A stability result for balanced dictatorships in Sn

We prove that a balanced Boolean function on S_n whose Fourier transform is highly concentrated on the first two irreducible representations of S_n, is close in structure to a dictatorship, a function which is determined by the image or pre‐image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on S_n generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, [6] deals with Boolean functions of expectation O(1/ n) whose Fourier transform is highly concentrated on the first two irreducible representations of S_n. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point‐stabilizers. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 494–530, 2015     

A character approach to the multiplier conjecture and a new result for the case n = 3n1

In this article we present a character approach to the Multiplier Conjecture. Using this method we obtain a new result for the case n = 3n₁ which improves a result due to McFarland. We also give an application of our theorem. © 1997 John Wiley & Sons, Inc.     

A conjectural generalization of the n! result to arbitrary groups

The aim of this paper is to formulate a conjecture for an arbitrary simple Lie algebra g in terms of the geometry of principal nilpotent pairs. When g is specialized to sln, this conjecture readily implies the n! result and it is very likely that, in fact, it is equivalent to the n! result in this case. In addition, this conjecture can be thought of as generalizing an old result of Kostant. In another direction, we show that to prove the validity of the n! result for an arbitrary n and an arbitrary partition of n, it suffices to show its validity only for the staircase partitions.     

On Beauville structures on the groups S n and A n

It is known that the symmetric group S _n , for n ≥ 5, and the alternating group A _n , for large n, admit a Beauville structure. In this paper we prove that A _n admits a Beauville (resp. strongly real Beauville) structure if and only if n ≥ 6 (resp n ≥ 7). We also show that S _n admits a strongly real Beauville structure for n ≥ 5.     

Willmore Surfaces in S n

A surface x> : M S ⁿ is called a Willmore surface if it is a critical surface of the Willmore functional. It is well known that any minimal surface is a Willmore surface and that many nonminimal Willmore surfaces exists. In this paper, we establish an integral inequality for compact Willmore surfaces in S ⁿ and obtain a new characterization of the Veronese surface in S ⁴ as a Willmore surface. Our result reduces to a well-known result in the case of minimal surfaces.     

A relaxation result for micromagnetics in SBV

An integral representation for the functional

A Characterization of Minimal Legendrian Submanifolds in S2n+1

Let x: L ⁿ S²ⁿ⁺¹ R²ⁿ⁺² be a minimal submanifold in S²ⁿ⁺¹. In this note, we show that L is Legendrian if and only if for any A su(n + 1) the restriction to L of Ax, (–1)x satisfies f = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian with multiplicity at least (n(n + 3)). Moreover if the multiplicity equals to ;(n(n + 3)), then L ⁿ is totally geodesic.     

A surjectivity result for quasibounded operators

Using a degree theory for countably 1-contractive operators, we show a surjectivity theorem for such quasibounded operators. Moreover, the existence of an eigenvalue for these operators is presented.     

A finiteness result for Heegaard splittings

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g−3 disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to fact that in general there can exist infinitely many such systems of disks which satisfy just the Casson-Gordon rectangle condition.     

A boundedness result for twisted convolution

We consider twisted convolution operators with kernels having singularities on a sphere and having as Fourier transform the oscillatory symbol m_α(|ξ|) = |ξ|^–αe^i|ξ|, 0 ≤ ??α < 2n. We give integral representations for such operators and, as a principal result, we study L_p–L_q estimates for them.     

A Bochner-Riesz result for a cone

In this paper we use the curvature and torsion of a curve inR ³, together with a standard argument to give a near optimal result for the Bochner-Riesz problem for certain conical hypersurfaces inR ⁴. The main interest in this result lies in the close relation between such Bochner-Riesz problems and certain Maximal function problems in R³. Supported by an E.P.S.R.C. grant.     

A sensitivity result for semidefinite programs

We study the sensitivity of solutions of linear semidefinite programs under small arbitrary perturbations of the data. We present an elementary and self-contained proof of the differentiability of the solutions as functions of the perturbations, and we characterize the derivative as the solution of a system of linear equations.