The classification of solutions for semilinear partial differential equations, as well as the classification ofcritical points of the corresponding functionals, have wide applications in the study of partial differential equationsand differential geometry. The classical moving plane method and the moving sphere method on $\mathbb{R}^n$ provide aneffective approach to capturing the symmetry of solutions. In this talk, we focus on the equation\begin{equation*} P_k u = f(u)\end{equation*}on hyperbolic spaces $\mathbb{H}^n$, where $P_k$ denotes the GJMS operators and $f : \mathbb{R} \to \mathbb{R}$ satisfies certain growth conditions. I will introduce a moving sphere approach on $\mathbb{H}^n$, to obtain the symmetry property as well as the classification result towards positive solutions. Our methods also rely on Helgason-Fourier analysis and Hardy-Littlewood-Sobolev inequalities on hyperbolic spaces together with a newly introduced Kelvin-type transform on $\mathbb{H}^n$.
Events this week
– Kidder 237
– Kidd 280
There is a classical problem to determine whether a manifold admits r linearly independent tangent vector fields. In the case of one everywhere non-zero vector field, this problem was solved by Hopf, and the obstruction is the Euler characteristic of the manifold. Bokstedt, Dupont and Svane approached this problem by instead determining the obstruction to finding a cobordant manifold with r vector fields.
We extend their results by looking at obstructions to finding linearly independent complex sections of the tangent bundle of almost complex manifolds. In this case, we are able to describe the rational obstruction for almost complex manifolds. This obstruction is given in terms of Chern characteristic numbers. Moreover, we are able to give certain bounds for r under which the torsion obstruction vanishes.
– STAG 263
In 1947 Skolem proved that the multiplicative group of an algebraic number field K modulo its torsion subgroup is a free abelian group. We outline a proof that this remains true for infinite algebraic extensions of the rationals provided the infinite extension satisfies the Bogomolov property. In contrast to these results, the multiplicative group of all nonzero algebraic numbers modulo its torsion subgroup is known to be a vector space over the rationals, and therefore it is a divisible abelian group.
Events next week
– N/A
– Kidd 280
In this talk, I discuss an infinite-dimensional Lagrange-multipliers problem that first appeared in Donaldson and Segal’s paper “Gauge Theory in Higher Dimensions II”. The longterm goal is to apply Floer theory to a functional whose critical points are generalizations of three (real) dimensional, special Lagrangian submanifolds. I will discuss a transversality theorem related to the moduli space of solutions to the Lagrange multiplers problem.
– TBA
Double saddle-point systems are drawing increasing attention in the past few years, due to the importance of multiphysics and other relevant applications and the challenge in developing efficient iterative numerical solvers. In this talk we describe some of the numerical properties of the matrices arising from these problems. We derive eigenvalue bounds and analyze the spectrum of preconditioned matrices, and it is shown that if Schur complements are effectively approximated, the eigenvalue structure gives rise to rapid convergence of Krylov subspace solvers. A few numerical experiments illustrate our findings.