We introduce the notion of viscosity solutions for second order fully nonlinear elliptic/parabolic equations. A viscosity solution, u, is defined through the condition that whenever a test function, φ, touches u from below/above at a point then φ should be a classical super/sub-solutions of the equation at this point. Since u is apriori merely continuous, it is natural to study whether the equation finally impose any regularizing effects on u. We start by giving an overview of the theory that have been developed the last decades on the interior regularity properties of viscosity solutions, introducing the main tools used, namely the ABP and Harnack inequalities. Then we shall concentrate on oblique derivative problems (where the oblique derivative condition is understood only in the viscosity sense as well) and discuss about the up to the boundary (first order) Hölder regularity of the solutions giving quantitative bounds for the corresponding Hölder norms in terms of the data.

We consider a sequence of linear-quadratic optimization problems defined in an abstract functional framework. Each problem is accompanied by the constraint of reaching a given target with some precision. We show that problems are well posed and we characterise their solutions. The main result provides the conditions under which these solutions converge to the minimizer of the limit problem. The theory can be applied to a wide range of problems: elliptic, parabolic, control ones etc. In the talk, the following examples will be presented: 1. optimal controllability of the heat equation 2. optimal solvability of the Poisson equation.

Classroom discussions—dialogic interactions in which the ideas of students and teachers are collectively shared and developed—can increase student learning in mathematics classrooms. Yet, multilingual students in US schools often experience mathematics classrooms where most talk is teacher-led lecture. In this seminar, I will report on results from a five-year design research intervention in linguistically diverse 9th grade mathematics classrooms in the US-Mexico border region. Our goal was to promote student participation in dialogic interactions. I will 1) Outline the design framework, 2) Describe the design principles that were developed in the course of the research, and 3) Present key qualitative and quantitiave findings from the research. I will close with a discussion of how language policies and language ideologies shape the quality of student participation in classroom discussions, and consider future directions for work in this area of investigation.

Am Beginn des Mathematikstudiums stellt der Umgang mit mathematischen Beweisen in der Hochschulmathematik für Studierende eine neuartige Anforderung dar. Diese liegt einerseits im eigenständigen Führen von Beweisen und andererseits im Verstehen von vorgegebenen Beweisen. Der Vortrag konzentriert sich auf Letzeres, d.h. das Verstehen von Beweisen, die Studierenden in Vorlesungen und in Lehrtexten präsentiert werden. Der Vortragende stellt zwei Ansätze aus seiner eigenen Lehrpraxis vor, mit der Studierende beim Beweisverstehen unterstützt werden sollen: Peer Instruction und Audioguides. Die zwei Ansätze sind für unterschiedliche Settings entworfen: Während sich Peer Instruction -- besonders in der Variante der ``Mid-Proof Peer Instruction`` -- für den Einsatz in synchronen Lehrveranstaltungen eignet, sind Audioguides eine Möglichkeit zur Unterstützung in der asynchronen Lehre. Im Vortrag werden Hintergrund und Einsatzweisen sowie Ergebnisse aus Untersuchungen zu beiden Methoden vorgestellt.