Trees through specified vertices

We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1≤k≤n−2 if G is a connected graph with A⊂V(G) such that d_G(v)≥k for all v∈A, then there exists a subtree T of G such that V(T)⊃A and for all v∈A.     

(DFS)-spaces of holomorphic functions invariant under differentiation

Let be a bounded convex domain, A^−∞(G) be the (DFS)-space of all holomorphic functions of polynomial growth on G and A^∞(G) be the Fréchet space of C^∞-functions on closure of G which are holomorphic on G. With the help of the Laplace transform we describe the strong dual of A^−∞(G) and prove that A^−∞(G) is the unique (DFS)-space H such that the space A^∞(G) is contained in H, H is embedded continuously in A^−∞(G) and H is invariant under differentiation.     

Modules of derivations for toral arrangements

Let G be a connected semi-simple algebraic group over . Fix a maximal torus T in G with coordinate ring _T. Let Φ⁺ be the set of positive roots of G with respect to T. The pair (T, A), where A = {kerα}_α?φ⁺, is a toral arrangement. We show that if G is simply connected then the module of A-derivations D(A) is a free _T-module.     

Filtered ends of infinite covers and groups

Let f:A→B be a covering map. We say that A has e filtered ends with respect to f (or B) if, for some filtration {K_n} of B by compact subsets, A−f⁻¹(K_n) “eventually” has e components. The main theorem states that if Y is a (suitable) free H-space, if K<H has infinite index, and if Y has a positive finite number of filtered ends with respect to H?Y, then Y has one filtered end with respect to K?Y. This implies that if G is a finitely generated group and K<H<G are subgroups each having infinite index in the next, then implies that , where is the number of filtered ends of a pair of groups in the sense of Kropholler and Roller.     

Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian

Let G be a simple algebraic group defined over C and T be a maximal torus of G. For a dominant coweight λ of G, the T-fixed point subscheme of the Schubert variety in the affine Grassmannian GrG is a finite scheme. We prove that for all such λ if G is of type A or D and for many of them if G is of type E, there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to λ and the ring of functions (twisted by certain line bundle on GrG) of . We use this fact to give a geometrical proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of A,D,E type and lattice vertex algebras.     

KKM-Fan’s lemma for solving nonlinear vector evolution equations with nonmonotonic perturbations

In this paper we are interested in the existence of solutions of the following initial value problem: on (0,T) with u(0)=u₀ where A:V→V^′ is a monotone operator, G:V→V^′ is a nonlinear nonmonotone operator and f:(0,T)→V^′ is a measurable function, by means of a recent generalization of the famous KKM-Fan’s lemma.     

Quantisation commutes with reduction at discrete series representations of semisimple groups

Using the analytic assembly map that appears in the Baum-Connes conjecture in noncommutative geometry, we generalise the Guillemin-Sternberg conjecture that ‘quantisation commutes with reduction’ to (discrete series representations of) semisimple groups G with maximal compact subgroups K acting cocompactly on symplectic manifolds. We prove this generalised statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements , the set of elements of g^∗ with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that M=GK_×N, for a compact Hamiltonian K-manifold N. The proof comes down to a reduction to the compact case. This reduction is based on a ‘quantisation commutes with induction’-principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion K?G.     

The biinvariant diagonal class for Hamiltonian torus actions

Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs {(x,y)∈X×X|y=gx for some g∈G} defines a nonzero equivariant cohomology class . We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a Hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [ΔG] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.     

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence of cardinals such that

     

γ-Bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C^∗(G)→B(X) on the group C^∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α).     

Stable equivariant abelianization, its properties, and applications

Let G be a finite group. For a based G-space X and a Mackey functor M, a topological Mackey functor is constructed, which will be called the stable equivariant abelianization of X with coefficients in M. When X is a based G-CW complex, is shown to be an infinite loop space in the sense of G-spaces. This gives a version of the RO(G)-graded equivariant Dold-Thom theorem. Applying a variant of Elmendorf's construction, we get a model for the Eilenberg-Mac Lane spectrum HM. The proof uses a structure theorem for Mackey functors and our previous results.     

Operator space structure and amenability for Figà-Talamanca-Herz algebras

Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′∈(1,∞) with , we use the operator space structure on to equip the Figà-Talamanca-Herz algebra A_p(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p?q?2 or 2?q?p and amenable G, the canonical inclusion A_q(G)⊂A_p(G) is completely bounded (with cb-norm at most , where is Grothendieck's constant). As an application, we show that G is amenable if and only if A_p(G) is operator amenable for all—and equivalently for one—p∈(1,∞); this extends a theorem by Ruan.     

On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

The classical Weyl-von Neumann theorem states that for any self-adjoint operator A₀ in a separable Hilbert space H there exists a (non-unique) Hilbert-Schmidt operator C=C^? such that the perturbed operator A₀+C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering the set ExtA of self-adjoint extensions of a given densely defined symmetric operator A in H and some fixed . We show that the ac-parts and of and A₀ are unitarily equivalent provided that the resolvent difference is compact and the Weyl function M(⋅) of the pair {A,A₀} admits weak boundary limits M(t):=w-lim_y→+0M(t+iy) for a.e. t∈R. This result generalizes the classical Kato-Rosenblum theorem. Moreover, it demonstrates that for such pairs {A,A₀} the Weyl-von Neumann theorem is in general not true in the class ExtA.     

Turán's theorem for pseudo-random graphs

The generalized Turán number ex(G,H) of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph Km on m vertices, the value of ex(Km,H) is , where o(1)→0 as m→∞, by the Erd?s-Stone-Simonovits theorem.In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs '85, Poznań, 1985, North-Holland, Amsterdam, 1987, pp. 307-331. MR 89d:05158]. A graph G is (q,β)-bi-jumbled if

     

Cyclic-type cohomology of strict inductive limits of Fréchet algebras

We present methods for the computation of the Hochschild and cyclic-type continuous homology and cohomology of some locally convex strict inductive limits of Fréchet algebras A_m. In the pure algebraic case it is known that, for the cyclic homology of A, for all n?0 [Cyclic Homology, Springer, Berlin, 1992, E.2.1.1]. We show that, for a locally convex strict inductive system of Fréchet algebras such that

0→A_m→A_m+1→A_m+1/A_m→0     

A functional equation characterizing the second derivative

Consider an operator T:C²(R)→C(R) and isotropic maps A₁,A₂:C¹(R)→C(R) such that the functional equation

     

On Euler characteristic of equivariant sheaves

Let k be an algebraically closed field of characteristic p>0 and let ? be another prime number. Gabber and Looser proved that for any algebraic torus T over k and any perverse ?-adic sheaf on T the Euler characteristic is non-negative.We conjecture that the same result holds for any perverse sheaf on a reductive group G over k which is equivariant with respect to the adjoint action. We prove the conjecture when is obtained by Goresky-MacPherson extension from the set of regular semi-simple elements in G. From this we deduce that the conjecture holds for G of semi-simple rank 1.     

Invariant almost Hermitian structures on flag manifolds

Let G be a complex semi-simple Lie group and form its maximal flag manifold where P is a minimal parabolic (Borel) subgroup, U a compact real form and T=U∩P a maximal torus of U. We study U-invariant almost Hermitian structures on . The (1,2)-symplectic (or quasi-Kähler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler, except in the A₂ case.