Jacobian quotients, an algebraic proof

We give an algebraic proof of a theorem of H. Maugendre showing how the jacobian quotients of a pair of germs of plane curve may be computed from their simultaneous immerged resolution, thus proving in particular their topological invariance.     

Spectrum,entropy, and geometric structures for smooth actions of kazhdan groups

We show that many natural classes of actions of discrete subgroups of semisimple Lie groups have discrete spectrum, i.e., are measurably conjugate to isometric actions. Research partially supported by NSF Grant.     

Cross-sections, quotients, and representation rings of semisimple algebraic groups

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny t:[^(G)] ? G \tau :\hat{G} \to G is bijective; this answers Grothendieck’s question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg’s theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G] ^G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G] ^G and that of the representation ring of G and answer two Grothendieck’s questions on constructing generating sets of k[G] ^G . We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T ⇢ G/T where T is a maximal torus of G and W the Weyl group.     

A note on strong protomodularity,actions and quotients

In order to study the problems of extending an action along a quotient of the acted object and along a quotient of the acting object, we investigate some properties of the fibration of points. In fact, we obtain a characterization of protomodular categories among quasi-pointed regular ones, and, in the semi-abelian case, a characterization of strong protomodular categories. Eventually, we return to the initial questions by stating the results in terms of internal actions.     

Rigid versus unique determination of protein structures with geometric buildup

We introduce a geometric buildup approach to the distance geometry problem in protein modeling, and discuss the necessary and sufficient conditions on the distances for rigid or unique determination of a protein structure. We describe a new buildup algorithm for determining protein structures rigidly instead of uniquely. The algorithm requires even fewer distance constraints than the general buildup algorithm. We present the test results from applying the algorithm to determining the protein structures with varying degrees of availability of the distances, and show that the new development increases the modeling ability of the geometric buildup method even more while retaining much of the computational feasibility of the method.     

Variation of geometric invariant theory quotients

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several natural questions arise. Is the set of equivalence classes, and hence the set of non-isomorphic quotients, finite? How does the quotient vary under change of the equivalence class? In this paper we give partial answers to these questions in the case of actions of reductive algebraic groups on nonsingular projective algebraic varieties. We shall show that among ample line bundles which give projective geometric quotients there are only finitely many equivalence classes. These classes span certain convex subsets (chambers) in a certain convex cone in Euclidean space, and when we cross a wall separating one chamber from another, the corresponding quotient undergoes a birational transformation which is similar to a Mori flip.     

On the partial quotients of algebraic integers

It is shown that there are algebraic integers, with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Davenport for algebraic numbers.     

Galois actions on Q-curves and winding quotients

We prove two ``large images' results for the Galois representations attached to a degree d Q-curve E over a quadratic field K: if K is arbitrary, we prove maximality of the image for every prime p>13 not dividing d, provided that d is divisible by q (but d≠q) with q=2 or 3 or 5 or 7 or 13. If K is real we prove maximality of the image for every odd prime p not dividing d D, where D= disc(K), provided that E is a semistable Q-curve. In both cases we make the (standard) assumptions that E does not have potentially good reduction at all primes p∤6 and that d is square free. The first author is supported by BFM2003-06092.     

Testing algebraic geometric codes

Property testing was initially studied from various motivations in 1990’s. A code C  GF (r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector’s coordinates. The problem of testing codes was firstly studied by Blum, Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs). How to characterize locally te...     

Copolarity of isometric actions

We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson.

     

Rigid extensions of algebraic frames

An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each ${h \in H}An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each h ? H{h \in H} there exists a g ? G{g \in G} such that h ^⊥⊥ = g ^⊥⊥. In this paper, we will define rigid extensions and some other generalizations in the context of algebraic frames satisfying the FIP. One of the main results is a characterization of rigid extensions using d-elements of the frame. We also show that a rigid extension between two algebraic frames satisfying the FIP will induce a homeomorphism between their corresponding minimal prime spaces with respect to both the hull-kernel topology and the inverse topology. Moreover, basic open sets map to basic open sets.     

Moduleability, algebraic structures, and nonlinear properties

We show that some pathological phenomena occur more often than one could expect, existing large algebraic structures (infinite dimensional vector spaces, algebras, positive cones or infinitely generated modules) enjoying certain special properties. In particular we construct infinite dimensional vector spaces of non-integrable, measurable functions, completing some recent results shown in García-Pacheco et al. (2009) [13], García-Pacheco and Seoane-Sepúlveda (2006) [15], Muñoz-Fernández et al. (2008) [20]. We prove, as well, the existence of dense and not barrelled spaces of sequences every non-zero element of which has a finite number of zero coordinates (giving partial answers to a problem originally posed by R.M. Aron and V.I. Gurariy in 2003).     

Isoperimetric inequalities,isometric actions and the higher Newman numbers

M will be a compact connected n-dimensional Riemannian manifold. If M contains a closed connected k-dimensional, 2 k < n, minimal immersed submanifold of M, we define the kth isoperimetric number of M, Ñ _k (M), as the infimum of the volumes of all such submanifolds. We obtain a number of interesting estimates for Ñ _k (M), for both general and special manifolds, which appear to be new.Next we turn to isometric actions and a 1931 theorem of M. H. A. Newman involving the size of orbits of group actions on manifolds. We introduce the higher Newman numbers N _k (M), 1 k n. Roughly speaking, if M admits isometric actions of compact connected Lie groups with k-dimensional principal orbits, N _k (M) is defined as the infimum over all such actions of the maximum volume of all maximal dimensional orbits. We observe that N _k (M) Ñ _k (M), 2 k < n, provided N _k (M) is defined; hence our prior estimates for the isoperimetric numbers of M apply directly to the higher Newman numbers.As a best possible candidate we conjecture that N _k (M) vol S ^k (i(M)/), 1 k n, where i(M) denotes the radius of injectivity of M and S ^k (i(M)/) denotes the standard k-sphere of radius i(M)/. We verify the conjecture for various special cases. We conclude the paper by studying Newman's theorem for compact connected Lie groups with invariant metrics and obtaining a lower bound for the size of small subgroups.     

On lightlike geometry: isometric actions,and rigidity aspects

Degenerate Riemannian metrics exist naturally in various contexts. Unfortunately, their study stops to the ‘admission of failure’ that they are too poor, for instance, to generate a coherent intrinsic or extrinsic differential geometry, e.g. a kind of Levi-Civita connection. In this first text, we start the investigation of rigidity aspects of these structures, from the point of view of isometric actions of ‘big’ (e.g. semi-simple) Lie groups. To cite this article: E. Bekkara et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Observable actions of algebraic groups

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ \Bbbk\Bbbk [X] ^G such that f| _Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if: