Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Nondegenerate Invariant Symmetric Bilinear Forms on Simple Lie Superalgebras in Characteristic 2
No full textAs is well-known, the dimension of the space of non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the ground field is distinct from 2. We prove that in characteristic 2, the superdimension of the space of NISes can be equal to 0, or 1, or 0|1, or 1|1. This superdimension is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple restricted Lie algebra with a NIS (for examples of such Lie algebras, although mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations with both even and odd parameter of several simple finite-dimensional Lie superalgebras in characteristic 2
Structure-Preserving Discretizations for Nonlinear Systems of
No full textBecause of the pandemia, the workshop on "Structure-Preserving Discretizations for Nonlinear Systems of Hyperbolic, Involution-Constrained Partial Differential Equations on Manifolds" could not be realized in the usual format or in the new hybrid format. Only one participant (P. Helluy) was able to physically come the MFO. He worked remotely with C. Klingenberg on a structure preserving time integration for kinetic models. This allows to building efficient schemes for solving conservation laws. A preprint describing this work, with applications to MHD, can be found here: https://hal.archives-ouvertes.fr/hal-02965967. Meanwhile, C. Klingenberg organized a remote seminar on the topics of the workshop. From September 2020 to December 2020, 13 talks were given online by participants to the workshop and other personalities
Computational Inverse Problems for Partial Differential Equations (hybrid meeting)
No full textInverse problems in partial differential equations (PDEs) consist in reconstructing
some part of a PDE such as a coefficient, a boundary condition, an initial condition, the shape
of a domain, or a singularity from partial knowledge of solutions to the PDE.
This has numerous applications in nondestructive testing, medical imaging, seismology, and optical
imaging. Whereas classically mostly boundary or far field data of solutions to deterministic PDEs were considered,
more recently also statistical properties of solutions to random PDEs have been studied.
The study of numerical reconstruction methods of inverse problems in PDEs is at the interface of
numerical analysis, PDE theory, functional analysis, statistics, optimization, and differential geometry.
This workshop has mainly addressed five related topics of current interest:
model reduction, control-based techniques in inverse problems,
imaging with correlation data of waves, fractional diffusion,
and model-based approaches using machine learning
Discrete Geometry (hybrid meeting)
No full textA number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
Representation Theory of Quivers and Finite Dimensional Algebras
No full textMethods and results from the representation theory of quivers and
finite dimensional algebras have led to many interactions with other
areas of mathematics. Such areas include the theory of Lie algebras
and quantum groups, commutative algebra, algebraic geometry and
topology, and in particular the theory of cluster algebras. The aim of
this workshop was to further develop such interactions and to
stimulate progress in the representation theory of algebras
Stochastic Processes under Constraints (hybrid meeting)
No full textThe analysis of random processes under various constraints and
conditions has been a
central theme in the theory of stochastic processes, which links
together several mathematical subdisciplines. The connection between
potential theory and a certain type of conditioning of Markov
processes via Doob's h-transform can be seen as a classical highlight.
The last decades have seen further exciting and highly interesting
developments which are related to the title of the workshop
such as the analysis of persistence exponents for various classes of
processes and various types of penalization problems. Many of these
problems are rooted in questions from statistical mechanics. The
workshop aims to investigate the topic stochastic processes under
constraints from all these different perspectives
Determinacy versus indeterminacy
No full textCan a continuous function on an interval be uniquely
determined if we know all the integrals of the function
against the natural powers of the variable? Following
Weierstrass and Stieltjes, we show that the answer is
yes if the interval is finite, and no if the interval is
infinite
Mini-Workshop: Relativistic Fluids at the Intersection of Mathematics and Physics (online meeting)
No full textRelativistic Hydrodynamics is the description of fluid motion in regimes where relativistic effects are important. This is the case for fluids moving at high velocities or interacting with very strong gravitational fields, such as in the physics of black hole accretion disks or neutron star mergers but also in the microscopic dynamics of high-energy heavy-ion collisions. Although the first formulation of hydrodynamic equations dates back to the beginning stages of relativity theory, many mathematical problems remain wide open. In particular, the development of the theory of relativistic "viscous" fluids was slow and mathematical progress only made recently. The purpose of this Mini-Workshop was to bring together a diverse group of researchers, including specialists in nonlinear PDEs and physicists, to jump-start the mathematical development of this field. This allowed for a vital exchange of ideas between mathematics and physics communities
Multivariate Hybrid Orthogonal Functions
No full textWe consider multivariate orthogonal functions satisfying hybrid orthogonality conditions
with respect to a moment functional. This kind of orthogonality means that the product of
functions of different parity order is computed by means of the moment functional, and the product of elements of the same parity order is computed by a modification of the original moment functional.
Results about existence conditions, three term relations with matrix coefficients, a Favard type theorem for this kind of hybrid orthogonal functions are proved. In addition, a method to construct bivariate hybrid
orthogonal functions from univariate orthogonal polynomials and univariate orthogonal
functions is presented. Finally, we give a complete description of a sequence of hybrid orthogonal
functions on the unit disk on , that includes, as particular case, the classical orthogonal polynomials on the disk
From Betti numbers to ℓ²-Betti numbers
No full textWe provide a leisurely introduction to ℓ²-Betti numbers,
which are topological invariants, by relating
them to their much older cousins, Betti numbers. In
the end we present an open research problem about
ℓ²-Betti numbers