Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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The Periodic Tables of Algebraic Geometry
No full textTo understand our world, we classify things. A famous example is the periodic table of elements, which describes the properties of all known chemical elements and gives us a classification of the building blocks we can use in physics, chemistry, and biology. In mathematics, and algebraic geometry in particular, there are many instances of similar “periodic tables”, describing fundamental classification results. We will go on a tour of some of these
Ansprachen, Grußworte, Festvortrag und Gastbeiträge anlässlich des 75-jährigen Bestehens des Mathematischen Forschungsinstituts Oberwolfach
No full textThe Festschrift that marks the 75th anniversary of the MFO in July 2019 includes many interesting contributions, documents and pictures. We would like to put your special focus on the lecture of Stefan Müller (Universität Bonn) and on the article of Volker Remmert (Bergische Universität Wuppertal) about the role of Szolem Mandelbroijt and John Todd for the maintenance of the Institute after World War II.The Festschrift that marks the 75th anniversary of the MFO in July 2019 includes many interesting contributions, documents and pictures. We would like to put your special focus on the lecture of Stefan Müller (Universität Bonn) and on the article of Volker Remmert (Bergische Universität Wuppertal) about the role of Szolem Mandelbroijt and John Todd for the maintenance of the Institute after World War II
Emergence in biology and social sciences
No full textMathematics is the key to linking scientific knowledge at different scales: from microscopic to macroscopic dynamics. This link gives us understanding on the emergence of observable patterns like flocking of birds, leaf venation, opinion dynamics, and network formation, to name a few. In this article, we explore how mathematics is able to traverse scales, and in particular its application in modelling collective motion of bacteria driven by chemical signalling
Mini-Workshop: Mathematical Foundations of Robust and Generalizable Learning
No full textMachine learning has become an highly active field of research,
but its mathematical underpinnings are still hardly understood.
This workshop identified key challenges,
and it discussed potential solutions.
Bringing together a diverse group of researchers,
the workshop established different views on the topic based on notions from statistics, probability theory, and optimization
Solving inverse problems with Bayes' theorem
No full textThe goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as the Bayesian approach. In this approach, the unknown parameter is modelled as a random variable to reflect its uncertain value. Bayes’ theorem is applied to update our knowledge given new information from noisy data
Deciding Non-Freeness of Rational Möbius Groups
No full textWe explore a new computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, ). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, ) for a localization = [ of . We prove that a Möbius subgroup is not free by showing that it has finite index in the relevant SL(2, ). Further information about the structure of is obtained; for example, we compute the minimal subgroup of finite index in SL(2, ) that contains
Large Scale Stochastic Dynamics
No full textThe goal of this workshop was to explore the recent advances in the
mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems.
The talks addressed the following topics:
randomness emerging from deterministic dynamics,
hydrodynamic limits, Markov chain mixing times and cut-off phenomenon, superdiffusivity in out-of-equilibrium 2-dimensional systems
A tale of three curves
No full textIn this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve
Arbeitsgemeinschaft: Geometric Representation Theory
No full textOur understanding of algebraic representations of reductive algebraic groups in positive characteristic has seen big advances in the last years and
has been largely transformed into the geometric theory of
studying parity sheaves on affine Grassmannians and affine flag varieties or, equivalently and
more combinatorially, the diagrammatic Hecke category.
This has led, among other things,
to a geometric proof of the linkage principle and a greatly simplified
proof of Lusztig's character formula for large characteristics
Nonlinear Waves and Dispersive Equations
No full textNonlinear dispersive equations are models for nonlinear waves in a wide
range of physical contexts. Mathematically they display an interplay between
linear dispersion and nonlinear interactions, which can result in a wide range of
outcomes from finite time blow-up to solitons and scattering. They are linked
to many areas of mathematics and physics, ranging from integrable systems and
harmonic analysis to fluid dynamics, geometry and general relativity and probability