A set of detailed lecture notes on six topics at the forefront of current research in numerical analysis and applied mathematics. Each set of notes presents a self-contained guide to a current research area. Detailed proofs of key results are provided. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. Current (unsolved) problems are also described and directions for future research are given. This book is also suitable for professional mathematicians.

Franco Brezzi and Donatella Marini: Subgrid Phenomena and Numerical Schemes.

- Franco Brezzi: Stability of Saddle-points in Finite Dimensions. Klaus Deckelnick and Gerhard Dziuk: Mean Curvature Flow.

- Nicholas I.M. Gould and Sven Leyffer: An Introduction to Algorithms for Nonlinear Optimization.

- Ernst Hairer and Martin Hairer: Matlab Programs for Geometric Numerical Integration.

- Thomas Y. Hou: Numerical Approximations to Multiscale Solutions in Partial Differential Equations.

- Volker Mehrmann: Numerical Methods for Eigenvalue and Control Problems.

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