Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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On Vietoris-Rips Complexes of Ellipses
No full textMSC: 05E45; 55U10; 68R05Research in Pairs 2015For a metric space and a scale parameter, the Vietoris–Rips complex
(resp. ) has as its vertex set, and a finite subset as a simplex whenever the
diameter of is less than (resp. at most ). Though Vietoris–Rips complexes have been studied at
small choices of scale by Hausmann and Latschev [12, 14], they are not well-understood at larger scale
parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses
of small eccentricity, meaning . Indeed, we show there
are constants such that for all , we have and ,
though only one of the two-spheres in is persistent. Furthermore, we show that for any scale
parameter , there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips
complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As
our main tool we link these homotopy types to the structure of infinite cyclic graphs
Cryptography
No full textThe Oberwolfach workshop Cryptography brought together scientists from cryptography with mathematicians specializing in the algorithmic problems underlying cryptographic security. The goal of the workshop was to stimulate interaction and collaboration that enables a holistic approach to designing cryptography from the mathematical foundations to practical applications. The workshop addressed fundamental research results leading to innovative cryptography for protecting security and privacy
Geometry of Free Loci and Factorization of Noncommutative Polynomials
No full textMSC 2010: 13J30; 15A22; 47A56 (Primary) | 14P10; 16U30; 16R30 (Secondary)Research in Pairs 2017The free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry
Representation Theory of Quivers and Finite Dimensional Algebras
No full textMethods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras
Looking Back on Inverse Scattering Theory
No full textResearch in Pairs 2017We present an essay on the mathematical development of inverse scattering theory for time-harmonic waves during the past fifty years together with some personal memories of our participation in these
events
A few shades of interpolation
No full textThe topic of this snapshot is interpolation. In the
ordinary sense, interpolation means to insert something
of a different nature into something else. In
mathematics, interpolation means constructing new
data points from given data points. The new points
usually lie in between the already-known points. The
purpose of this snapshot is to introduce a particular
type of interpolation, namely, polynomial interpolation.
This will be explained starting from basic ideas
that go back to the ancient Babylonians and Greeks,
and will arrive at subjects of current research activity
Mathematics plays a key role in scientific computing
No full textI attended a very interesting workshop at the research
center MFO in Oberwolfach on “Recent Developments
in the Numerics of Nonlinear Hyperbolic Conservation
Laws”. The title sounds a bit technical,
but in plain language we could say: The theme is
to survey recent research concerning how mathematics
is used to study numerical algorithms involving
a special class of equations. These equations arise
from computer simulations to solve application problems
including those in aerospace engineering, automobile
design, and electromagnetic waves in communications
as examples. This topic belongs to the general
research area called “scientific computing”
On an Effective Variation of Kronecker’s Approximation Theorem Avoiding Algebraic Sets
No full textResearch in Pairs 2017Let be an algebraic lattice, coming from a projective module over the ring of integers of a number field . Let be the zero locus of a finite collection of polynomials such that or a finite union of proper full-rank sublattices of . Let be the number field generated over by coordinates of vectors in , and let be linear forms in variables with algebraic coefficients satisfying an appropriate linear independence condition over . For each and , we prove the existence of a vector of explicitly bounded sup-norm such that
for each , where stands for the distance to the nearest integer. The bound on sup-norm of depends on , as well as on , , and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of under the linear forms in the -torus~. In the appendix, we also discuss a construction of badly approximable matrices, a subject closely related to our proof of effective Kronecker's theorem, via Liouville-type inequalities and algebraic transference principles
Analysis, Geometry and Topology of Positive Scalar Curvature Metrics
No full textRiemannian manifolds with positive scalar curvature play an important role in mathematics and general relativity. Obstruction and existence results are connected to index theory, bordism theory and homotopy theory, using methods from partial differential equations and functional analysis. The workshop led to a lively interaction between mathematicians working in these areas
Analysis and Simulation of a New Multi-Component Two-Phase Flow Model with Phase Transitions and Chemical Reactions
No full textResearch in Pairs 2016A class-II model for multi-component mixtures recently introduced in D. Bothe, W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757–1805 is investigated for simple mixtures. Bothe and Dreyer were aiming at deriving physically admissible closure conditions. Here the focus is on mathematical properties of this model. In particular, hyperbolicity of the inviscid flux Jacobian is verified for non-resonance states. Although the eigenvalues cannot be determined explicitly but have to be computed numerically an eigenvector basis is constructed depending on the eigenvectors. This basis is helpful to apply standard numerical solvers for the discretization of the model. This is verified by numerical computations for two- and three-component mixtures with and without phase transition and chemical reactions