Inlämningsomgång 3

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Lämnas in (via Canvas) senast onsdagen den 14 maj, 2025. Skriv namn och personnummer på din text.

Målet med denna uppgift är att ni ska bekanta er lite med den aktuella matematiklitteraturen. Matematik är ett levande ämne och det bedrivs intensiv forskning inom väldigt många inriktningar. Det är viktigt för lärare att förmedla detta till sina elever, så att inte dessa ska tro att matematikämnet är ett ”dött” ämne som redan är färdigutvecklat. 

Välj ett av förslagen nedan och skriv en rapport. En fullständig rapport bör innehålla en introduktion där problemformuleringen och bakgrund beskrivs, följt av en huvuddel där den teoretiska bakgrunden och redogörelsen av arbetet finns. Skriv sedan en kort diskussion där egna reflektioner och tankar kring projektet ingår. Avsluta med en referenslista. Var noggrann med att besvara alla frågor som finns med i projektbeskrivningen du valt. 

Omfång: 3-6 sidor (exklusive referenser och bilder).

Diskussion med andra är tillåtet, men när sådana förekommit skall så anges genom en referens.

1) Learn about The Ultimate Challenge: The 3x + 1 Problem Links to an external site. Links to an external site.This is a very old open problem which is very easy to state. Still - no definitive solution! Report on what the question is, how you understand it, try to find a visual way to explain (to fellow students) what the problem is, what were some advances and give some directions from which the problem has been attacked in the past.

2) We are used to thinking that there are 1 dimensional, 2 dimensional, 3 dimensional objects, and in our course we work in 1,2, or 3 (or higher) dimensional space. But there are sets which have a dimension which is somewhere between 1 and 2, or 2 and 3. In this reading you will learn what type os sets these are, how to construct them, how they appear in nature, and how to distinguish their size by using the concept of "box dimension". Read about fractals. Links to an external site.Report should include: at least 5 examples of fractals, at least 5 examples of fractal looking structure in nature, your understanding of what fractals are, a definition of how we measure the size of fractals (the box dimension) Links to an external site., and include a computation of a box dimension of some  Cantor set Links to an external site.of your choice.

3) Journey among numbers and a lesson in persistence:  Read about the twin prime conjecture and the fascinating story Links to an external site. of how it developed (follow the links in the text to read about the conjecture and the developments). Report should include: what the conjecture is, what were (during the history of the problem) some important stages in resolving the problem, what is your impression on why it is such a difficult question.  

4) Read about Groups, addition, sets, sumsets..a success story Links to an external site.. Desribe the conjecture discussed in the text, follow the links to read and report on its history, how do you see the difficulties which arouse, and what were the successful attacks on the problem, how would you visually present the gist of the problem to a fellow student?

5) Read how A first year graduate student finds a paradoxical number set Links to an external site.Follow the links in the article and report on: what is the paradox about? What was the approach the student took? What is the history of the question? How would you visually describe the problem? 

6) Read about billiards   Links to an external site.Follow the links in the text and report on: what is a mathematical billiard? Links to an external site. How we describe the motion of the billiard ball mathematically? Give at least four examples of billiard tables which demonstrate how different the bouncing will seem if we change the shape of the table, what are some questions which are still puzzling when it comes to mathematical billiards, what are some real-world models of billiard-like motion (think about gas molecules Links to an external site. in a box) 

 

 

 

 

 

 

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