Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations
No full textIn multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations
Flat Surfaces and Algebraic Curves
No full textThis workshop brought together two distinct communities: “flat” geometers, studying the moduli of flat surfaces, and Teichmüller dynamics, and algebraic geometers studying the moduli space of curves. While both communities study similar or often the same objects, very different viewpoints and toolboxes lead to different questions being addressed, and different progress being made. The workshop sought to educate each community about the techniques of the other, and to foster communication between the two groups. One particular focus was enumerative geometry
Mini-Workshop: Chromatic Phenomena and Duality in Homotopy Theory and Representation Theory
No full textThis mini-workshop focused on chromatic phenomena and duality as unifying themes in algebra, geometry, and topology. The overarching goal was to establish a fruitful exchange of ideas between experts from various areas, fostering the study of the local and global structure of the fundamental categories appearing in algebraic geometry, homotopy theory, and representation theory. The workshop started with introductory talks to bring researches from different backgrounds to the same page, and later highlighted recent progress in these areas with an emphasis on the interdisciplinary nature of the results and structures found. Moreover, new directions were explored in focused group work throughout the week, as well as in an evening discussion identifying promising long-term goals in the subject. Topics included support theories and their applications to the classification of localizing ideals in triangulated categories, equivariant and homotopical enhancements of important structural results, descent and Galois theory, numerous notions of duality, Picard and Brauer groups, as well as computational techniques
Algebraische Zahlentheorie
No full textThe origins of Algebraic Number Theory can be traced to over two centuries ago, wherein algebraic techniques are used to glean information about integers and rational numbers. It continues to be at the forefront of modern research as it evolves to straddle wider areas of Mathematics
Number theory in quantum computing
No full textAlgorithms are mathematical procedures developed
to solve a problem. When encoded on a computer,
algorithms must be "translated" to a series of simple
steps, each of which the computer knows how
to do. This task is relatively easy to do on a classical
computer and we witness the benefits of this
success in our everyday life. Quantum mechanics,
the physical theory of the very small, promises to enable
completely novel architectures of our machines,
which will provide specific tasks with higher computing
power. Translating and implementing algorithms
on quantum computers is hard. However, we will
show that solutions to this problem can be found and
yield surprising applications to number theory
Arbeitsgemeinschaft: Topological Cyclic Homology
No full textIntroduced by Bökstedt-Hsiang-Madsen in the nineties, topological cyclic homology is a manifestation of the dual visions of Connes-Tsygan and Waldhausen to extend de Rham cohomology to a noncommutative setting and to replace algebra by higher algebra. The cohomology theory that ensues receives a denominator-free Chern character from algebraic K-theory, used by Hesselholt-Madsen to evaluate the p-adic K-theory of p-adic fields. More recently, Bhatt-Morrow-Scholze have defined a “motivic” filtration of topological cyclic homology and its variants, the filtration quotients of which give rise to their denominator-free p-adic Hodge theory AΩ
Nonlinear Data: Theory and Algorithms
No full textTechniques and concepts from differential geometry are used in many parts of applied mathematics today. However, there is no joint community for users of such techniques. The workshop on Nonlinear Data assembled researchers from fields like numerical linear algebra, partial differential equations, and data analysis to explore differential geometry techniques, share knowledge, and learn about new ideas and applications
The mathematics of aquatic locomotion
No full textAquatic locomotion is a self-propelled motion through
a liquid medium. It can be of biological nature (fish,
microorganisms,. . .) or performed by robotic swimmers.
This snapshot aims to introduce the reader to
some of the challenges raised by the mathematical
modelling of aquatic locomotion, even in seemingly
very simple cases
Topological recursion
No full textIn this snapshot we present the concept of topological
recursion – a new, surprisingly powerful formalism
at the border of mathematics and physics, which has
been actively developed within the last decade. After
introducing necessary ingredients – expectation values,
random matrices, quantum theories, recursion
relations, and topology – we explain how they get
combined together in one unifying picture
Mini-Workshop: Asymptotic Invariants of Homogeneous Ideals
No full textRecent decades have witnessed a shift in interest from isolated objects to families of objects and their limit behavior, both in algebraic geometry and in commutative algebra. A series of various invariants have been introduced in order to measure and capture asymptotic properties of various algebraic objects motivated by geometrical ideas. The major goals of this workshop were to refine these asymptotic ideas, to articulate unifying themes, and to identify the most promising new directions for study in the near future. We expect the ideas discussed and originated during this workshop to be poised to have a broad impact beyond the areas of algebraic geometry and commutative algebra