Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Shock-avoiding Slicing Conditions: Tests and Calibrations
No full textWhile the 1+log slicing condition has been extremely successful in numerous numerical relativity simulations, it is also known to develop "gauge-shocks" in some examples. Alternative "shockavoiding" slicing conditions suggested by Alcubierre prevent these pathologies in those examples, but have not yet been explored and tested very broadly. In this paper we compare the performance of shock-avoiding slicing conditions with those of 1+log slicing for a number of "text-book" problems, including black holes and relativistic stars. While, in some simulations, the shock-avoiding slicing conditions feature some unusual properties and lead to more "gauge-dynamics" than the 1+log slicing condition, we find that they perform quite similarly in terms of stability and accuracy, and hence provide a very viable alternative to 1+log slicing
Biological shape analysis with geometric statistics and learning
No full textThe advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold the key to unlocking outstanding mysteries in biomedicine. This snapshot introduces the mathematical framework of geometric statistics and learning and its applications to biomedicine
Character Theory and Categorification
No full textOver a hundred years after the work of Frobenius and Schur, the sheer enormity
of what is not known about the character theory of symmetric and alternating groups continues to surprise and awe the uninitiated.
How does one decompose the tensor product of a pair of complex characters?
Or the restriction of a complex character to a Sylow or wreath product subgroup?
Can we understand the vanishing sets of complex characters?
What about the asymptotic behaviour of complex characters?
What are the dimensions of the modular characters?
These questions have been hailed as some of the definitive open problems in representation theory and algebraic combinatorics, they have deep connections with Lie theory, group theory, local-global conjectures in representation theory of finite groups, symplectic geometry, complexity theory, statistical mechanics and quantum information theory.
The overarching theme of this proposal is the use of hidden, richer representation theoretic structures arising in modular, local-global, and categorical representation theory in order to prove and disprove conjectures concerning characters of symmetric and alternating groups
Recent Developments in Representation Theory and Mathematical Physics
No full textThis mini-workshop was devoted to foster the interactions between mathematicians and mathematical physicists who are working on questions related to representation theory. This includes for example the representation theory of supergroups, vertex operator algebras and quantum groups. Another focus was on link and manifold invariants and TQFTs
Space-Time Methods for Time-Dependent Partial Differential Equations
No full textModern discretization and solution methods for time-dependent PDEs
consider the full problem in space and time simultaneously and aim to
overcome limitations of classical approaches by first discretizing in
space and then solving the resulting ODE, or first discretizing in
time and then solving the PDE in space.
The development of space-time methods for hyperbolic and parabolic
differential equation is an emerging and rapidly growing field in
numerical analysis and scientific computing. At the first Workshop on
this topic in 2017 a large variety of interesting and challenging
concepts, methods, and research directions have been presented; now we
exchange the new developments.
The focus is on the optimal convergence of
discretizations and on efficient error control for space-time methods for
hyperbolic and parabolic problems, and on solution methods with optimal
complexity. This is complemented by applications in the field of
time-dependent stochastic PDEs, non-local material laws in space and time,
optimization with time-dependent PDE
constraints, and multiscale methods for time-dependent PDEs
Descriptive Combinatorics, LOCAL Algorithms and Random Processes
No full textThe aim of this mini-workshop was to discover and deepen connections between the fields of descriptive combinatorics, distributed computing and random processes.
The common link is played by the so-called local coloring problems on graphs, where the validity of solution can be checked locally, and the common interest can be phrased as the following central question: Is it possible to produce a solution to a given local problem efficiently?
While all three areas possess a solid background that was achieved by decades of intense research, a systematic study of formal connections between them is a recent emerging phenomenon.
This approach has already proved to be very fruitful: several open questions in each of the fields were solved by means and techniques of the other two.
The purpose of this meeting is to bring together researchers in all three fields in order to explore these exciting connections
Geometric Structures in Group Theory
No full textThe conference was in the area of geometric group theory, the field of mathematics in which one studies infinite groups (finitely generated, or more generally locally compact, countable etc.) via actions on spaces endowed with various structures (geometric, measurable, analytic etc.).
The surging current activity in the field is drawing more and more connections with other mathematical areas, and this was successfully reflected in the program of this week, during which problems in algebraic topology, representation theory and functional analysis, to name just a few, featured prominently alongside core topics in the area
Coorbit Spaces and Dual Molecules: the Quasi-Banach Case
No full textThis paper provides a self-contained exposition of coorbit spaces associated with integrable group representations and quasi-Banach function spaces. It extends the theory in [Studia Math., 180(3):237–253, 2007] to locally compact groups that do not necessarily possess a compact, conjugation-invariant unit neighborhood. Furthermore, the present paper establishes the existence of dual molecules of frames and Riesz sequences as in [J. Funct. Anal., 280(10):56, 2021] for the setting of quasi-Banach spaces. To ensure the direct applicability to various well-studied examples, the theory is developed for possibly projective and reducible unitary representations
Seeing through rock with help from optimal transport
No full textGeophysicists and mathematicians work together to detect geological structures located deep within the earth by measuring and interpreting echoes from manmade earthquakes. This inverse problem naturally involves the mathematics of wave propagation, but we will see that a different mathematical theory – optimal transport – also turns out to be very useful
Complex Geometry and Dynamical Systems
No full textThe workshop focused on recent developments in
holomorphic dynamics, several complex variables and complex geometry.
The topics of the talks included dynamics of holomorphic and rational maps,
the theory of currents, the Bergman kernel, together with applications in
geometry, dynamics, foliations and mathematical physics