Oberwolfach Seminars vol. 51 and 52 published
In order to make the Oberwolfach Seminars available to an even larger audience, the MFO supports the publication within the book series Oberwolfach Seminars, published in the Birkhäuser program of Springer Basel. We should like to express our sincere thanks for the combined efforts of the organizers to publish these lecture notes.
Interfaces: Modeling, Analysis, Numerics
Oberwolfach Seminars vol. 51
Eberhard Bänsch, Klaus Deckelnick, Harald Garcke, Paola Pozzi (2023)
https://doi.org/10.1007/978-3-031-35550-9
These lecture notes are dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems appearing in geometry and in various applications, ranging from crystal growth, tumour growth, biological membranes to porous media, two-phase flows, fluid-structure interactions, and shape optimization.
We first give an introduction to classical methods from differential geometry and systematically derive the governing equations from physical principles. Then we will analyse parametric approaches to interface evolution problems and derive numerical methods which will be thoroughly analysed. In addition, implicit descriptions of interfaces such as phase field and level set methods will be analysed. Finally, we will discuss numerical methods for complex interface evolutions and will focus on two phase flow problems as an important example of such evolutions.
Tropical and Logarithmic Methods in Enumerative Geometry
Oberwolfach Seminars vol. 52
Renzo Cavalieri, Hannah Markwig, Dhruv Ranganathan (2023)
https://doi.org/10.1007/978-3-031-39401-0
Logarithmic Gromov-Witten theory lies at the heart of modern approaches to mirror symmetry, but also opens up a number of new directions in enumerative geometry of a more classical flavour. Tropical geometry forms the calculus through which calculations in this subject are carried out. These notes cover the foundational aspects of this tropical calculus, geometric aspects of the degeneration formula for Gromov-Witten invariants, and the practical nuances of working with and enumerating tropical curves. Readers will get an assisted entry route to the subject, focusing on examples and explicit calculations.