Using online STACK assessment to teach complex analysis: a prototype course design?

We describe a new course design, informed by our experience of the pandemic, that we think could be used in other high-level mathematics courses. The course’s main resource was a set of interactive STACK workbooks containing the course notes, automatically-marked comprehension and practice questions for self-assessment, and short videos of examples, calculations, and high-level motivation. This freed up synchronous class time to address conceptual understanding using interactive polling. We describe the course and discuss how it worked in practice

Singular minimizers in the calculus of variations

This thesis examines the possible failure of regularity for minimizers of onedimensional variational problems. The direct method of the calculus of variations gives rigorous assurance that minimizers exist, but necessarily admits the possibility that minimizers might not be smooth. Regularity theory seeks to assert some extra smoothness of minimizers. Tonelli's partial regularity theorem states that any absolutely continuous minimizer has a (possibly infinite) classical derivative everywhere, and this derivative is continuous as a function into the extended real line. We examine the limits of this theorem. We find an example of a reasonable problem where partial regularity fails, and examples where partial regularity holds, but the infinite derivatives of minimizers permitted by the theorem occur very often, in precise senses. We construct continuous Lagrangians, strictly convex and superlinear in the third variable, such that the associated variational problems have minimizers nondifferentiable on dense second category sets. Thus mere continuity is an insufficient smoothness assumption for Tonelli's partial regularity theorem. Davie showed that any compact null set can occur as the singular set of a minimizer to a problem given via a smooth Lagrangian with quadratic growth. The proof relies on enforcing the occurrence of the Lavrentiev phenomenon. We give a new proof of the result, but constructing also a Lagrangian with arbitrary superlinear growth, and in which the Lavrentiev phenomenon does not occur in the problem. Universal singular sets record how often a given Lagrangian can have minimizers with infinite derivative. Despite being negligible in terms of both topology and category, they can have dimension two: any compact purely unrectifiable set can lie inside the universal singular set of a Lagrangian with arbitrary superlinearity. We show this also to be true of Fσ purely unrectifiable sets, suggesting a possible characterization of universal singular sets.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Variations, approximation, and low regularity in one dimension

We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general of the form of a standard Lipschitz "variation". Part of this investigation, but of interest in its own right, is an example of a nowhere locally Lipschitz minimizer which serves as a counter-example to any putative Tonelli partial regularity statement. Under these low assumptions we find it nonetheless remains possible to derive necessary conditions for minimizers, in terms of approximate continuity and equality of the one-sided derivatives.Comment: v3, 60 pages. To appear in CoVPDE. Minor cosmetic correction

Singular minimizers in the calculus of variations

This thesis examines the possible failure of regularity for minimizers of onedimensional variational problems. The direct method of the calculus of variations gives rigorous assurance that minimizers exist, but necessarily admits the possibility that minimizers might not be smooth. Regularity theory seeks to assert some extra smoothness of minimizers. Tonelli's partial regularity theorem states that any absolutely continuous minimizer has a (possibly infinite) classical derivative everywhere, and this derivative is continuous as a function into the extended real line. We examine the limits of this theorem. We find an example of a reasonable problem where partial regularity fails, and examples where partial regularity holds, but the infinite derivatives of minimizers permitted by the theorem occur very often, in precise senses. We construct continuous Lagrangians, strictly convex and superlinear in the third variable, such that the associated variational problems have minimizers nondifferentiable on dense second category sets. Thus mere continuity is an insufficient smoothness assumption for Tonelli's partial regularity theorem. Davie showed that any compact null set can occur as the singular set of a minimizer to a problem given via a smooth Lagrangian with quadratic growth. The proof relies on enforcing the occurrence of the Lavrentiev phenomenon. We give a new proof of the result, but constructing also a Lagrangian with arbitrary superlinear growth, and in which the Lavrentiev phenomenon does not occur in the problem. Universal singular sets record how often a given Lagrangian can have minimizers with infinite derivative. Despite being negligible in terms of both topology and category, they can have dimension two: any compact purely unrectifiable set can lie inside the universal singular set of a Lagrangian with arbitrary superlinearity. We show this also to be true of Fσ purely unrectifiable sets, suggesting a possible characterization of universal singular sets