LOCALLY CONFORMALLY SYMPLECTIC STRUCTURES ON COMPACT NON- KAHLER COMPLEX SURFACES

International audienceWe prove that every compact complex surface with odd first Betti num-ber admits a locally conformally symplectic 2-form which tames the underlying almost complex structure

Algebraic Multigrid for Markov Chains and Tensor Decomposition

The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes. This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required

Non-compact Einstein manifolds with symmetry

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result: The nilradical N of G acts polarly, and the N-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.Comment: 57 page

완전 비선형 포물 편미분 방정식의 정칙 이론과 그 응용

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 8. 이기암.이 학위 논문에서는 비발산 구조를 갖는 완전 비선형 포물 방정식의 해의 정칙 이론과 그 응용에 대하여 연구하였다. 첫번째 장은 완전 비선형 고른 포물형 및 퇴화된 포물형 방정식의 해의 점근적 행동 양상에 대한 연구이다. 먼저, 포물 방정식의 정규화 된 해가 시간이 흐름에 따라 방정식과 관련된 완전 비선형 타원 작용소의 제 1 고유 함수로 수렴함을 증명하였다. 또한 볼록한 영역에서 오목한 완전 비선형 제차 작용소가 주어졌을때, 포물형 해의 초기 기하적 구조-특정한 오목성(log-concavity, power concavity)-가 보존되는 것을 보였다. 위의 수렴성을 이용하면 제 1 고유 함수 또한 같은 기하적 구조를 가짐을 알 수 있다. 두번째 장에서는 완전 리만 다양체 위에서 완전 비선형 포물 방정식의 해를 다루었는데, 특히 정칙 이론의 초석이 되는 포물형 Harnack 부등식을 증명하였다. 선형 작용소에 대해서는 거리 함수로 정의된 특정한 조건을 가정하고 정칙인 해의 대역적 Harnack 부등식을 얻었다. 또 단면 곡률의 하한을 가지는 다양체에 대해 비선형 작용소의 국소적 Harnack 부등식을 보였다. 마지막으로 Jensen의 sup- and inf-convolution을 이용하여, 연속 해인 viscosity 해에 대한 Harnack 부등식을 증명하였다.Abstract 1 Introduction 1 1.1 Long-time asymptotics for parabolic equations 2 1.2 Parabolic Harnack inequality on Riemannian manifolds 4 2 Preliminaries 8 2.1 Viscosity solutions 8 2.1.1 Uniformly elliptic operator 8 2.1.2 Viscosity solutions 10 2.1.3 Regularity for uniformly elliptic and parabolic equations 11 2.2 Riemannian geometry 12 2.2.1 Variation formulas and Volume comparison 15 2.2.2 Semi-concavity 18 2.2.3 Viscosity solutions on Riemannian manifolds 19 3 Asymptotic behavior of parabolic equations 22 3.1 Uniformly parabolic equations 22 3.1.1 Elliptic eigenvalue problem 22 3.1.2 Long-time asymptotics for uniformly parabolic equations 23 3.1.3 Log-concavity 29 3.2 Degenerate parabolic equations 39 3.2.1 Sub-linear elliptic eigenvalue problems 39 3.2.2 Long-time asymptotics for degenerate parabolic equations 42 3.2.3 Square-root concavity of thepressure 47 4 Harnack inequality on Riemannian manifolds 69 4.1 Harnack inequality for linear parabolic operators 69 4.1.1 ABP-Krylov-Tso type estimate 70 4.1.2 Barrier functions 78 4.1.3 Parabolic version of the Calderon-Zygmund decomposition 90 4.1.4 Proof of parabolic Harnack inequality 94 4.1.5 Weak Harnack inequality 107 4.2 Harnack inequality for nonlinear parabolic operators 110 4.3 Harnack inequality for viscosity solutions 121 4.3.1 Sup-and inf-convolution 121 4.3.2 Proof of parabolic Harnack inequality 132 Abstract (in Korean)Docto

Quanta of Maths

The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics