In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres \({\mathbb {S}}^n\). First, we propose a new approach to isoperimetric eigenvalue inequalities based on energy index. Using this approach we show that for any positive k, the k-th non-zero eigenvalue of the Laplacian on the real projective plane endowed with a metric

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in \(\mathbb {R}_{\ge 0}\). In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and \(\Theta \)-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension \(n \ge 3\). For every bounded open subset \(\Omega \subset M\) with smooth boundary, we prove that $$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-\(A_\infty \) property) of harmonic measure with respect to surface measure, on the boundary of an open set \( \Omega \subset \mathbb {R}^{n+1}\) with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in \(\Omega \), with data in \(L^p(\partial \Omega )\) for some

In the classical theory of multiple zeta values (MZV’s), Furusho proposed a conjecture asserting that the p-adic MZV’s satisfy the same \({\mathbb {Q}}\)-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho’s conjecture in the sense that we are able to deal with all linear relations over an

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the planar stochastic hyperbolic triangulations (PSHT) defined in Curien (Probab Theory Relat Fields 165(3):509–540, 2016). The proof relies on a combinatorial argument and the Goulden–Jackson recurrence relation to obtain tightness, and

In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions

We show that the set of not uniquely ergodic d-IETs with permutation in the Rauzy class of the hyperelliptic permutation has Hausdorff dimension \(d-\frac{3}{2} \) [in the \((d-1)\)-dimension space of d-IETs] for \(d\ge 5\). For \(d=4\) this was shown by Athreya–Chaika and for \(d\in \{2,3\}\) the set is known to have dimension \(d-2\). This provides lower bounds on the Hausdorff dimension of non-weakly

We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin (in: Algebra, geometry, and physics in the 21st century. Birkhäuser/Springer, Cham, pp 99–129, 2017). In particular, we show that there exists a minimal

We show 10/8-type inequalities for some end-periodic 4-manifolds which have positive scalar curvature metrics on the ends. As an application, we construct a new family of closed 4-manifolds which do not admit positive scalar curvature metrics.

We construct the first example of a \(C^*\)-algebra A with the properties in the title. This gives a new example of non-nuclear A for which there is a unique \(C^*\)-norm on \(A \otimes A^{op}\). This example is of particular interest in connection with the Connes–Kirchberg problem, which is equivalent to the question whether \(C^*({\mathbb {F}}_2)\), which is known to have the LLP, also has the WEP

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of

In this paper we obtain a localization formula in differential K-theory for \(S^1\)-actions. We establish a localization formula for equivariant \(\eta \)-invariants by combining this result with our extension of Goette’s result on the comparison of two types of equivariant \(\eta \)-invariants. An important step in our approach is to construct a pre-\(\lambda \)-ring structure in differential K-theory

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic varieties. As a first application, we provide a general incidence estimate that is tight in its dependence on the size, degree and dimension of the varieties involved

We show that the Masur–Veech volumes and area Siegel–Veech constants can be obtained using intersection theory on strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel–Veech constants for all strata. We also show that the same results hold for the spin and hyperelliptic components of the strata

We give an alternative proof of Faltings’s theorem (Mordell’s conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings’s original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of p-adic Galois representations

For a finite subgroup \(\Gamma \subset \mathrm {SL}(2,\mathbb {C})\) and for \(n\ge 1\), we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity \(\mathbb {C}^2/\Gamma \). It is well known that \(X:={{\,\mathrm{{\mathrm {Hilb}}}\,}}^{[n]}(S)\) is a projective, crepant resolution

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey–Stewartson II equation. We then use it to prove global well-posedness and scattering in \(L^2\) for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension 2, for conductivities

We study the twisted Ruelle zeta function \(\zeta _X(s)\) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove the Fried conjecture, relating Reidemeister torsion and \(\zeta _X(0)\). In higher dimensions, we show more generally that \(\zeta _X(0)\) is locally constant with respect to the vector field X under a spectral condition. As

We bound the generation level of the Hodge filtration on the localization along a hypersurface in terms of its minimal exponent. As a consequence, we obtain a local vanishing theorem for sheaves of forms with log poles. These results are extended to \({\mathbf {Q}}\)-divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing

In this paper we consider the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo

We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil’s Murphy’s Law holds up to retraction for this scheme. Our main tool is a generalized version of the Białynicki-Birula decomposition.

