In this thesis, we study two different kinds of approximation of plurisubharmonic functions. The first one is a Mergelyan type approximation for plurisubharmonic functions. That is, we study which domains in C^n have the property that every continuous plurisubharmonic function can be uniformly approximated with continuous and plurisubharmonic functions defined on neighborhoods of the domain. We will improve a result by Fornaess and Wiegerinck and show that domains with C^0-boundary have this property. We will also use the notion of plurisubharmonic functions on compact sets when trying to characterize those continuous and plurisubharmonic functions that can be approximated from outside. Here a new kind of convexity of a domain comes in handy, namely those domains in C^n that have a negative exhaustion function that is plurisubharmonic on the closure. For these domains, we prove that it is enough to look at the boundary values of a plurisubharmonic function to know whether it can be approximated from outside. The second type of approximation is the following: we want to approximate functions u that are defined on bounded hyperconvex domains Omega in C^n and have essentially boundary values zero and bounded Monge-Ampère mass, with increasing sequences of certain functions u_j that are defined on strictly larger domains. We show that for certain conditions on Omega, this is always possible. We also generalize this to functions with given boundary values. The main tool in the proofs concerning this second approximation is subextension of plurisubharmonic functions.

Avhandlingen består av två ganska olika delar som ändå har en del gemensamt. Del A är baserad på två artiklar i matematik och del B är baserad på två matematikdidaktiska artiklar. De matematiska artiklarna utgår från ett begrepp som heter polynomkonvexitet. Grundidén är att man skulle kunna se vissa ytor som en sorts ”tak” (tänk på taket till en carport). Alla punkter, eller positioner, ”under taket” (ungefär som de platser som skyddas från regn av carporttaket) ligger i något som kallas ”polynomkonvexa höljet.” Tidigare forskning har visat att för ett givet tak och en given punkt så finns det ett sätt att avgöra om punkten ligger ”under taket”. Det finns nämligen i så fall alltid en sorts matematisk funktion med vissa egenskaper. Finns det ingen sådan funktion så ligger inte punkten under taket och tvärt om; ligger punkten utanför taket så finns det heller ingen sådan funktion. Jag visar i min första artikel att det kan finnas flera olika sådana funktioner till en punkt som ligger under taket. I den andra artikeln visar jag några exempel på hur man kan konstruera sådana funktioner när man vet hur taket ser ut och var under taket punkten ligger. De matematikdidaktiska artiklarna i avhandlingen handlar om vad som krävs av studenterna när de gör universitetstentor i matematik. Vissa uppgifter kan gå att lösa genom att studenterna lär sig någonting utantill ur läroboken och sen skriver ner det på tentan. Andra går kanske att lösa med hjälp en algoritm, ett ”recept,” som studenterna har övat på att använda. Båda dessa sätt att resonera kallas imitativt resonemang. Om uppgiften kräver att studenterna ”tänker själva” och skapar en (för dem) ny lösning, så kallas det kreativt resonemang. Forskning visar att elever i stor utsträckning väljer att jobba med imitativt resonemang, även när uppgifterna inte går att lösa på det sättet. Mycket pekar också på att de svårigheter med att lära sig matematik som elever ofta har är nära kopplat till detta arbetssätt. Det är därför viktigt att undersöka i vilken utsträckning de möter olika typer av resonemang i undervisningen. Den första artikeln består av en genomgång av tentauppgifter där det noggrant avgörs vilken typ av resonemang som de kräver av studenterna. Resultatet visar att studenterna kunde bli godkända på nästan alla tentorna med hjälp av imitativt resonemang. Den andra artikeln baserades på intervjuer med sex av de lärare som konstruerat tentorna. Syftet var att ta reda på varför tentorna såg ut som de gjorde och varför det räckte med imitativt resonemang för att klara dem. Det visade sig att lärarna kopplade uppgifternas svårighetsgrad till resonemangstypen. De ansåg att om uppgiften krävde kreativt resonemang så var den svår och att de uppgifter som gick att lösa med imitativt resonemang var lättare. Lärarna menade att under rådande omständigheter, t.ex. studenternas försämrade förkunskaper, så är det inte rimligt att kräva mer kreativt resonemang vid tentamenstillfället. / This dissertation consists of two different but connected parts. Part A is based on two articles in mathematics and Part B on two articles in mathematics education. Part A mainly focus on properties of positive currents in connection to polynomial convexity. Earlier research has shown that a point z0 lies in the polynomial hull of a compact set K if and only if there is a positive current with compact support such that ddcT = μ−δz0. Here μ is a probability measure on K and δz0 denotes the Dirac mass at z0. The main result of Article I is that the current T does not have to be unique. The second paper, Article II, contains two examples of different constructions of this type of currents. The paper is concluded by the proof of a proposition that might be the first step towards generalising the method used in the first example. Part B consider the types of reasoning that are required by students taking introductory calculus courses at Swedish universities. Two main concepts are used to describe the students’ reasoning: imitative reasoning and creative reasoning. Imitative reasoning consists basically of remembering facts or recalling algorithms. Creative reasoning includes flexible thinking founded on the relevant mathematical properties of ob jects in the task. Earlier research results show that students often choose imitative reasoning to solve mathematical tasks, even when it is not a successful method. In this context the word choose does not necessarily mean that the students make a conscious and well considered selection between methods, but just as well that they have a subconscious preference for certain types of procedures. The research also show examples of how students that work with algorithms seem to focus solely on remembering the steps, and researchers argue that this weakens the students’ understanding of the underlying mathematics. Article III examine to what extent students at Swedish universities can solve exam tasks in introductory calculus courses using only imitative reasoning. The results show that about 70 % of the tasks were solvable by imitative reasoning and that the students were required to use creative reasoning in only one of 16 exams in order to pass. In Article IV, six of the teachers that constructed the analysed exams in Article III were interviewed. The purpose was to examine their views and opinions on the reasoning required in the exams. The analysis showed that the teachers are quite content with the present situation. The teachers expressed the opinion that tasks demanding creative reasoning are usually more difficult than tasks solvable with imitative reasoning. They therefore use the required reasoning as a tool to regulate the tasks’ degree of difficulty, rather than as a task dimension of its own. The exams demand mostly imitative reasoning since the teachers believe that they otherwise would, under the current circumstances, be too difficult and lead to too low passing rates.

This thesis consists of a comprehensive summary and six scientific papers dealing with the boundary behaviour of functions related to parabolic partial differential equations and several complex variables. Paper I concerns solutions to non-linear parabolic equations of linear growth. The main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It is also shown that the Riesz measure associated with such solutions has the doubling property. Paper II is concerned with solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are that non-negative solutions which vanish continuously on the lateral boundary of an NTA cylinder satisfy a backward Harnack inequality and that the quotient of two such functions is Hölder continuous up to the boundary. Another result is that the parabolic measure associated to such equations has the doubling property. In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölder for each 0<α<1, is hyperconvex. Global estimates of the exhaustion function are given. In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graph of a continuous function, all plurisubharmonic functions with continuous boundary values can be uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoods of the closure of the domain. Paper V studies  Poletsky’s notion of plurisubharmonicity on compact sets. It is shown that a function is plurisubharmonic on a given compact set if, and only if, it can be pointwise approximated by a decreasing sequence of smooth plurisubharmonic functions defined in neighbourhoods of the set. Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such a domain, both the Dirichlet problem with respect to functions plurisubharmonic on the closure of the domain, and the problem of approximation by smooth plurisubharmoinc functions in neighbourhoods of the closure of the domain have satisfactory answers in terms of plurisubharmonicity on the boundary.