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I felt this course was a success. The course was a continuation of MA3403 - Algebraic Topology 1 and all of the students who completed the course had attended MA3403 the previous semester. Though the material in this course is significantly more demanding, the students demonstrated excellent facility with both computations and theory in the exams. The students were engaged and asking questions throughout the semester. The format and content were fairly similar to last year's course. Class time was devoted to lectures and students were encouraged to ask questions and/or contribute steps of proofs or computations. Exercises were assigned in class and from the texts, but not collected. There was occasional group work near the beginning of the course. The final assessment was an oral exam. For exam review, we had an in-class activity using Kahoot. Additionally, as in the previous year and in MA3403, the students were given an exercise sheet in advance of the exam. The exams began with a question from this sheet. The course this year was quite small. This made it easier to tailor the course to student preferences in terms of material and format, e.g. students generally preferred to discuss as a class, rather than work individually or in pairs. However, this suggests more work may be needed to encourage students to take the course. Feedback from the students who attended was generally quite positive. The students seemed to enjoy the topics, the range of the material and the balance of computations vs theory, the variety of resources, and the level of difficulty. The biggest obstacles seemed to be the facilities, as there were several technical issues in both classrooms we used. I felt the exercise sheet and the Kahoot review activity went well, and students seemed to appreciate both.

NTNU:TIA:REST

Institutt for matematiske fag

Department of Mathematical Sciences

D: Ingen trykte eller håndskrevne hjelpemidler tillatt. Bestemt, enkel kalkulator tillatt.

Engelsk

Norsk

Anbefalte forkunnskaper

Det er sterkt anbefalt å ha tatt MA3403 Algebraisk topologi 1.

Link til informasjon om emnet

Faglig innhold

Emnet bygger på MA3403 Algebraisk topologi 1. Emnet vil innføre grunnleggende homotopiteori av rom (fibrasjoner og kofibrasjoner, homotopigrupper) samt dekke andre klassiske tema innen algebraisk topologi slik som: spektralfølger (spesielt Serres spektralfølge), vektorbunter og karakteristiske klasser, kohomologioperasjoner.

Kursmateriell

Læringsutbytte

1. Kunnskap. Studenten kan gi definisjoner av de viktige nøkkelbegrepene, formulere og bevise hovedresultatene, samt greie ut om eksempler på tema dekket i emnet. 2. Ferdigheter. Studenten har en forståelse av algebraisk topologi og homotopiteorien av rom. Studenten kan formulere og løse problemer, samt utføre enkle beregninger ved å bruke metodene gjennomgått i emnet.

Læringsformer og aktiviteter

Læringsformer og aktiviter kan avhenge av foreleser. Emnet kan gis på engelsk.

Algebraisk topologi 2

Topologi

Topologi og geometri

Matematikk

Matematiske fag - doktorgrad (ph.d.)

Mathematical Sciences (Master's Programme)

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Overall, this course went quite well. The course was a continuation of the MA3403 - Algebraic Topology 1 course in Autumn 2022, focusing on higher homotopy groups and spectral sequences. The majority of the students in the course had taken MA3043 the previous semester. The exams were excellent, and students demonstrated very impressive facility with challenging material, both computational and theoretical. A number of students were engaged and asking questions and/or contributing throughout. At several points, the course content was adapted to student questions and interests. As in the previous semester, the format of the class was largely lecture based. Students were encouraged to ask questions and occasionally contribute to computations or proofs during lectures. Exercises were assigned throughout, but not collected. There was very occasional group work in class, mostly early on in the semester. The final assessment was an oral exam. As in MA3403, students were given an exercise sheet in advance of the exam, and told the first question on the exam would come from the exercise sheet. The exam exercise sheet was successful this term as well. It seemed to help students focus their preparation, facilitate cooperative learning, and clarify the expectations for the exam. In the future, I would like to provide more and consistent opportunities for active learning in class, including occasional group work and mini-presentations to help students prepare for the oral exam. The report from the reference group suggests the students were happy with the course, but would also appreciate more in-class group work. It was a challenge for students that the material jumped around between many texts. This may be somewhat unavoidable given the content, and possibly good training for preparing for research in the field. Nevertheless, I will continue to look for good sources. I will also considering shuffling the order of some of the material to make more computations/exercises accessible earlier in the course.

Ingen trykte eller håndskrevne hjelpemidler tillatt. Bestemt, enkel kalkulator tillatt.