We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations

Kontsevich’s 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich’s integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof

We show that a factor M is full if and only if the \(C^*\)-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type \(\mathrm{III}_1\) factor is always ergodic. As a consequence, for any type \(\mathrm{III}_1\) factor M and any \(\lambda \in ]0,1]\), there exists an irreducible AFD type \(\mathrm{III}_\lambda \) subfactor

We formulate the “real integral Hodge conjecture”, a version of the integral Hodge conjecture for real varieties, and raise the question of its validity for cycles of dimension 1 on uniruled and Calabi–Yau threefolds and on rationally connected varieties. We relate it to the problem of determining the image of the Borel–Haefliger cycle class map for 1-cycles, with the problem of deciding whether a

We prove that every countably infinite group with Kazhdan’s property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.

Due to an oversight in the Acknowledgment the grant number from Samsung Science and Technology Foundation is wrong, it should read SSTF-BA1301-06 and SSTF-BA1301-51.

The original version of this article unfortunately contains a mistake.

Given a vector field \(\rho (1,\mathbf {b}) \in L^1_\mathrm{loc}(\mathbb {R}^+\times \mathbb {R}^{d},\mathbb {R}^{d+1})\) such that \({{\,\mathrm{div}\,}}_{t,x} (\rho (1,\mathbf {b}))\) is a measure, we consider the problem of uniqueness of the representation \(\eta \) of \(\rho (1,\mathbf {b}) {\mathcal {L}}^{d+1}\) as a superposition of characteristics \(\gamma : (t^-_\gamma ,t^+_\gamma ) \rightarrow

A circle domain \(\Omega \) in the Riemann sphere is conformally rigid if every conformal map from \(\Omega \) onto another circle domain is the restriction of a Möbius transformation. We show that circle domains satisfying a certain quasihyperbolic condition, which was considered by Jones and Smirnov (Ark Mat 38, 263–279, 2000), are conformally rigid. In particular, Hölder circle domains and John

Let \(\lambda \) denote the Liouville function. We show that as \(X \rightarrow \infty \), $$\begin{aligned} \int _{X}^{2X} \sup _{\alpha } \left| \sum _{x < n \le x + H} \lambda (n) e(-\alpha n) \right| dx = o ( X H) \end{aligned}$$ for all \(H \ge X^{\theta }\) with \(\theta > 0\) fixed but arbitrarily small. Previously, this was only known for \(\theta > 5/8\). For smaller values of \(\theta \)

We prove the Topological Mirror Symmetry Conjecture by Hausel–Thaddeus for smooth moduli spaces of Higgs bundles of type \(SL_n\) and \(PGL_n\). More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from \(\ell \)-adic cohomology, as well as a Bombieri–Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke–Maass cusp form of trivial nebentypus, the eigenvalue of the n-th Hecke operator does not coincide with the Kloosterman

Let E be a CM elliptic curve over the rationals and \(p>3\) a good ordinary prime for E. We show that $$\begin{aligned} {\mathrm {corank}}_{{\mathbb {Z}}_{p}} {\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})=1 \implies {\mathrm {ord}}_{s=1}L(s,E_{/{\mathbb {Q}}})=1 \end{aligned}$$ for the \(p^{\infty }\)-Selmer group \({\mathrm {Sel}}_{p^{\infty }}(E_{/{\mathbb {Q}}})\) and the complex L-function

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences \(\mathcal {F}_a\): $$\begin{aligned} \left( \frac{aw-1}{w-1}\right) ^2+\left( \frac{aw-1}{w-1}\right) \left( \frac{az+1}{z+1}\right) +\left( \frac{az+1}{z+1}\right) ^2=3 \end{aligned}$$ and proved that for every value of \(a \in [4,7] \subset \mathbb {R}\) the correspondence \(\mathcal

This paper concerns the cohomological aspects of Donaldson–Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula, and furthermore realise the isomorphism in both of these theorems as Poincaré–Birkhoff–Witt isomorphisms for the associated

Abstract We consider the Korteweg–de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise

Abstract We prove Zimmer’s conjecture for \(C^2\) actions by finite-index subgroups of \(\mathrm {SL}(m,{\mathbb {Z}})\) provided \(m>3\). The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in \(\mathrm {SL}(m,{\mathbb {R}})\) (Brown et al. in Zimmer’s conjecture: subexponential growth, measure rigidity, and strong property (T), 2016. arXiv:1608

Abstract Given a point \(\xi \) on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of \(\xi \). When \((A, \xi )\) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important

Abstract We show that Lubin–Tate spectra at the prime 2 are Real oriented and Real Landweber exact. The proof is by application of the Goerss–Hopkins–Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for \(E_n\) with its \(C_2\)-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable

Abstract Let I be a homogeneous ideal of \(S=K[x_1,\ldots , x_n]\) and let J be an initial ideal of I with respect to a term order. We prove that if J is radical then the Hilbert functions of the local cohomology modules supported at the homogeneous maximal ideal of S/I and S/J coincide. In particular, \({\text {depth}} (S/I)={\text {depth}} (S/J)\) and \({\text {reg}} (S/I)={\text {reg}} (S/J)\).

