Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

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T contains all weakly Lindelöf Banach spaces;

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l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

•

T is stable under weak homeomorphisms;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Inversion arrangements and Bruhat intervals

Let W be a finite Coxeter group. For a given w∈W, the following assertion may or may not be satisfied:

(?)

The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w.

We present a type independent combinatorial criterion which characterises the elements w∈W that satisfy (?). A couple of immediate consequences are derived:

(1)

The criterion only involves the order ideal of w as an abstract poset. In this sense, (?) is a poset-theoretic property.

(2)

For W of type A, another characterisation of (?), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjöstrand. We obtain a short and simple proof of that result.

(3)

If W is a Weyl group and the Schubert variety indexed by w∈W is rationally smooth, then w satisfies (?).

     

A criterion for unitary similarity of upper triangular matrices in general position

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A=[a_ij] and B=[b_ij] be upper triangular n×n matrices that

•

are not similar to direct sums of square matrices of smaller sizes, or

•

are in general position and have the same main diagonal.

We prove that A and B are unitarily similar if and only if

     

On some contributions of John Horváth to the theory of distributions

Our main task is a presentation of J. Horváth's results concerning

•

singular and hypersingular integral operators,

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the analytic continuation of distribution-valued meromorphic functions, and

•

a general definition of the convolution of distributions.

At some instances minor supplements to his results are given.     

Canonical matrices of isometric operators on indefinite inner product spaces

We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases:

•

F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric;

•

F is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F.

These classification problems are wild if B may be degenerate. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (3) (1988) 481-501].     

Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered:

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we improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets;

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we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances;

•

we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte.

We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.     

Properties of four partial orders on standard Young tableaux

Let SYTn be the set of all standard Young tableaux with n cells. After recalling the definitions of four partial orders, the weak, KL, geometric and chain orders on SYTn and some of their crucial properties, we prove three main results:

•

Intervals in any of these four orders essentially describe the product in a Hopf algebra of tableaux defined by Poirier and Reutenauer.

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The map sending a tableau to its descent set induces a homotopy equivalence of the proper parts of all of these orders on tableaux with that of the Boolean algebra ²[n−1]. In particular, the Möbius function of these orders on tableaux is (−1)ⁿ⁻³.

•

For two of the four orders, one can define a more general order on skew tableaux having fixed inner boundary, and similarly analyze their homotopy type and Möbius function.

     

A Lagrangian heuristic algorithm for a real-world train timetabling problem

The train timetabling problem (TTP) aims at determining an optimal timetable for a set of trains which does not violate track capacities and satisfies some operational constraints.In this paper, we describe the design of a train timetabling system that takes into account several additional constraints that arise in real-world applications. In particular, we address the following issues:

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Manual block signaling for managing a train on a track segment between two consecutive stations.

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Station capacities, i.e., maximum number of trains that can be present in a station at the same time.

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Prescribed timetable for a subset of the trains, which is imposed when some of the trains are already scheduled on the railway line and additional trains are to be inserted.

•

Maintenance operations that keep a track segment occupied for a given period.

We show how to incorporate these additional constraints into a mathematical model for a basic version of the problem, and into the resulting Lagrangian heuristic. Computational results on real-world instances from Rete Ferroviaria Italiana (RFI), the Italian railway infrastructure management company, are presented.     

Finite Eulerian posets which are binomial, Sheffer or triangular

In this paper we study finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows:

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We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets.

•

We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases.

•

In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets.

We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.     

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:

(i)

AM is relatively weakly compact;

(ii)

B is a strict contraction;

(iii)

.

Then A+B has at least one fixed point in M.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.     

Galois representations modulo p and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention:

•

control of the image of Galois representations modulo p,

•

Hida's congruence criterion outside an explicit set of primes,

•

freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge-Tate (resp. Fontaine-Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.     

The signature operator at 2

It is well known that the signature operator on a manifold defines a K-homology class which is an orientation after inverting 2. Here we address the following puzzle: What is this class localized at 2, and what special properties does it have? Our answers include the following:

•

the K-homology class Δ_M of the signature operator is a bordism invariant;

•

the reduction mod 8 of the K-homology class of the signature operator is an oriented homotopy invariant;

•

the reduction mod 16 of the K-homology class of the signature operator is not an oriented homotopy invariant.

     

A canonical form for nonderogatory matrices under unitary similarity

A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms for

•

nonderogatory complex matrices up to unitary similarity, and

•

pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues.

The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles.     

The Mukai pairing—II: the Hochschild-Kostant-Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are:

•

we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds;

•

we give a proof of the fact that the natural Chern character map K₀(X)→HH₀(X) becomes, after the HKR isomorphism, the usual one ; and

•

we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.

     

Strong zero-dimensionality of hyperspaces

For a space X, X² denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X². The following are known:

•

ω² is not normal, where ω denotes the discrete space of countably infinite cardinality.

•

For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.

In this paper, we will prove:

(1)

ω² is strongly zero-dimensional.

(2)

K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.

In (2), we use the technique of elementary submodels.     

Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:

the generalized Bethe tree B,

a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,

a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,

a graph obtained from the cycle C_r by attaching B, by its root, to each vertex of the cycle, and

a tree obtained from the path P_r by attaching B, by its root, to each vertex of the path,

is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.

     

A new generation of applied probability textbooks

This article reviews the following books:

•

S. Asmussen, Applied Probability and Queues, second ed., Springer, Berlin, 2003, ISBN 0-387-00211-1, xii+438pp., EUR 85.55.

•

H. Chen, D. Yao, Fundamentals of Queueing Networks, Springer, Berlin, 2003, ISBN 0-387-95166-0, xviii+405pp., EUR 74,95.

•

W. Whitt, Stochastic-Process Limits, Springer, Berlin, 2002, ISBN 0-387-95358-2, xxiv+602pp., EUR 106,95.

     

Spectral sequences in combinatorial geometry: Cheeses, inscribed sets, and Borsuk-Ulam type theorems

Algebraic topological methods are especially well suited for determining the non-existence of continuous mappings satisfying certain properties. In combinatorial problems it is sometimes possible to define a mapping from a space X of configurations to a Euclidean space Rm in which a subspace, a discriminant, often an arrangement of linear subspaces A, expresses a target condition on the configurations. Add symmetries of all these data under a group G for which the mapping is equivariant. If we remove the discriminant from Rm, we can pose the problem of the existence of an equivariant mapping from X to the complement of the discriminant in Rm. Algebraic topology may sometimes be applied to show that no such mapping exists, and hence the image of the original equivariant mapping must meet the discriminant.We introduce a general framework, based on a comparison of Leray-Serre spectral sequences. This comparison can be related to the theory of the Fadell-Husseini index. We apply the framework to:

•

solve a mass partition problem (antipodal cheeses) in Rd,

•

determine the existence of a class of inscribed 5-element sets on a deformed 2-sphere,

•

obtain two different generalizations of the theorem of Dold for the non-existence of equivariant maps which generalizes the Borsuk-Ulam theorem.