Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Tropical geometry
No full textWhat kind of strange spaces hide behind the enigmatic
name of tropical geometry? In the tropics, just
as in other geometries, one of the simplest objects is
a line. Therefore, we begin our exploration by considering
tropical lines. Afterwards, we take a look at
tropical arithmetic and algebra, and describe how to
define tropical curves using tropical polynomials
Scaling Limits in Models of Statistical Mechanics
No full textThis conference (part of a long running series) aims to cover the interplay between probability and mathematical statistical mechanics. Specific topics addressed during the 22 talks include: Universality and critical phenomena, disordered models, Gaussian free field (GFF), random planar graphs and unimodular planar maps, reinforced random walks and non-linear σ-models, non-equilibrium dynamics. Less stress is given to topics which have running series of Oberwolfach conferences devoted to them specifically, such as random matrices or integrable models and KPZ universality class. There were 50 participants, including 9 postdocs and graduate students, working in diverse intertwining areas of probability, statistical mechanics and mathematical physics
Sur le Minimum de la Fonction de Brjuno
No full textThe Brjuno function attains a strict global minimum at the golden section
Criteria for Algebraicity of Analytic Functions
No full textWe consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing trough a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper
Topologie
No full textThe talks covered advances in algebraic K-theory and topological cyclic homology, geometric group theory, low dimensional topology relying on a mixture of combinatorial and analytic methods, classification of highdimensional manifolds and more. Special emphasis was given to a recent breakthrough on the question of triangulability of high-dimensional manifolds
Mixed volumes and mixed integrals
No full textIn recent years, mathematicians have developed new
approaches to study convex sets: instead of considering
convex sets themselves, they explore certain functions
or measures that are related to them. Problems
from convex geometry become thereby accessible to
analytic and probabilistic tools, and we can use these
tools to make progress on very difficult open problems.
We discuss in this Snapshot such a functional extension
of some “volumes” which measure how “big”
a set is. We recall the construction of “intrinsic volumes”,
discuss the fundamental inequalities between
them, and explain the functional extensions of these
results
Emergence of Structures in Particle Systems: Mechanics, Analysis and Computation
No full textThe meeting focused on the last advances in particle systems. The talks covered a broad range of topics, ranging from questions in crystallization and atomistic systems to mesoscopic models of defects to machine learning approaches and computational aspects
The Sylow Structure of Scalar Automorphism Groups
No full textFor any locally compact abelian periodic group A its automorphism group contains as a subgroup those automorphisms that leave invariant every closed subgroup of A, to be denoted by SAut(A). This subgroup is again a locally compact abelian periodic group in its natural topology and hence allows a decomposition into its p-primary subgroups for p the primes for which topological p-elements in this automorphism subgroup exist. The interplay between the p-primary decomposition of SAut(A) and A can be encoded in a bipartite graph, the mastergraph of A. Properties and applications of this concept are discussed
Demailly’s Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps
No full textDemailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffiths-Lang conjecture on hyperbolic projective varieties
Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
No full textThe existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed