Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Reflection Positivity and Hankel Operators- the Multiplicity Free Case
No full textWe analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples , where is a group, an involutive automorphism of and a subsemigroup with . For the triples , corresponding to reflection positive operators, and , corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method
consists in using the measure on defined by a positive Hankel operator on to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for
Quantum Groups - Algebra, Analysis and Category Theory (hybrid meeting)
No full textThe meeting was devoted to discussing the state of the art of different branches of tensor categories and quantum groups, with emphasis on the exchange of ideas between the purely algebraic and operator algebraic sides of these theories
Mathematics and its Ancient Classics Worldwide: Translations, Appropriations, Reconstructions, Roles (hybrid meeting)
No full textThe workshop analyzes the constitution, recovery, and role of the classical texts in mathematical practice throughout history. It aims at problematizing the notion of "classic", to make it a historical category and to study the rhetorical, pedagogical, and institutional mechanisms that contribute to secure the status of classic to specific texts. So far, the focus of the historiography has dealt mostly with Greek classics and their impact on Western European societies. We aim to expand the focus of our enquiry culturally and chronologically in two ways. We want to address the reception and transformation of these "classics" outside Europe in different historical periods. We are particularly interested in the roles played by this classical tradition within Islamicate societies, South-East and East Asia. Secondly, we are interested in the ancient mathematical writings in Arabic, Chinese, Sanskrit and other languages that, at certain time periods in these other parts of the world and elsewhere, were perceived as classics. Widening the focus should allow us to inquire into questions such as: what did classical texts mean for various types of actors? How were they available to them? How did they read them? In the contexts of which institutions and with which expectations? The important role classical works have played in mathematical history pose deep methodological questions with far-reaching implications for the history and philosophy of mathematics. In mathematics conceptual and methodological innovations are thought to be legitimized only by appeal to mathematical arguments and consistency. Yet, legitimation has involved in many crucial episodes giving a prominent role to classical works. The mathematical classics have repeatedly been the source and grounds for new ideas and techniques. There is therefore a deep, complex tension between innovation and tradition. We are interested in how innovation has often been legitimized by re-reading old texts, concepts, and methods-old texts whose principles and methods were utterly different from the ones they contributed to sustain. What can this teach us about the nature of mathematical argument, and mathematical practice
Homotopic and Geometric Galois Theory (online meeting)
No full textIn his "Letter to Faltings'', Grothendieck lays the foundation of what will become part of his multi-faceted legacy to arithmetic geometry. This includes the following three branches discussed in the workshop: the arithmetic of Galois covers, the theory of motives and the theory of anabelian Galois representations. Their geometrical paradigms endow similar but complementary arithmetic insights for the study of the absolute Galois group of the field of rational numbers
that initially crystallized into a functorially group-theoretic unifying approach. Recent years have seen some new enrichments based on modern geometrical constructions - e.g. simplicial homotopy, Tannaka perversity, automorphic forms - that endow some higher considerations and outline new geometric principles. This workshop brought together an international panel of young and senior experts of arithmetic geometry who sketched the future desire paths of homotopic and geometric Galois theory
Small Collaboration: Numerical Analysis of Electromagnetic Problems (hybrid meeting)
No full textThe classical theory of electromagnetism describes the interaction of electrically charged particles through electromagnetic forces, which are carried by the electric and magnetic fields. The propagation of the electromagnetic fields can be described by Maxwell's equations. Solving Maxwell's equations numerically is a challenging problem which appears in many different technical applications. Difficulties arise for instance from material interfaces or if the geometrical features are much larger than or much smaller than a typical wavelength. The spatial discretization needs to combine good geometrical flexibility with a relatively high order of accuracy.
The aim of this small-scale, week-long interactive mini-workshop jointly organized by the University of Duisburg-Essen and the University of Twente, and kindly hosted at the MFO, is to bring together experts in non-standard and mixed finite elements methods with experts in the field of electromagnetism
Combinatorial Optimization (hybrid meeting)
No full textCombinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years
Mini-Workshop: Scattering Amplitudes, Cluster Algebras, and Positive Geometries (hybrid meeting)
No full textCluster algebras were developed by Fomin and Zelevinsky about twenty years ago.
While the initial motivation came from within algebra (total positivity, canonical bases), it quickly became clear that cluster algebras possess deep links to a host of other subjects in mathematics and physics.
In a separate vein, starting about ten years ago, Arkani-Hamed and his collaborators began a program of reformulating the bases of quantum field theory, motivated by a desire to discover the basic rules of quantum mechanics and spacetime as arising from deeper mathematical principles. Their approach to the fundamental problem of particle scattering amplitudes entails encoding the solution in geometrical objects, "positive geometries'' and "amplituhedra''. Surprisingly, cluster algebras have been found to be tightly woven into the mathematics needed to describe these geometries. The purpose of this workshop is to explore the various
connections between cluster algebras, scattering amplitudes, and positive geometries
From the dollar game to the Riemann-Roch Theorem
No full textWhat is the dollar game? What can you do to win
it? Can you always win it? In this snapshot you
will find answers to these questions as well as several
of the mathematical surprises that lurk in the background,
including a new perspective on a century-old theorem
Geometric Methods of Complex Analysis (hybrid meeting)
No full textThe purpose of this workshop was to discuss recent results in Several
Complex Variables, Complex Geometry and Complex Dynamical Systems
with a special focus on the exchange of ideas and methods among these areas. The
main topics of the workshop included Holomorphic Dynamics and Nevanlinna's Theory; -methods and Cohomologies; Plurisubharmonic Functions and Pluripotential Theory; Geometric Questions of Complex Analysis
Foundations of Bayesian Inference for Complex Statistical Models (hybrid meeting)
No full textIn this virtual workshop a variety of recent developments in Bayesian high-dimensional and nonparametric statistics were discussed in depth. There were 12 in depth talks delivered via zoom in the afternoons (to allow for US attendance), and several informal evening time meetings on wonder.me, where follow up discussions of the most important mathematical developments took place