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Mini-Workshop: Women in Mathematics: Historical and Modern Perspectives
The aim of the workshop is to build a bridge between research on the situation of women in mathematics at the beginning of coeducative studies and the current circumstances in academia. The issue of women in mathematics has been a recent political and social hot topic in the mathematical community. As thematic foci we place a double comparison: besides shedding light on differences and similarities in several European countries, we complete this investigation by comparing the developments of women studies from the beginnings. This shall lead to new results on tradition and suggest improvements on the present situation.
Algebraic Geometry: Birational Classification, Derived Categories, and Moduli Spaces
The workshop covered a number of active areas of research in algebraic geometry with a focus on derived categories, moduli spaces (of varieties and sheaves) and birational geometry (often in positive characteristic) and their interactions. Special emphasis was put on hyperkähler manifolds and singularity theory.
Arbeitsgemeinschaft: Additive Combinatorics, Entropy, and Fractal Geometry
The aim of the workshop was to survey recent developments in fractal geometry, specifically those related to projections and slices of planar self-similar sets, and dimension and absolute continuity of self-similar measures on the line, in particular Bernoulli convolutions. The methods combine ergodic theory, additive combinatorics, and algebraic number theory. Talks were high-level descriptions of the results, aimed at a mixed audience with minimal background in real analysis, ergodic theory and dimension theory.
Spaces of Riemannian metrics
Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the "shape" of a manifold. The space of Riemannian metrics is a mathematical object that encodes the many possible ways in which we can geometrically deform the shape of a manifold.
Algebraic Statistics
Algebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research.
Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs
Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising.
Automorphic Forms and Arithmetic
The workshop brought together leading experts and young researchers at the interface of automorphic forms and analytic number theory to disseminate, discuss and develop important recent methods and results. A particular focus was on higher rank groups, as well as their arithmetic applications. This includes, for instance, the study of various aspects of $L$-functions, the fine distribution properties of their Fourier coefficients and Hecke eigenvalues, the mass distribution of automorphic forms on general symmetric spaces, and applications of results of algebraic geometry to automorphic forms.
Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics
The mini-workshop focused on central questions and new results concerning spaces and moduli spaces of Riemannian metrics with lower or upper curvature bounds on open and closed manifolds and, moreover, related themes from Anosov geometry. These are all described in detail below. The event brought together young and senior researchers working about (moduli) spaces of negative and nonnegative sectional, nonnegative Ricci and positive scalar curvature as well as Anosov metrics, and the talks and discussions brought about many new and inspiring research problems to pursue.
Differentialgeometrie im Großen
The topics discussed at the meeting were Kähler geometry, geometric evolution equations, manifolds of nonnegative curvature, metric geometry and geometric representations of groups. The choice of topics reflects current trends in the development of differential geometry.
On the Markov inequality in the L2-norm with the Gegenbauer weight
Let wλ(t):=(1−t2)λ−1/2, where λ>−12, be the Gegenbauer weight function, let ∥⋅∥wλ be the associated L2-norm, |f∥wλ={∫1−1|f(x)|2wλ(x)dx}1/2, and denote by Pn the space of algebraic polynomials of degree ≤n. We study the best constant cn(λ) in the Markov inequality in this norm ∥p′n∥wλ≤cn(λ)∥pn∥wλ,pn∈Pn, namely the constant cn(λ):=suppn∈Pn∥p′n∥wλ∥pn∥wλ. We derive explicit lower and upper bounds for the Markov constant cn(λ), which are valid for all n and λ.