Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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Field Arithmetic
No full textField Arithmetic studies the interrelation between arithmetic properties of fields and their absolute Galois groups. It is an interdisciplinary area that uses methods of algebraic number theory, commutative algebra, algebraic geometry, arithmetic geometry, finite and profinite groups, and nonarchimedean analysis. Some of the results are motivated by questions of model theory and used to establish results in (un-)decidability
The Tutte Polynomial of Ideal Arrangements
No full textThe Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo
Computing Congruence Quotients of Zariski Dense Subgroups
No full textWe obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group for . More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of for
Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups
No full textThe geometric study of the absolute Galois group of the rational numbers has been a highly active research topic since the first milestones: Hilbert’s Irreducibility Theorem, Noether’s program, Riemann’s Existence Theorem. It gained special interest in the last decades with Grothendieck’s “Esquisse d’un programme”, his “Letter to Faltings” and Fried’s introduction of Hurwitz spaces. It grew on and thrived on a wide range of areas, e.g. formal algebraic geometry, Diophantine geometry, group theory. The recent years have seen the development and integration in algebraic geometry and Galois theory of new advanced techniques from algebraic stacks, ℓ-adic representations and homotopy theories. It was the goal of this mini-workshop, to bring together an international panel of young and senior experts to draw bridges towards these fields of research and to incorporate new methods, techniques and structures in the development of geometric Galois theory
A Well-Posedness Result for Viscous Compressible Fluids with Only Bounded Density
No full textWe are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension and for general pressure laws
Mini-Workshop: Mathematical and Numerical Analysis of Maxwell's Equations
No full textIn this mini-workshop 17 leading mathematicians from Europe and United States met at the MFO to discuss and present new developments in the mathematical and numerical analysis of Maxwell’s equations and related systems of partial differential equations. The report at hand offers the extended abstracts of their talks
A Deformed Quon Algebra
No full textThe quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators , , on an infinite dimensional vector space satisfying the deformed -mutator relations . We prove the realizability of our model by showing that, for suitable values of , the vector space generated by the particle states obtained by applying combinations of 's and 's to a vacuum state is a Hilbert space. The proof particularly needs the investigation of the new statistic
and representations of the colored permutation group
Geometry behind one of the Painlevé III differential equations
No full textThe Painlevé equations are second order differential
equations, which were first studied more than 100
years ago. Nowadays they arise in many areas in
mathematics and mathematical physics. This snapshot
discusses the solutions of one of the Painlevé
equations and presents old results on the asymptotics
at two singular points and new results on the global
behavior
The Mathematics of Mechanobiology and Cell Signaling
No full textThe workshop focussed on the mathematical modeling and analysis of the mutual interaction among living cells, their interaction with the environment, and the resulting morphogenetic processes. The interplay of bio-mechanical processes and molecular signaling and their combined effect on the emergence of shape and function in cell clusters, tissues, and organs was addressed. Classical methods of continuum mechanics and necessary extensions were discussed at a formal and a rigorous mathematical level. Several introductory talks were given by experimentalists
An Optimal Bound for the Ratio Between Ordinary and Uniform Exponents of Diophantine Approximation
No full textWe provide a lower bound for the ratio between the ordinary and uniform exponents of both simultaneous Diophantine approximation to n real numbers and Diophantine approximation for one linear form in n variables. This question was first considered in the 50’s by V. Jarník who solved the problem for two real numbers and established certain bounds in higher dimension. Recently different authors reconsidered the question, solving the problem in dimension three with different methods. Considering a new concept of parametric geometry of numbers, W. M. Schmidt and L. Summerer conjectured that the optimal lower bound is reached at regular systems. It follows from a remarkable result of D. Roy that this lower bound is then optimal. In the present paper we give a proof of this conjecture by W. M. Schmidt and L. Summerer