Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Searching for structure in complex data: a modern statistical quest
No full textCurrent research in statistics has taken interesting
new directions, as data collected from scientific studies
has become increasingly complex. At first glance,
the number of experiments conducted by a scientist
must be fairly large in order for a statistician to draw
correct conclusions based on noisy measurements of
a large number of factors. However, statisticians may
often uncover simpler structure in the data, enabling
accurate statistical inference based on relatively few
experiments. In this snapshot, we will introduce the
concept of high-dimensional statistical estimation via
optimization, and illustrate this principle using an
example from medical imaging. We will also present
several open questions which are actively being studied
by researchers in statistics
Mathematical Foundations of Machine Learning (hybrid meeting)
No full textMachine learning has achieved
remarkable successes in various applications, but there is wide agreement that a mathematical theory for deep learning is missing. Recently, some first mathematical results have been derived in different areas such as mathematical statistics and statistical learning. Any mathematical theory of machine learning will have to combine tools from different fields such as nonparametric statistics, high-dimensional statistics, empirical process theory and approximation theory. The main objective of the workshop was to bring together leading researchers contributing to the mathematics of machine learning.
A focus of the workshop was on theory for deep neural networks. Mathematically speaking, neural networks define function classes with a rich mathematical structure that are extremely difficult to analyze because of non-linearity in the parameters. Until very recently, most existing theoretical results could not cope with many of the distinctive characteristics of deep networks such as multiple hidden layers or the ReLU activation function. Other topics of the workshop are procedures for quantifying the uncertainty of machine learning methods and the mathematics of data privacy
Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)
No full textThis small collaborative workshop brought together
experts from the Sino-German project working in the field of advanced numerical methods for
hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the
convergence of numerical methods and proper solution concepts were addressed as well
Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)
No full textThis workshop brought together
leading experts, as well as the most
promising young researchers, working on nonlinear
hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in
modeling, analysis, and numerics. Particular topics included ill-/well-posedness,
randomness and multiscale modeling, flows in a moving domain, free boundary problems,
games and control
Computation and Learning in High Dimensions (hybrid meeting)
No full textThe most challenging problems in science often involve the learning and
accurate computation of high dimensional functions.
High-dimensionality is a typical feature for a multitude of problems
in various areas of science.
The so-called curse of dimensionality typically negates the use of
traditional numerical techniques for the solution of
high-dimensional problems. Instead, novel theoretical and
computational approaches need to be developed to make them tractable
and to capture fine resolutions and relevant features. Paradoxically,
increasing computational power may even serve to heighten this demand,
since the wealth of new computational data itself becomes a major
obstruction. Extracting essential information from complex
problem-inherent structures and developing rigorous models to quantify
the quality of information in a high-dimensional setting pose
challenging tasks from both theoretical and numerical perspective.
This has led to the emergence of several new computational methodologies,
accounting for the fact that by now well understood methods drawing on
spatial localization and mesh-refinement are in their original form no longer viable.
Common to these approaches is the nonlinearity of the solution method.
For certain problem classes, these methods have
drastically advanced the frontiers of computability.
The most visible of these new methods is deep learning. Although the use of deep neural
networks has been extremely successful in certain
application areas, their mathematical understanding is far from complete.
This workshop proposed to deepen the understanding of
the underlying mathematical concepts that drive this new evolution of
computational methods and to promote the exchange of ideas emerging in various
disciplines about how to treat multiscale and high-dimensional problems
Computational Group Theory (hybrid meeting)
No full textThis was the eighth Oberwolfach Workshop on Computational Group Theory.
It demonstrated how an increasing number and variety of deep theoretical
results are being used to devise powerful and practical algorithms in
Computational Group Theory.
The talks also presented connections with and applications to
Number Theory, Combinatorics, Geometry, and Geometric Group Theory
Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (hybrid meeting)
No full textThe investigation of Riemannian metrics with lower scalar curvature bounds has been a central topic in differential geometry for decades.
It addresses foundational problems, combining ideas and methods from global analysis, geometric topology, metric geometry and general relativity.
Seminal contributions by Gromov during the last years have led to a significant increase of activities in the area which have produced a number of impressive results.
Our workshop reflected the state of the art of this thriving field of research
Weak*-Continuity of Invariant Means on Spaces of Matrix Coefficients
No full textWith every locally compact group , one can associate several interesting bi-invariant subspaces of the weakly almost periodic functions on , each of which captures parts of the representation theory of . Under certain natural assumptions, such a space carries a unique invariant mean and has a natural predual, and we view the weak-continuity of this mean as a rigidity property of . Important examples of such spaces , which we study explicitly, are the algebra of -completely bounded multipliers of the Figà-Talamanca-Herz algebra and the -Fourier-Stieltjes algebra . In the setting of connected Lie groups , we relate the weak-continuity of the mean on these spaces to structural properties of . Our results generalise results of Bekka, Kaniuth, Lau and Schlichting
The C-Map as a Functor on Certain Variations of Hodge Structure
No full textWe give a new manifestly natural presentation of the supergravity c-map. We achieve this by giving a more explicit description of the correspondence between projective special Kähler manifolds and variations of Hodge structure, and by demonstrating that the twist construction of Swann, for a certain kind of twist data, reduces to a quotient by a discrete group. We combine these two ideas by showing that variations of Hodge structure give rise to the aforementioned kind of twist data and by then applying the twist realisation of the c-map due to Macia and Swann. This extends previous results regarding the lifting, along the c-map, of infinitesimal automorphisms to the lifting of general isomorphisms
Nonstandard Finite Element Methods (hybrid meeting)
No full textFinite element methodologies dominate the computational approaches for the solution to partial differential equations and nonstandard finite element schemes most urgently require mathematical insight in their design. The hybrid workshop vividly enlightened and discussed innovative nonconforming and polyhedral methods, discrete complex-based finite element methods for tensor-problems, fast solvers and adaptivity, as well as applications to challenging ill-posed and nonlinear problems