Let \(G=\mathrm{GL}_{2n}\) over a totally real number field F and \(n\ge 2\). Let \(\Pi \) be a cuspidal automorphic representation of \(G(\mathbb {A})\), which is cohomological and a functorial lift from SO\((2n+1)\). The latter condition can be equivalently reformulated that the exterior square L-function of \(\Pi \) has a pole at \(s=1\). In this paper, we prove a rationality ...

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general \(L^p\) lower bound on the Calabi functional involving test configurations and their associated numerical ...

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to ...

We construct and study the moduli of continuous representations of a profinite group with integral p-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a p-adic local field, we show that these moduli spaces admit ...

Let K be the function field of a smooth curve over an algebraically closed field k. Let X be a scheme, which is smooth and projective over K. Suppose that the cotangent bundle \(\Omega _{X/K}\) is ample. Let \(R:=\mathrm{Zar}(X(K)\cap X)\) be the Zariski closure of the set of all K-rational points of X, endowed with its reduced induced structure. We prove that for each irreducible ...

For any \(H \in (0,\frac{1}{2})\), we construct complete, non-proper, stable, simply-connected surfaces embedded in \({\mathbb H}^2\times {\mathbb R}\) with constant mean curvature H.

Jun O’Hara invented a family of knot energies \(E^{j,p}\), \(j,p \in (0, \infty )\), O’Hara in Topology Hawaii (Honolulu, HI, 1990). World Science Publication, River Edge 1992. We study the negative gradient flow of the sum of one of the energies \(E^\alpha = E^{\alpha ,1}\), \(\alpha \in (2,3)\), and a positive multiple of the length. Showing that the gradients of these knot ...

We extend Igusa’s description of the relation between invariants of binary sextics and Siegel modular forms of degree 2 to a relation between covariants and vector-valued Siegel modular forms of degree 2. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree 2.

We prove that the pseudoisotopy stable range for manifolds of dimension 2n can be no better than \((2n-2)\). In order to do so, we define new characteristic classes for block bundles, extending our earlier work with Ebert, and prove their non-triviality. We also explain how similar methods show that \(\mathrm {Top}(2n)/\mathrm {O}(2n)\) is rationally \((4n-5)\)-connected.

Assuming that T is a potential blow up time for the Navier–Stokes system in \(\mathbb {R}^{3}_{+}\), we show that the norm of the velocity field in the Lorenz space \(L^{3,q}\) with \(q<\infty \) goes to \(\infty \) as time t approaches T.

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite.

We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also ...

Let G be a simple algebraic group over an algebraically closed field K of characteristic \(p\geqslant 0\), let H be a proper closed subgroup of G and let V be a nontrivial irreducible KG-module, which is p-restricted, tensor indecomposable and rational. Assume that the restriction of V to H is irreducible. In this paper, we study the triples (G, H, V) of this form when G is a ...

We study hyperbolic systems with multiplicities and smooth coefficients. In the case of non-analytic, smooth coefficients, we prove well-posedness in any Gevrey class and when the coefficients are analytic, we prove \(C^\infty \) well-posedness. The proof is based on a transformation to block Sylvester form introduced by D’Ancona and Spagnolo (Boll UMI 8(1B):169–185, 1998) which ...

We obtain a complete solution to the problem of classifying all two-dimensional ideal fluid flows with harmonic Lagrangian labelling maps; thus, we explicitly provide all solutions, with the specified structural property, to the incompressible two-dimensional Euler equations (in Lagrangian variables).

This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors’ previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393–423, 2012), where dispersive equations were treated. For operators \(a(D_x)\) of order m satisfying the dispersiveness condition ...

We consider a short time existence problem motivated by a conjecture of Joyce (Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. arXiv:1401.4949, 2014). Specifically we prove that given any compact Lagrangian \(L\subset \mathbb {C}^n\) with a finite number of singularities, each asymptotic ...

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice \({\mathbb Z}^{2}\), gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all ...

For an ordinary abelian variety X, \(F^e_*\mathcal {O}_X\) is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and the Kodaira dimension of X is non-negative, then X is an ordinary abelian variety.

We prove that every non-trivial structure of a rationally connected fibre space on a generic (in the sense of Zariski topology) hypersurface V of degree M in the \((M+1)\)-dimensional projective space for \(M\ge 16\) is given by a pencil of hyperplane sections. In particular, the variety V is non-rational and its group of birational self-maps coincides with the group of biregular ...

Let \(\Gamma \) be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold X. We show that a normal subgroup \(\Gamma _0\) has critical exponent equal to the critical exponent of \(\Gamma \) if and only if \(\Gamma /\Gamma _0\) is amenable. We prove a similar result for the exponential growth rate of closed geodesics on \(X/\Gamma \). These ...

Markov’s inequality is a certain estimate for the norm of the derivative of a polynomial in terms of the degree and the norm of this polynomial. It has many interesting applications in approximation theory, constructive function theory and in analysis (for instance, to Sobolev inequalities or Whitney-type extension problems). One of the purposes of this paper is to give a solution ...