Integral bounds on curvature, elliptic estimates and rectifiability of singular sets

((no abstract)) . Submitted: February 2002, Final version: October 2002.     

Logarithmic Sobolev Inequalities,Matrix Models and Free Entropy

We give two applications of logarithmic Sobolev inequalities to matrix models and free probability. We also provide a new characterization of semi-circular systems through a Poincare-type inequality.     

Quivers, Floer cohomology, and braid group actions

We consider the derived categories of modules over a certain family () of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type We show that each of these two rather different objects encodes the topology of curves on an -punctured disc. We prove that the braid group acts faithfully on the derived category of -modules, and that it injects into the symplectic mapping class group of the Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should be able to associate to the Milnor fibre a triangulated category (its construction has not been carried out in detail yet). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of -modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.

     

Irreducibility, Brill-Noether loci, and Vojta's inequality

This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover of the projective line defined over a number field , there exist infinitely many -rational points on the projective line such that the fiber of over is irreducible over . In this paper, we consider similar statements about algebraic points of higher degree on curves of any genus. We prove that Hilbert's irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus, via a theorem of Abramovich and Harris, to points of degree 3 or less on any curve. We also present examples that show that this generalization does not hold for points of degree 4 or more. These examples come from an earlier geometric construction of Debarre and Fahlaoui; some additional necessary facts about this construction can be found in the appendix provided by Debarre. We exhibit a connection between these irreducibility questions and the sharpness of Vojta's inequality for algebraic points on curves. In particular, we show that Vojta's inequality is not sharp for the algebraic points arising in our examples.

     

Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

Involutions fixing , I

This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space with its normal bundle nonbounding and a Dold manifold with 0$"> and 0$">. For odd , the complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on codimension of may not be best possible; for even , the problem may be reduced to the problem for even projective spaces.

     

One‐factorizations of complete multigraphs and quadrics in PG(n,q)

A new infinite family of simple indecomposable one‐factorizations of the complete multigraphs is constructed by using quadrics of finite projective spaces. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 139–143, 2002; DOI 10.1002/jcd.997     

Compatible Dubrovin–Novikov Hamiltonian Operators,Lie Derivative,and Integrable Systems of Hydrodynamic Type [Erratum]

Correction to the paper published in Vol. 133, No. 2, November, 2002.     

New examples of non-slice, algebraically slice knots

For 1$">, if the Seifert form of a knotted -sphere in has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three. However, in the three-dimensional case it is true that if the metabolizer has a basis represented by a strongly slice link, then is slice. The question has been asked as to whether it is sufficient that each basis element is represented by a slice knot to assure that is slice. For genus one knots this is of course true; here we present genus two counterexamples.

     

Primary decomposition: Compatibility, independence and linear growth

For finitely generated modules over a Noetherian ring , we study the following properties about primary decomposition: (1) The Compatibility property, which says that if and is a -primary component of for each , then ; (2) For a given subset , is an open subset of if and only if the intersections for all possible -primary components and of ; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals of there exists a such that for any there exists a primary decomposition of such that every -primary component of that primary decomposition contains .

     

On a problem of Byrnes concerning polynomials with restricted coefficients, II

As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length of a polynomial with all coefficients in which has a zero of a given order at . In that paper we showed that for all and showed that the extremal polynomials for were those conjectured by Byrnes, but for that rather than . A polynomial with was exhibited for , but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that is one of the 7 values or . Here we prove that without determining all extremal polynomials. We also make some progress toward determining . As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if is a primitive th root of unity where is a prime, then the condition that all coefficients of be in , together with the requirement that be divisible by puts severe restrictions on the possible values for the cyclotomic integer .

     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

Author Index,Volume 22 (2002)

Authors Index

Author Index, Volume 21 (2002)     

Author Index, Volume 22 (2002)

Authors Index

Author Index, Volume 22 (2002)     

Isomorphisms of function modules, and generalized approximation in modulus

For a function algebra we investigate relations between the following three topics: isomorphisms of singly generated -modules, Morita equivalence bimodules, and ``real harmonic functions' with respect to . We also consider certain groups which are naturally associated with a uniform algebra . We illustrate the notions considered with several examples.

     

Meet,discuss, and segregate!

We present a model of opinion dynamics in which agents adjust continuous opinions as a result of random binary encounters whenever their difference in opinion is below a given threshold. High thresholds yield convergence of opinions toward an average opinion, whereas low thresholds result in several opinion clusters. The model is further generalized to network interactions, threshold heterogeneity, adaptive thresholds, and binary strings of opinions. © 2002 Wiley Periodicals, Inc.     

Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Let be a connected semisimple Lie group with finite center. Let be the maximal compact subgroup of corresponding to a fixed Cartan involution . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary -module contains a -type with highest weight , then has infinitesimal character . Here is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary -modules with non-zero Dirac cohomology, provided has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant's cubic Dirac operator.