Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Geometry and Physics of Higgs Bundles
No full textThis workshop focused on interactions between the various perspectives on the moduli space of Higgs bundles over
a Riemann surface. This subject draws on algebraic geometry, geometric topology, geometric analysis and mathematical
physics, and the goal was to promote interactions between these various branches of the subject. The main
current directions of research were well represented by the participants, and the talks included many from both senior and
junior participants
Mixed-dimensional models for real-world applications
No full textWe explore mathematical models for physical problems
in which it is necessary to simultaneously consider
equations in different dimensions; these are called
mixed-dimensional models. We first give several examples,
and then an overview of recent progress made
towards finding a general method of solution of such
problems
On a Group Functor Describing Invariants of Algebraic Surfaces
No full textLiedtke (2008) has introduced group functors and , which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate and to a group functor arising in the construction of the non-abelian exterior square of a group. In contrast to , there exist efficient algorithms for constructing , especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when is a quotient of , and when and are isomorphic
Diophantine equations and why they are hard
No full textDiophantine equations are polynomial equations whose
solutions are required to be integer numbers. They
have captured the attention of mathematicians during
millennia and are at the center of much of contemporary
research. Some Diophantine equations are easy,
while some others are truly difficult. After some time
spent with these equations, it might seem that no
matter what powerful methods we learn or develop,
there will always be a Diophantine equation immune
to them, which requires a new trick, a better idea, or
a refined technique. In this snapshot we explain why
Is it possible to predict the far future before the near future is known accurately?
No full textIt has always been the dream of mankind to predict
the future. If the future is governed by laws
of physics, like in the case of the weather, one can
try to make a model, solve the associated equations,
and thus predict the future. However, to make accurate
predictions can require extremely large amounts
of computation. If we need seven days to compute
a prediction for the weather tomorrow and the day
after tomorrow, the prediction arrives too late and
is thus not a prediction any more. Although it may
seem improbable, with the advent of powerful computers
with many parallel processors, it is possible to
compute a prediction for tomorrow and the day after
tomorrow simultaneously. We describe a mathematical
algorithm which is designed to achieve this
Foundations and New Horizons for Causal Inference
No full textWhile causal inference is established in some disciplines such as econometrics and biostatistics, it is only starting to emerge as a
valuable tool in areas such as machine learning and artificial intelligence. The mathematical foundations of causal inference are fragmented at present.
The aim of the workshop "Foundations and new horizons for causal inference" was to
unify existing approaches and mathematical foundations as well as exchange ideas between different fields.
We regard this workshop as successful in that
it brought together researchers from different disciplines
who
were able to
learn from each other not only about
different formulations of related problems,
but also about solutions and methods that exist
in the different fields
Mini-Workshop: Recent Progress in Path Integration on Graphs and Manifolds
No full textEver since Richard Feynman's PhD thesis, path integrals have played a decisive role in mathematical physics. While it is well-known that such formulae can hold only formally, it was Mark Kac who realized that by replacing the unitary group by the heat semigroup, one obtains well-defined and rigorous formulae. Following this pioneering work, Feynman-Kac path integral formulae have been adapted to several situations and generalized into several directions providing the central focus of this workshop
A surprising connection between quantum mechanics and shallow water waves
No full textWe describe a connection between quantum mechanics
and nonlinear wave equations and highlight a few
problems at the forefront of modern research in the
intersection of these areas
Reflective Prolate-Spheroidal Operators and the KP/KdV Equations
No full textCommuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned
concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the KdV equation. We prove a general theorem that the integral operator associated to every wave function in the infinite dimensional Adelic Grassmannian Gr of Wilson always reflects a differential operator (in the sense of Definition 1 below). This intrinsic property is shown to follow from the symmetries of Grassmannians of KP wave functions, where the direct commutativity property holds for operators associated to wave functions fixed by Wilson's sign involution but is violated in general. Based on this result, we prove a second main theorem that the integral operators in the computation of the singular values of the truncated generalized Laplace transforms associated to all bispectral wave functions of rank 1 reflect a differential operator. A 90 rotation argument is used to prove a third main theorem that the integral operators in the computation of the singular values of the truncated generalized Fourier transforms associated to all such KP wave functions commute with a differential operator. These methods produce vast collections of integral operators with prolate-spheroidal
properties, including as special cases the integral operators associated to all rational solutions of the KdV and KP hierarchies considered by Airault-McKean-Moser and Krichever, respectively, in the late 70's. Many novel examples are presented
Geometric, Algebraic, and Topological Combinatorics
No full textThe 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz