Uzi Vishne

Explicit constructions of Ramanujan complexes of type A d

In this paper we present for every d ≥ 2 and every local field F of positive characteristic, explicit constructions of Ramanujan complexes which are quotients of the Bruhat--Tits building Bd(F) associated with PGLd(F).

Pattern matching with address errors: rearrangement distances

Historically, approximate pattern matching has mainly focused at coping with errors in the data, while the order of the text/pattern was assumed to be more or less correct. In this paper we consider a class of pattern matching problems where the content is assumed to be correct, while the locations may have shifted/changed. We formally define a broad class of problems of this type, capturing situations in which the pattern is obtained from the text by a sequence of rearrangements. We consider several natural rearrangement schemes, including the analogues of the l 1 and l 2 distances, as well as two distances based on interchanges. For these, we present efficient algorithms to solve the resulting string matching problems.

Ramanujan complexes of typeÃd

We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS]). They are obtained as quotients of the buildings of typeÃd−1 associated with PGLd(F) whereF is a local field of positive characteristic.

Open problems on central simple algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

Efficient Linear Feedback Shift Registers with Maximal Period

We introduce and analyze an efficient family of linear feedback shift registers (LFSRs) with maximal period. This family is word-oriented and is suitable for implementation in software, thus provides a solution to a recent challenge 8]. The classical theory of LFSRs is extended to provide efficient algorithms for generation of irreducible and primitive LFSRs of this new type.

Pattern matching with address errors: Rearrangement distances

Historically, approximate pattern matching has mainly focused at coping with errors in the data, while the order of the text/pattern was assumed to be more or less correct. In this paper we consider a class of pattern matching problems where the content is assumed to be correct, while the locations may have shifted/changed. We formally define a broad class of problems of this type, capturing situations in which the pattern is obtained from the text by a sequence of rearrangements. We consider several natural rearrangement schemes, including the analogues of the l1 and l2 distances, as well as two distances based on interchanges. For these, we present efficient algorithms to solve the resulting string matching problems.

Division algebras and noncommensurable isospectral manifolds

A. Reid (R) showed that if 1 and 2 are arithmetic lattices in G = PGL2(R) or in PGL2(C) which give rise to isospec- tral manifolds, then 1 and 2 are commensurable (after conju- gation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S = PGLd(C)/PUd(C), the situation is quite different: there are arbitrarily large finite families of isospectral non-commensurable compact manifolds covered by S. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but dif- ferent invariants.

Coxeter covers of the symmetric groups

Abstract We study Coxeter groups from which there is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set T of transpositions. These quotients, denoted here by CY(T ), are a special case of the generalized Coxeter groups defined in [5], and also arise in the computation of certain invariants of surfaces. We use a surprising action of Sn on the kernel of the surjection CY(T ) → Sn to show that this kernel embeds in the direct product of ncopies of the free group π 1(T ), except when Tis the full set of transpositions in S 4. As a result, we show that each group CY(T ) either is virtually Abelian or contains a non-Abelian free subgroup.

CLASSES OF WIRING DIAGRAMS AND THEIR INVARIANTS

Wiring diagrams usually serve as a tool in the study of arrangements of lines and pseudolines. In this paper we go in the opposite direction, using known properties of line arrangements to motivate certain equivalence relations and actions on sets of wiring diagrams, which preserve the incidence lattice and the fundamental groups of the affine and projective complements of the diagrams. These actions are used in [GTV] to classify real arrangements of up to 8 lines and show that in this case, the incidence lattice determines both the affine and the projective fundamental groups.

CHARACTERS AND SOLUTIONS TO EQUATIONS IN FINITE GROUPS

The number of ways an element of a finite group can be expressed as a square, a commutator, or more generally in the form w(x1, …, xr), where w is a word in the free group, defines a natural class function. We investigate some properties of these class functions, in particular their tendency to be characters or virtual characters of the underlying group. Generalizing classical results of Frobenius and others, we prove that generalized commutators yield characters in this manner, and use this to exhibit a criterion for nilpotency based on a certain equation associated with the irreducible characters.