Geometric and Functional Analysis

We prove that a sequence of solutions of the Seiberg–Witten equation with multiple spinors in dimension three can degenerate only by converging (after rescaling) to a Fueter section of a bundle of moduli spaces of ASD instantons.

Let G be a semisimple Lie group with finite center and $ \Gamma $ a discrete Zariski dense subgroup of G. We use here the indicator of growth of $ \Gamma $ , introduced in [Q3], to build analoguous of Patterson—Sullivan measures used in $ {\Bbb R} $ -rank 1.¶¶Résumé. Soient G un groupe de Lie semi-simple de centre fini et $ \Gamma $ un sous-groupe discret Zariski dense de G. Nous utilisons ici l'indicateur de croissance de $ \Gamma $ , introduit dans [Q3], pour construire des analogues des mesures de Patterson—Sullivan utilisées en $ {\Bbb R} $ -rang 1.

We introduce a new iterative amalgamated free product construction of II $$_1$$ factors, and use it to construct a separable II $$_1$$ factor which does not have property Gamma and is not elementarily equivalent to the free group factor $$\text {L}(\mathbb F_n)$$ , for any $$2\le n\le \infty $$ . This provides the first explicit example of two non-elementarily equivalent II $$_1$$ factors without property Gamma. Moreover, our construction also provides the first explicit example of a II $$_1$$ factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory.

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is $$o(\log n)$$ . However, both results become useless when the total influence of the function is $$\omega (\log n)$$ . The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of $$S_n$$ . In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time $$2^{O(K\log K)}$$ using membership queries. Thus, the class of functions with total influence $$O(\log n/\log \log n)$$ is agnostically learnable in $$\mathsf{poly}(n)$$ time.

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For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x 1, x 2. Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples.

We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its $${\ell_2}$$ norm. Our results pertain to a wide class of random matrices, including matrices with independent entries, symmetric and skew-symmetric matrices, as well as some other naturally arising ensembles. The matrices can be real and complex; in the latter case we assume that the real and imaginary parts of the entries are independent.