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Differential equations. --- 517.91 Differential equations --- Differential equations --- Geometria algebraica --- Topologia --- Geometria de Riemann --- Equacions diferencials --- Anàlisi harmònica --- Geometria algèbrica --- Geometria --- Anàlisi p-àdica --- Cicles algebraics --- Espais algebraics --- Esquemes (Geometria algebraica) --- Esquemes de grups (Matemàtica) --- Geometria algebraica aritmètica --- Grups algebraics lineals --- Geometria analítica --- Geometria biracional --- Geometria enumerativa --- Geometria tropical --- Homologia --- Singularitats (Matemàtica) --- Superfícies algebraiques --- Teoria de mòduls --- Teoria de la intersecció --- Teoria de Hodge --- Varietats abelianes --- Varietats algebraiques --- Corbes algebraiques --- Funcions abelianes --- Àlgebres de Banach --- Càlcul --- Àlgebres de mesura --- Harmòniques esfèriques --- Ondetes (Matemàtica) --- Anàlisi de Fourier --- Anàlisi de sèries temporals --- Funcions de Bessel --- Àlgebra diferencial --- Càlcul integral --- Càlcul operacional --- Equacions d'evolució --- Equacions de camp d'Einstein --- Equacions de Lagrange --- Equacions de Pfaff --- Equacions diferencials algebraiques --- Equacions diferencials ordinàries --- Equacions en derivades parcials --- Funcions de Green --- Funcions de Lyapunov --- Problemes de contorn --- Problemes inversos (Equacions diferencials) --- Teoria d'estabilitat (Matemàtica) --- Transformació de Laplace --- Geometria el·líptica --- Geometria riemanniana --- Espais generalitzats --- Geometria diferencial --- Geometria diferencial global --- Geometria no euclidiana --- Poliedres --- Teoria de conjunts --- Aplicacions (Matemàtica) --- Categories (Matemàtica) --- Compactificacions --- Conjunts de Borel --- Dinàmica topològica --- Espais mètrics --- Espais vectorials topològics --- Grups de transformacions --- Grups topològics --- Jocs d'estratègia (Matemàtica) --- Politops --- Teoria de l'homotopia --- Teoria de la dimensió (Topologia) --- Teoria de la dualitat (Matemàtica) --- Teoria de grafs --- Teoria dels reticles --- Topologia algebraica --- Topologia combinatòria --- Topologia diferencial --- Varietats (Matemàtica) --- Varietats topològiques --- Xarxes (Matemàtica) --- Àlgebra lineal

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This book consists of five chapters presenting problems of current research in mathematics, with its history and development, current state, and possible future direction. Four of the chapters are expository in nature while one is based more directly on research. All deal with important areas of mathematics, however, such as algebraic geometry, topology, partial differential equations, Riemannian geometry, and harmonic analysis. This book is addressed to researchers who are interested in those subject areas. Young-Hoon Kiem discusses classical enumerative geometry before string theory and improvements after string theory as well as some recent advances in quantum singularity theory, Donaldson-Thomas theory for Calabi-Yau 4-folds, and Vafa-Witten invariants. Dongho Chae discusses the finite-time singularity problem for three-dimensional incompressible Euler equations. He presents Kato's classical local well-posedness results, Beale-Kato-Majda's blow-up criterion, and recent studies on the singularity problem for the 2D Boussinesq equations. Simon Brendle discusses recent developments that have led to a complete classification of all the singularity models in a three-dimensional Riemannian manifold. He gives an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3. Hyeonbae Kang reviews some of the developments in the Neumann-Poincare operator (NPO). His topics include visibility and invisibility via polarization tensors, the decay rate of eigenvalues and surface localization of plasmon, singular geometry and the essential spectrum, analysis of stress, and the structure of the elastic NPO. Danny Calegari provides an explicit description of the shift locus as a complex of spaces over a contractible building. He describes the pieces in terms of dynamically extended laminations and of certain explicit "discriminant-like" affine algebraic varieties. .

Differential geometry. Global analysis --- Topology --- Geometry --- Harmonic analysis. Fourier analysis --- Differential equations --- Mathematical analysis --- differentiaalvergelijkingen --- landmeetkunde --- analyse (wiskunde) --- differentiaal geometrie --- Fourierreeksen --- mathematische modellen --- wiskunde --- topologie