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On the geometry of regular maps from a quasi-projective surface to a curve
We explore consequences of the triviality of the monodromy group, using the condition of purity of the mixed Hodge structure on the cohomology of the surface X.
Infeasibility certificates for linear matrix inequalities
Farkas' lemma is a fundamental result from linear programming providing linear certi cates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality L(x)⪰0 is infeasible if and only if −1 lies in the quadratic module associated to L. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certi cate, we use a more involved algebraic certificate motivated by the real radical and Prestel's theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination.
Random permutations
100 people leave their hats at the door at a party and pick up a completely random hat when they leave. How likely is it that at least one of them will get back their own hat? If the hats carry name tags, how difficult is it to arrange for all hats to be returned to their owner? These classical questions of probability theory can be answered relatively easily. But if a geometric component is added, answering the same questions immediately becomes very hard, and little is known about them. We present some of the open questions and give an overview of what current research can say about them.
Noncompact harmonic manifolds
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szab´o [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR] provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient. The classification of all noncompact harmonic spaces is still a very difficult open problem. In this paper we provide a survey on recent results on noncompact simply connected harmonic manifolds, and we also prove many new results, both for general noncompact harmonic manifolds and for noncompact harmonic manifolds with purely exponential volume growth.
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.
Matrixfaktorisierungen
Im Folgenden soll ein kurzer Abriss des Themas Matrixfaktorisierungen gegeben werden. Wir werden darlegen, warum dieses recht simple Konzept zu erstaunlich tiefen mathematischen Gedankengängen führt und auch in der modernen theoretischen Physik wichtige Anwendungen hat.
The codimension
In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one.
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.
Mehler-Heine asymptotics of a class of generalized hypergeometric polynomials
We obtain a Mehler–Heine type formula for a class of generalized hypergeometric polynomials. This type of formula describes the asymptotics of polynomials scale conveniently. As a consequence of this formula, we obtain the asymptotic behavior of the corresponding zeros. We illustrate these results with numerical experiments and some figures.