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There were no suggested imrpovements from the previous course, but as this course is more or less a follow on from MA3403 which I was also responsible for, I took on board some of the suggestions from that course:<ul><li>As per a suggestion of the students, the last 45 minutes of the lecture on Friday were dedicated to the weekly exercise sheet (last year I spent 10-15 minutes at the start of the week reviewing the solutions). Although this obviously allowed less time for lecturing, I still had no problem getting through all the material that I wanted to, and I know that the students (at least those who showed up for the lecture class) found it very useful. My impression was that the majority of the students would not otherwise do the lecture sheet. One could consider a more formal addition to the course, of say one additional hour a week for a dedicated exercise class. </li><li>I did set up a discourse server for the class. Sadly this was not used. The feedback was there was enough opportunity in class, in particular during the exercise class, to go over any problems. Moreover, as opposed to MA3403, I provided typed solutions for the exercise sheets during this class (which was one of the complaints about MA3403). </li></ul>

I again feel that the course was a success overall. The performance on the exam was very high, despite me at least attempting to make it harder than last year. Again I feel in the future there should be more questions added that could differentiate the very strong (A) students, from the merely strong (B) students. On the other hand, MA3408 is one of the last topology courses, and most of the students who took it were students who got an A in MA3403, while the lower performing students in that tended to not attend the course. So it may be that it is somewhat self-selecting; the students who are more interested in algebra topology are of course more likely to attend the second course. As for the lectures themselves, they were mostly successful. For the second year in a row, the student struggled a lot with the definition and basic properties of fiber bundles. I'm not sure if it is a reflection on how I teach this material, or if it is just complicated stuff, but in the end I had to dedicate an entire 2-hour lecture to reviewing the previous two weeks we had done before the students felt comfortable. On the other hand, they were able to successfully answer the questions on fiber bundles on the exam.

The main comment from the students was that the lectures strayed too far from the lecture notes, and they would have preferred more guidance. It is an ongoing task of mine to type up all my handwritten lecture notes, so whoever teaches in the future can at least have them as a starting point. I had hoped to finish that after last semester, but it is a long and slow process. I am currently about 1/3 of the way through typing them up, so hopefully, in the future, the notes will be available for use. I think the students will appreciate this!

Mer om vurdering

Dersom studenten også etter utsatt eksamen har sluttkarakteren F/ikke-bestått, må studenten gjenta emnet i sin helhet. Arbeider som teller med i sluttkarakteren må gjentas. For mer informasjon om vurdering, se «Læringsformer og aktiviteter».

1. Kunnskap. Studenten kan gi definisjoner av de viktige nøkkelbegrepene, formulere og bevise hovedresultatene, samt greie ut om eksempler på tema dekket i emnet. 2. Ferdigheter. Studenten har en forståelse av algebraisk topologi og homotopiteorien av rom. Studenten kan formulere og løse problemer, samt utføre enkle beregninger ved å bruke metodene gjennomgått i emnet. 

Læringsformer og aktiviter kan avhenge av foreleser. Emnet kan gis på engelsk. 

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The course was run either as a hybrid or fully digital course. Despite this, we managed to cover all the relevant material in the course, and I consider it as success.

I would consider that the students are now ready to do some (basic) research in algebraic topology and in homotopy theory. They experienced a broad range of topics, from higher homotopy groups, bundle theory, and even some K-theory.

As noted, the course was run in a hybrid form for the majority of the semester. We spent the first part of each Tuesday lecture going over a series of exercises that I had set the previous week. The students commented that they found this helpful.

The exams were oral, and fully online. The students were very strong, with all receiving an A except for a single B student. This is probably due to a combination of <ul><li>the exam being too easy (I had based it on the level of the algebraic topology I students, but many of these dropped out, so the level of student was quite high)</li><li>a high number of master/PhD students in the course, who were already familiar with a lot of the material. </li></ul>

1. Kunnskap. Studenten kan gi definisjoner av de viktige nøkkelbegrepene, formulere og bevise hovedresultatene, samt greie ut om eksempler på tema dekket i emnet.

2. Ferdigheter. Studenten har en forståelse av algebraisk topologi og homotopiteorien av rom. Studenten kan formulere og løse problemer, samt utføre enkle beregninger ved å bruke metodene gjennomgått i emnet.

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This course was taught for the first time. The learning was normal until the middle of March, when the campus was closed because of a virus. Given the changed circumstances of teaching, I believe that the course was as successful as I wished for.

As the students have suggested in their report, it might be better to rename this course to make it better visible that it is not a direct continuation of Algebraic Topology 1. The choice of topics itself should not change much, though. It was enough for a term and only things that we want every student to know.

The class was taught as a lecture course until the campus was closed due to a virus. Then classes were prepared as a mixture of lecture notes and video recordings. Finally, toward the end of the term, the video recordings were replaced by live meetings online (using Zoom). It it only fair to say that the in-person-meetings were much better for the students learning than anything where I was not present in person.

The exams were oral and online (using Zoom). All five (5) students passed and were graded B each.

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Algebraisk topologi 2 - MA3408