Abstract 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.

Abstract 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

Abstract 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

Abstract 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

Abstract 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

Abstract 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.

Abstract 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

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.

Abstract 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

Abstract 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

The original version of this article unfortunately contains a mistake.

Abstract 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

Abstract 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

Abstract 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

Abstract 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

Abstract 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

Abstract 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

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 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

Abstract 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

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

We develop a fairly explicit Kuznetsov formula on GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl's law, exceptional eigenvalues, a large sieve and L-functions are given.

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, we show that every irreducible smooth G-representation contains a rigid type

We prove a descent criterion for certain families of smooth representations of GL n ( F ) (F a p-adic field) in terms of the γ -factors of pairs constructed in Moss (Int Math Res Not 2016(16):4903-4936, 2016). We then use this descent criterion, together with a theory of γ -factors for families of representations of the Weil group W F (Helm and Moss in Deligne-Langlands gamma factors in families, arXiv:1510

Block and Göttsche have defined a q-number refinement of counts of tropical curves in R 2 . Under the change of variables q = e iu , we show that the result is a generating series of higher genus log Gromov-Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.