Contributions to the essential dimension of finite and algebraic groups

Essential dimension, introduced by Joe Buhler and Zinovy Reichstein and in its most general form by Alexander Merkurjev is a measure of complexity of algebraic objects such as quadratic forms, hermitian forms, central simple algebras and étale algebras. Informally, the essential dimension of an algebraic object is the number of parameters needed to define it. Often isomorphism classes of objects of some type are in one to one bijection with isomorphism classes of G-torsors. The maximal essential dimension of a G-torsor (called essential dimension of G) gives an invariant of algebraic groups, which will be of primary interest in this thesis. The text is subdivided into four chapters as follows: Chapter I+II: Multihomogenization of covariants and its application to covariant and essential dimension The essential dimension of a linear algebraic group G can be expressed via G-equivariant rational maps phi: A(V) --> A(W), so called covariants, between generically free G-modules V and W. In these two chapters we explore a new technique for dealing with covariants, called multihomogenization. This technique was jointly introduced with Hanspeter Kraft and Gerald Schwarz in an already published paper, which forms the second chapter. Applications of the multihomogenization technique to the essential dimension of algebraic groups are given by results on the essential dimension of central extensions, direct products, subgroups and the precise relation of essential dimension and covariant dimension (which is a variant of the former with polynomial covariants). Moreover the multihomogenization technique allows one to extend a twisting construction introduced by Matthieu Florence from the case of irreducible representations to completely reducible representations. This relates Florence's work on the essential dimension of cyclic p-groups to recent stack theoretic approaches by Patrick Brosnan, Angelo Vistoli and Zinovy Reichstein and by Nikita Karpenko and Alexander Mekurjev. Chapter III: Faithful and p-faithful representations of minimal dimension The study of essential dimension of finite and algebraic groups is closely related to the study of its faithful resp. generically free representations. In general the essential dimension of an algebraic group is bounded above by the least dimension of a generically free representation minus the dimension of the algebraic group. In some prominent cases this upper bound or a variant of it is strict. In this chapter we are guided by the following general questions: What do faithful representations of the least possible dimension look like? How can they be constructed? How are they related to faithful representations of minimal dimension of subgroups? Along the way we compute the minimal number of irreducible representations needed to construct a faithful representation. Chapter IV: Essential p-dimension of algebraic tori This chapter is joint work with Mark MacDonald, Aurel Meyer and Zinovy Reichstein. We study a variant of essential dimension which is relative to a prime number p. This variant, called essential p-dimension, disregards effects resulting from other primes than p. In a recent paper Nikita Karpenko and Alexander Merkurjev have computed the essential dimension of p-groups. We extend their result and find the essential p-dimension for a class of algebraic groups, which includes all algebraic tori and twisted finite p-groups

Application of multihomogeneous covariants to the essential dimension of finite groups

We investigate the essential dimension of finite groups using the multihomogenization technique introduced in [KLS09], for which we provide new applications in a more general setting. We generalize the central extension theorem of Buhler and Reichstein [BR97, Theorem 5.3] and use multihomogenization as a substitute to the stackinvolved part of the theorem of Karpenko and Merkurjev [KM08] about the essential dimension of p-group

Compression of Finite Group Actions and Covariant Dimension, II

Let $G$ be a finite group and $\phi : V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $\phi$ is faithful if $G$ acts faithfully on $\phi(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{\phi(V)}$ taken over all faithful $\phi$. In \cite{KS07} we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of \cite{KS07}, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of \cite{KS07}

Essential p-dimension of algebraic groups whose connected component is a torus

Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G^0 is a torus.Comment: 23 pages, no figures. arXiv admin note: text overlap with arXiv:0910.557

Simulation-based medical education for Ambulance Jet and Helicopter Emergency Medical Services: A program description and evaluation

Introduction: In aviation, crew resource management trainings are established methods to enhance safety, a method that also gained popularity in medicine. In 2015, the Swiss Air Rescue (Rega) Helicopter Emergency Medical Services decided to start a simulation-based medical education program for its helicopter and ambulance jet crews (emergency physicians, paramedics/flight nurses and pilots). The aim of this program was to improve technical skills and the application of human factors during rescue missions. This report shows a five-year summary of the participants’ course evaluation. Methods: A 1-day high-fidelity simulation on crisis resource management with video-assisted debriefing took place at 3 centres, two in Switzerland; one in Germany. Crew members participated once per year. Simulation covered critical situations in the helicopter or jet, during handovers at an intensive care unit or in ambulances. Extra Corporeal Membrane Oxygenation and Intra-Aortic Balloon Pump use was simulated during helicopter transports. Additionally, four times per year flight crews rehearsed basic and advanced life support skills using low-fidelity equipment between missions. Participants answered an anonymized course evaluation survey. Answers were rated on a Numeric Rating Scale ranging from 1=no agreement to 5=total agreement. Results: 329 participated and answered the questionnaire; 50% were emergency physicians, 40% paramedics, 9% flight nurses, and 1% pilots. Participants agreed that the course taught competencies that were useful for their clinical practice. However, confidence to apply Extra Corporeal Membrane Oxygenation or Intra-Aortic Balloon Pump skills was significantly lower compared to other emergency competencies. Instructors were rated as experienced, engaged and motivated, as well as responsive to course participants. Conclusions: This simulation-based medical education program, with the goal to increase patient’s safety and outcome,was launched successfully. Participants especially valued the time to reflect on clinical performance as well as on crew interaction and ways to apply human factors to improve their team performance and task management

The slice method for G-torsors

The notion of a (G,N)(G,N)-slice of a G-variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G -torsors. Namely we show that any (G,N)(G,N)-slice of a versal G -scheme gives surjective maps H1(L,N)→H1(L,G)H1(L,N)→H1(L,G) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a (G,NG(H))(G,NG(H))-slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E7E7

A generalized composition of quadratic forms based on quadratic pairs

For quadratic spaces which represent 1 there is a characterization of hermitian compositions in the language of algebras-with-involutions using the even Clifford algebra. We extend this notion to define a generalized composition based on quadratic pairs and determine the degrees of minimal compositions for any given quadratic pair

Application of multihomogeneous covariants to the essential dimension of finite groups

We investigate essential dimension of finite groups over arbitrary fields and give a systematic treatment of multihomogenization, introduced by H.Kraft, G.Schwarz and the author. We generalize the central extension theorem of Buhler and Reichstein and use multihomogenization to substitute and generalize the stack-involved part of the theorem of Karpenko and Merkurjev about the essential dimension of p-groups. One part of this paper is devoted to the study of completely reducible faithful representations. Amongst results concerning faithful representations of minimal dimension there is a computation of the minimal number of irreducible components needed for a faithful representation