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Recent progress in mathematics / Nam-Gyu Kang, Jaigyoung Choe, Kyeongsu Choi, Sang-hyun Kim, editors.

9789811937088 (electronic bk.)
9811937087 (electronic bk.)
9789811937071
9811937079

Singapore : Springer, 2022.

English

1 online resource : illustrations (black and white, and color).

10.1007/978-981-19-3708-8 doi

QA564

516.3/5

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, DonaldsonThomas theory for CalabiYau 4-folds, and VafaWitten invariants. Dongho Chae discusses the finite-time singularity problem for three-dimensional incompressible Euler equations. He presents Kato's classical local well-posedness results, BealeKatoMajda'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 classication of all the singularity models in a three-dimensional Riemannian manifold. He gives an alternative proof of the classication of noncollapsed steady gradient Ricci solitons in dimension 3. Hyeonbae Kang reviews some of the developments in the NeumannPoincare 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" ane algebraic varieties. .

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KIAS Springer Series in Mathematics Ser.

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