connes and D. Kreimer, the Birkhoff decomposition is related to renormalization in quantum field theory and we give an application of our results in the last section. To cite this article: F. Menous, C. R. Acad. Sci. Paris, Ser. I 342 (2006).