We develop a new technique for proving concentration inequalities which relate the variance and influences of Boolean functions. Using this technique, we 1. Settle a conjecture of Talagrand (Combinatorica 17(2):275–285, 1997), proving that $$\begin{aligned} \int _{\left\{ -1,1\right\} ^{n}}\sqrt{h_{f}\left( x\right) }d\mu \left( x\right) \ge C\cdot \mathrm {Var}\left( f\right) \cdot \left( \log \left( \frac{1}{\sum \mathrm {Inf}_{i}^{2}\left( f\right) }\right) \right) ^{1/2}, \end{aligned}$$ where \(h_{f}\left( x\right) \) is the number of edges at x along which f changes its value, \(\mu \left( x\right) \) is the uniform measure on \(\left\{ -1,1\right\} ^{n}\), and \(\mathrm {Inf}_{i}\left( f\right) \) is the influence of the i-th coordinate. 2. Strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, \(\mathrm {Var}\left( f\right) \le C\sum _{i=1}^{n}\frac{\mathrm {Inf}_{i}\left( f\right) }{1+\log \left( 1/\mathrm {Inf}_{i}\left( f\right) \right) }\). We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn–Kalai–Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. 3. Improve a quantitative relation between influences and noise stability given by Keller and Kindler. Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.

We prove that \(m\)-dimensional Lipschitz graphs with anisotropic mean curvature bounded in \(L^p\), \(p>m\), are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a \(C^{1,1}\) neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a \(C^{1,1}\) (resp. \(C^1\)) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.

In this paper we solve an open problem concerning the characterization of those measurable sets Omega subset {mathbb {R}}^{2d} that, among all sets having a prescribed Lebesgue measure, can trap the largest possible energy fraction in time-frequency space, where the energy density of a generic function fin L^2({mathbb {R}}^d) is defined in terms of its Short-time Fourier transform (STFT) {mathcal {V}}f(x,omega ), with Gaussian window. More precisely, given a measurable set Omega subset {mathbb {R}}^{2d} having measure s> 0, we prove that the quantity ΦΩ=max{∫Ω|Vf(x,ω)|2dxdω:f∈L2(Rd),‖f‖L2=1},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Phi _\\Omega =\\max \\Big \\{\\int _\\Omega |{\\mathcal {V}}f(x,\\omega )|^2\\,dxd\\omega : f\\in L^2({\\mathbb {R}}^d),\\ \\Vert f\\Vert _{L^2}=1\\Big \\}, \\end{aligned}$$\\end{document}is largest possible if and only if Omega is equivalent, up to a negligible set, to a ball of measure s, and in this case we characterize all functions f that achieve equality. This result leads to a sharp uncertainty principle for the “essential support” of the STFT (when d=1, this can be summarized by the optimal bound Phi _Omega le 1-e^{-|Omega |}, with equality if and only if Omega is a ball). Our approach, using techniques from measure theory after suitably rephrasing the problem in the Fock space, also leads to a local version of Lieb’s uncertainty inequality for the STFT in L^p when pin [2,infty ), as well as to L^p-concentration estimates when pin [1,infty ), thus proving a related conjecture. In all cases we identify the corresponding extremals.

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold Sigma with Betti number b_1, the order of vanishing of the Ruelle zeta function at zero equals 4-b_1, while in the hyperbolic case it is equal to 4-2b_1. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle SSigma with harmonic 1-forms on Sigma .

Let G be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on G in which the role of the quasi-simple admissible G-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of G. We show that they produce eigenstates for spin versions of quantum hyperbolic Calogero–Moser systems. In the second part of the paper we define and study special subclasses of global and formal elementary spherical functions, which we call global and formal N-point spherical functions. Formal N-point spherical functions arise as limits of correlation functions for boundary Wess–Zumino–Witten conformal field theory on the cylinder when the position variables tend to infinity. We construct global N-point spherical functions in terms of compositions of equivariant differential intertwiners associated with principal series representations, and express them in terms of Eisenstein integrals. We show that the eigenstates of the quantum spin Calogero–Moser system associated to N-point spherical functions are also common eigenfunctions of a commuting family of first-order differential operators, which we call asymptotic boundary Knizhnik–Zamolodchikov–Bernard operators. These operators are explicitly given in terms of folded classical dynamical r-matrices and associated dynamical k-matrices.

We construct a zig–zag from the once delooped space of pseudoisotopies of a closed 2n-disc to the once looped algebraic K-theory space of the integers and show that the maps involved are p-locally (2n-4)-connected for n,{>},3 and large primes p. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik–Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa’s stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of mathrm {BDiff}_partial (D^{2n+1}) in degrees up to 2n-5.

In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products of conjugate self-dual automorphic representations, within the framework of the Gan–Gross–Prasad conjecture. We show that if the central critical value of the Rankin–Selberg L-function does not vanish, then the Bloch–Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch–Kato Selmer group constructed from a certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin–Selberg L-function, then the Bloch–Kato Selmer group is of rank one.

We show that the integer cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by Berglund, we compute the rational homotopy Lie algebras of those spaces, and obtain some estimates for Betti numbers of their free loop spaces in case of torsion coefficients. We also prove and conjecture some generalisations of our main result.

For a random matrix of entries sampled independently from a fairly general distribution in \({{\mathbf {Z}}}\) we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only \(n^{-1+\varepsilon }\).

Let \(E/{\mathbb {Q}}\) be an elliptic curve and p an odd prime where E has good reduction, and assume that E admits a rational p-isogeny. In this paper we study the anticyclotomic Iwasawa theory of E over an imaginary quadratic field in which p splits, which we relate to the anticyclotomic Iwasawa theory of characters by a variation of the method of Greenberg–Vatsal. As a result of our study we obtain proofs (under relatively mild hypotheses) of Perrin-Riou’s Heegner point main conjecture, a p-converse to the theorem of Gross–Zagier and Kolyvagin, and the p-part of the Birch–Swinnerton-Dyer formula in analytic rank 1, for Eisenstein primes p.