Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Mixed-integer Nonlinear Optimization: A Hatchery for Modern Mathematics
No full textMixed-integer nonlinear programming (MINLP) is concerned with finding optimal solutions to mathematical formulations of
optimization problems combining discrete and nonlinear phenomena.
The scientific program was organized around three areas:
convex envelopes and relaxation hierarchies, mixed-integer optimal
control, and current trends. These topics were addressed with a
variety of tutorials, talks, and short research announcements
Computer Algebra with GAP
No full textThis monograph includes the following topics: a basic introduction to the language, basic arithmetic, permutations, matrices, polynomial rings, finite fields, finite and finitely presented groups, small groups, group representations and character theory, and simple groups. Advanced topics include testing several open conjectures and theorems. In addition, each chapter ends with an extensive list of problems. We hope the reader will find some problems challenging and exciting as they are based on outstanding research papers. Selected solutions can be found at the end of the book
4 = 2 × 2, or the Power of Even Integers in Fourier Analysis
No full textWe describe how simple observations related to vectors of length 1 recently led to the proof of an important mathematical fact: the sharp Stein–Tomas inequality from Fourier restriction theory, a pillar of modern harmonic analysis with surprising applications to number theory and geometric measure theory
Low-dimensional Topology
No full textThe workshop brought together experts from across all areas of low-dimensional topology, including knot theory, computational topology, three-manifolds and four-manifolds. In addition to the standard research talks we had two survey talks by Marc Lackenby and Joel Hass, leading to discussions of open problems. Furthermore we had three sessions of five-minute talks by a total of roughly thirty participants
Geproci Sets: a New Perspective in Algebraic Geometry
No full textGeproci sets arise from applying the perspective of inverse scattering problems to algebraic geometry. Analogous to the reconstruction of an object from multiple X-ray images, we aim at a classification of sets with certain algebraic properties under multiple projections
Combinatorics
No full textCombinatorics is an area of mathematics primarily concerned with
counting and studying properties of discrete objects such as graphs,
set systems, partial orders, polyhedra, etc. Combinatorial problems
naturally arise in many areas of mathematics, such as algebra,
geometry, probability theory, and topology, and in theoretical
computer science. Historically, such questions were often studied
using ad hoc arguments. However, over the last few
decades, the development of general and powerful methods have
elevated combinatorics to a thriving branch of
mathematics with many connections to other subjects.
The workshop brought together
the established leading experts and the brightest young talents from
different parts of this very broad area in order to discuss the most
exciting recent developments, current themes and trends, and the
most promising new directions for future research
Multi-Dimensional Summation-by-Parts Operators for General Function Spaces: Theory and Construction
No full textSummation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.This research was supported by AFOSR #F9550-18-1-0316, the US DOD (ONR MURI) grant #N00014-20-1-2595, the US DOE (SciDAC program) grant #DE-SC0012704, Vetenskapsrådet Sweden grant 2018-05084 VR and 2021-05484, the Swedish e-Science Research Center (SeRC), and the Gutenberg Research College, JGU Mainz. Furthermore, it was supported through the program “Oberwolfach Research Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2022. We also thank Maximilian Winkler for helpful discussions on the POCS algorithm
Semantic Factorization and Descent
No full textLet be a -category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the higher cokernel of is up to isomorphism the same as the semantic factorization of , either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou-Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of trivially hold whenever has a left adjoint and, hence, in this case, we find monadicity to be a -dimensional exact condition on , namely, to be an effective faithful morphism of the -category
Interfaces: Modeling, Analysis, Numerics
No full textThese 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