Introduction to Homological Algebra I: Motivation

The first talk in series on Homological Algebra. The goal of this talk is to introduce some of the primary motivations and concepts of the series.

[infinitedimensions]

Hi Jeremy,
I wanted to say thank you for your series on Homological Algebra. I particularly agree with your statement that the existing literature is hard to understand, so it was great to come across a clear exposition of the subject.
What got me started on homological algebra was (please correct me if I’m wrong here) that it appears to give a bunch of useful tools for simplifying computations in homology and cohomology. Do you have any suggestions for which theorems are of particular importance in homological algebra, from a more applied perspective?

[jakebaer]

Jeremy please more! The world needs these

[malicksoumare]

One minute is just what I needed to like the video.

[feraudyh]

The formal expression of the boundary : final - initial is a WTF moment.

[dennisnh]

Excellent motivational intro; the graph theory example was especially well timed for what I'm currently working on. About one of your last statements: all of the fields you mentioned are characteristic zero.

[mayankvashishtha]

Bro, please complete this series, i loved it

[smoosq]

absolutely amazing lecture, thank you for making these!

[rajathradhakrishnan]

Hi Jeremy. I am really enjoying your videos. Thanks for making them! Could you please let me know the software that you use to make these videos? In particular, how are you recreating a chalk board? Looks cool!

[daigakunobaku]

4:45 >> which can be expressed as the following matrix

Shouldn't this matrix be a vertical 2x1 rather than horizontal? It acts on one-dimensional space and produces two-dimensional vectors.

Thanks a lot for the video!

[zy]

Hey good video. Would be helpful if you put in the description that the motivation you will cover is analysis and physics oriented. It was nice to see there are other uses of Hom Algebra, but some of us may be looking for videos showing the original motivation behind the development of the basic objects of Homological Algebra (exact sequences of modules, Hom, Ext, Tor functors). That later went to have applications to things like Group Cohomology/Number Theory (no continuos functions nor derivatives there)

[bartek_ewertowski]

Hi Jeremy, just wondering if there is a reason why you write the matrix for the boundary map \partial in your example at 4:45 as a 1x2 matrix? Shouldn't it be a 2x1 matrix since it is mapping from a 1-dimensional vector space (the edges) to a 2-dimensional vector space (the vertices).

[lakshaymd]

About your notation for the free vector space over F generated by a set S: Doesn't F^S clash with the notation for the set of all set maps from S to F? For finite S, this is fine, but for infinite S, they don't seem to match up, because \omega is not a Hamel basis for F^\omega.

[maciej]

10:47 why in algebra charakterystyk of structure is 0 not ininfity ? Infinity is not a number. 0 is number?

[mzg]

Why do you call homological algebra "higher linear algebra"? I don't get it. Chain complexes seem like a foreign concept in comparison to linear maps - or at least, weirdly specific objects for them to take such an important place. Can you give some hints?

[JournalKannada]

Is this the voice behind "MIT OPENCOURSEWARE"...
looks similar

[aleksandrkalinin]

Jeremy, if we are talking about linear algebra other rings, then the humanity is far from complete understanding.