Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Real Algebraic Geometry with a View toward Koopman Operator Methods
No full textThis workshop was dedicated to the newest developments
in real algebraic geometry and their interaction with convex optimization and operator theory.
A particular effort was invested in exploring the interrelations
with the Koopman operator methods in dynamical systems and their applications.
The presence of researchers from different scientific communities
enabled an interesting dialogue
leading to new exciting and promising synergies
Mini-Workshop: Flavors of Rabinowitz Floer and Tate Homology
No full textRabinowitz Floer homology originated 15 years ago in symplectic geometry. Recent developments have related it to algebraic topology via string toplogy and Tate homology, and to mirror symmetry via Fukaya categories. This mini-workshop brought together researchers from these different communities, in order to foster exchange and collaborations across research fields
Dynamische Systeme
No full textThis workshop continues a series of workshops whose current format originated in 1981 under then-organizers Moser and Zehnder, and whose latest iteration took place in July 2023. The general goal of this series of workshops is to discuss the latest developments in the field of dynamical systems, broadly construed, and its connections with neighboring areas of mathematics such as differential geometry, partial differential equations, and more recently contact and symplectic geometry. We continued this tradition, bringing in new participants working in areas of dynamical systems and its connections with other areas of mathematics that are currently highly active and/or showing great promise for future development. Key focus areas for the 2023 workshop include spectral rigidity for planar domains, chaotic and oscillatory motions in celestial mechanics, conformal symplectic dynamics, and relations between dynamics
Edifices: Building-like Spaces Associated to Linear Algebraic Groups; In memory of Jacques Tits
No full textGiven a semisimple linear algebraic -group , one has a spherical building , and one can interpret the geometric realisation of in terms of cocharacters of . The aim of this paper is to extend this construction to the case when is an arbitrary connected linear algebraic group; we call the resulting object the spherical edifice of . We also define an object which is an analogue of the vector building for a semisimple group; we call the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on and show they are all bi-Lipschitz equivalent to each other; with this extra structure, becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture
New Techniques in Resolution of Singularities
No full textResolution of singularities is notorious as a difficult topic within algebraic geometry. Recent work, aiming at resolution of families and semistable reduction, infused the subject with logarithmic geometry and algebraic stacks, two techniques essential for the current theory of moduli spaces. As a byproduct a short, a simple and efficient functorial resolution procedure in characteristic 0 using just algebraic stacks was produced.
The goals of the book, the result of an Oberwolfach Seminar, are to introduce readers to explicit techniques of resolution of singularities with access to computer implementations, introduce readers to the theories of algebraic stacks and logarithmic structures, and to resolution in families and semistable reduction methods
Arbeitsgemeinschaft: QFT and Stochastic PDEs
No full textQuantum field theory (QFT) is a fundamental framework for a wide range of phenomena is physics.
The link between QFT and SPDE was first observed by the physicists Parisi and Wu (1981), known as Stochastic Quantisation. The study of solution theories and properties of solutions to these SPDEs derived from the Stochastic Quantisation procedure has stimulated substantial progress of the solution theory of singular SPDE, especially the invention of the theories of regularity structures and paracontrolled distributions in the last decade. Moreover, Stochastic Quantisation allows us to bring in more tools including PDE and stochastic analysis to study QFT.
This Arbeitsgemeinschaft starts by covering some background material and then explores some of the advances made in recent years. The focus of this Arbeitsgemeinschaft is QFT models such as the
, sine-Gordon and Yang--Mills models as examples to discuss stochastic quantisation and SPDE methods and their applications in these models. We introduce the key ideas, results and applications of regularity structure and paracontrolled distributions, construction of solutions of the SPDEs corresponding to these models, and use the PDE method to study some qualitative behaviors of these QFTs, and connections with the corresponding lattice or statistical physical models. We also discuss some other topics of QFT, such as Wilsonian renormalisation group, log-Sobolev inequalities and their implications, and various connections between these topics and SPDEs
New Horizons in Motions in Random Media
No full textThe general topic of the mini-workshop "New Horizons in Motions in Random Media" was the study of random walks in random environments, both in their own right and in relation to stochastic homogenization and to models in statistical mechanics, in particular spin system. This is a subject at the intersection of probability, analysis and mathematical physics, and the workshop brought together leading researchers from those areas. While each of these areas has been quite active for decades with many remarkable breakthroughs obtained throughout the years, the workshop provided a unique opportunity to identify principal new objectives and initiate new collaborations
Model Theory: Combinatorics, Groups, Valued Fields and Neostability
No full textThe scope of contemporary model theory has expanded enormously over the last
several decades, helped by the development of new tools applicable to an ever wider range
of structures. In the spirit of the previous meetings in the series, this workshop will bring
together researchers from apparently separate subfields of model theory whose work is linked
by common themes, with a particular emphasis on intrinsic model theoretic
questions motivated by the classification of approriately 'tame' groups and fields and new
developments in asymptotic combinatorics
Tropical Methods in Geometry
No full textThe workshop "Tropical methods in geometry" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject including tropical methods in symplectic and Lagrangian geometry, topology of real algebraic varieties and tropical homology, tropical methods in algebraic, Berkovich analytic and log geometries, refined tropical enumerative geometry and enriched counting, and algebraic geometry and matroids
Lax Comma Categories of Ordered Sets
No full textLet be the category of (pre)ordered sets. Unlike , whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category . In this paper we show that the forgetful functor is topological if and only if is complete. Moreover, under suitable hypothesis, is complete and cartesian closed if and only if is. We end by analysing descent in this category. Namely, when is complete and cartesian closed, we show that, for a morphism in , being pointwise effective for descent in is sufficient, while being effective for descent in is necessary, to be effective for descent in