Recent Progress in Mathematics

1. Front Matter

   Pages i-vii

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2. Enumerative Geometry, Before and After String Theory

   • Young-Hoon Kiem

   Pages 1-42

3. On the Singularity Problem for the Euler Equations

   • Dongho Chae

   Pages 43-85

4. Singularity Models in the Three-Dimensional Ricci Flow

   • Simon Brendle

   Pages 87-118

5. Spectral Geometry and Analysis of the Neumann-Poincaré Operator, a Review

   • Hyeonbae Kang

   Pages 119-153

6. Sausages and Butcher Paper

   • Danny Calegari

   Pages 155-200

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 classicallocal 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. 

• Enumerative geometry
• shift locus
• Euler equations
• Ricci flow
• Neumann-Poincare operator

• School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea (Republic of)

  Nam-Gyu Kang, Kyeongsu Choi, Sang-hyun Kim

• Korea Institute for Advanced Study, Seoul, Korea (Republic of)

  Jaigyoung Choe

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