Abstract In this paper, we explicitly compute the semisimplifications of all Jacquet modules of irreducible representations with generic L-parameters of p-adic split odd special orthogonal groups or symplectic groups. Our computation represents them in terms of linear combinations of standard modules with rational coefficients. The main ingredient of this computation is to apply Mœglin’s explicit construction

Abstract We compute p-adic étale and pro-étale cohomologies of Drinfeld half-spaces. In the pro-étale case, the main input is a comparison theorem for p-adic Stein spaces; the cohomology groups involved here are much bigger than in the case of étale cohomology of algebraic varieties or proper analytic spaces considered in all previous works. In the étale case, the classical p-adic comparison theorems

Abstract We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational

Abstract If \(E \subset \mathbb {R}^2\) is a compact set of Hausdorff dimension greater than 5 / 4, we prove that there is a point \(x \in E\) so that the set of distances \(\{ |x-y| \}_{y \in E}\) has positive Lebesgue measure.

Abstract On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group G we show that its growth \(v_{G,S}(n)\) satisfies \(\lim _{n\rightarrow \infty }\log \log v_{G,S}(n)/\log n=\alpha _{0}\), where \(\alpha

Abstract We prove the following results: (1) There is a one-relator inverse monoid \(\mathrm {Inv}\langle A\,|\,w=1 \rangle \) with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining

Let M be a circle or a compact interval, and let \(\alpha =k+\tau \ge 1\) be a real number such that \(k=\lfloor \alpha \rfloor \). We write \({{\,\mathrm{Diff}\,}}_+^{\alpha }(M)\) for the group of orientation preserving \(C^k\) diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent \(\tau \). We prove that there exists a continuum of isomorphism types of finitely generated

Let H be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph G without isolated vertices, the weighted number of graph homomorphisms \(\hom (G, H)\) satisfies the inequality $$\begin{aligned} \hom (G, H ) \le \prod _{uv \in E(G)} \hom (K_{d_u,d_v}, H )^{1/(d_ud_v)}, \end{aligned}$$ where \(d_u\) denotes the degree of vertex u in G. In particular,

The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank \(n \ge 2\) in an asymptotic sense, under some weak assumptions reminiscent of nonpositive

Let \(p\in {\mathbb {Z}}\) be an odd prime. We prove a spectral version of Tate–Poitou duality for the algebraic K-theory spectra of number rings with p inverted. This identifies the homotopy type of the fiber of the cyclotomic trace \(K({\mathcal {O}}_{F})^{\scriptscriptstyle \wedge }_{p}\mathchoice{\longrightarrow }{\rightarrow }{\rightarrow }{\rightarrow }TC({\mathcal {O}}_{F})^{\scriptscriptstyle

Let G be a group acting on a tree with cyclic edge and vertex stabilizers. Then stable commutator length (scl) is rational in G. Furthermore, scl varies predictably and converges to rational limits in so-called “surgery” families. This is a homological analog of the phenomenon of geometric convergence in hyperbolic Dehn surgery.

Using our earlier results on polynomiality properties of plethystic logarithms of generating series of certain type, we show that Schiffmann’s formulas for various counts of Higgs bundles over finite fields can be reduced to much simpler formulas conjectured by Mozgovoy. In particular, our result implies the conjecture of Hausel and Rodriguez-Villegas on the Poincaré polynomials of twisted character

We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than 2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph \(\Gamma \) is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if \(\Gamma \) is realized as the 1-skeleton of a polyhedron

In this paper we give a new proof of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demonstrate that a two-dimensional map which lies in the stable set of the renormalization operator

We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological André-Quillen cohomology of the K(n)-local Spanier–Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable \(v_n\)-periodic homotopy groups of spheres from their Morava E-cohomology (as modules

Inspired by the Ax–Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the E(n, p)-local categories over any non-principal ultrafilter