In this article we define new flows on the Hitchin components for \(\mathrm {PGL}(V)\). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when \(n=2\). We determine a global coordinate system

We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety X, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification \((M,{\mathbf {D}})\) of X. We exhibit a broad class of pairs \((M,{\mathbf {D}})\) (characterized by the absence of relative holomorphic spheres or vanishing of certain

Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence between such manifold submetries and maximal Laplacian algebras, thus solving the Inverse Invariant Theory problem for this class of partitions

Singular vectors are those for which the quality of rational approximations provided by Dirichlet’s Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular vectors lying on self-similar fractals in \({\mathbb {R}}^d\) satisfying the open set condition. The bound is in terms of quantities which are closely tied to Frostman

We establish Anderson localization for general analytic k-frequency quasi-periodic operators on \({\mathbb {Z}}^d\) for arbitraryk, d.

We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich–Lishak–Nabutovsky–Rotman. We show also that for any \(C>0\) there is a Riemannian metric g on a 3-sphere such that \({\hbox {vol}}(S^3,g)=1\) and for any map \(f:S^3\rightarrow {\mathbb {R}}^2\) there is some \(x\in {\mathbb {R}}^2\)

I show that quasidiagonality and AF embeddability are equivalent properties for traceless \(\mathrm C^*\)-algebras and are characterised in terms of the primitive ideal space. For nuclear \(\mathrm C^*\)-algebras the same characterisation determines when Connes and Higson’s E-theory can be unsuspended.

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. We also construct the first known infinite family of almost symplectomorphic Weinstein domains

In this article we show that in any dimension there exist infinitely many pairs of formally contact isotopic isocontact embeddings into the standard contact sphere which are not contact isotopic. This is the first example of rigidity for contact submanifolds in higher dimensions. The contact embeddings are constructed via contact push-offs of higher-dimensional Legendrian submanifolds, a construction

Let \((M^n,g)\) be simply connected, complete, with non-positive sectional curvatures, and \(\Sigma \) a 2-dimensional closed integral current (or flat chain mod 2) with compact support in M. Let S be an area minimising integral 3-current (resp. flat chain mod 2) such that \(\partial S = \Sigma \). We use a weak mean curvature flow, obtained via elliptic regularisation, starting from \(\Sigma \), to

We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise is bounded and has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which

We prove two results on arithmetic quantum chaos for dihedral Maaß forms, both of which are manifestations of Berry’s random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level 1 forms, these results were previously known for Eisenstein series and conditionally on the generalised Lindelöf hypothesis for Hecke–Maaß eigenforms. A key aspect of

We prove that on a closed, orientable surface of genus g, the maximum cardinality of a set of simple loops with the property that no two are homotopic or intersect in more than k points grows as a function of g like \(g^{k+1}\), up to a factor of \(\log g\). The proof of the upper bound uses arguments from probabilistic combinatorics and a theorem of Scott related to the fact that surface groups are

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if \(A\subset {\mathbb {R}}^2\) is a Borel set of Hausdorff dimension \(s>1\), then its distance set has Hausdorff dimension at least \(37/54\approx 0.685\). Moreover, if \(s\in (1,3/2]\), then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned

This paper accomplishes two things. First, we construct a geometric analog of the rational Tits building for general noncompact, complete, finite volume n-manifolds M of bounded nonpositive curvature. Second, we prove that this analog has dimension less than \(\lfloor n/2\rfloor \).

All \(\mathrm {GL}(2, \mathbb {R})\) orbits in hyperelliptic components of strata of abelian differentials in genus greater than two are closed, dense, or contained in a locus of branched covers.

In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus \(\gamma \) and the area. The equality in Yang–Yau’s estimate is attained for \(\gamma =0\) by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for \(\gamma

In this article, we study the smallest gaps between eigenvalues of the Gaussian orthogonal ensemble (GOE). The main result is that the smallest gaps, after being normalized by n, will converge to a Poisson distribution, and the limiting density of the kth normalized smallest gap is \(2{}x^{2k-1}e^{-x^{2}}/(k-1)!\). The proof is based on the method developed in Feng and Wei (Small gaps of circular \(\beta

We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is inspired by the recent result (Gavrilov in A steady Euler flow with compact support. Geom Funct Anal 29(1):90–197, [Gav19]).

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for \(C^0\) metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from \(C^0\) initial data which is smooth for positive times, and that the weak lower scalar

We prove analogues of the Szemerédi–Trotter theorem and other incidence theorems using \(\delta \)-tubes in place of straight lines, assuming that the \(\delta \)-tubes are well spaced in a strong sense.