The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering ...

We initiate the systematic study of \(G_2\)-instantons with SU(2)\(^2\)-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on \(\mathbb {R}^4\times S^3\) with its two explicitly known distinct holonomy \(G_2\) metrics, which have different volume growths at infinity, exhibiting ...

We construct, using geometric invariant theory, a quasi-projective Deligne–Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded \(A_\infty \)-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a ...

We show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration.

Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and \(X \subseteq V\) some subset. A map from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, a continuous map \(f :X \rightarrow W\) is said to be piecewise-regular if there exists a ...

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh–Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these ...

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.