Posts tagged algebraic-geometry
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JMRA Video

06/09/20

I recently submitted a video to the Junior Mathematician Research Archive that gives a brief overview of the work in my PhD thesis. You can watch the video here but, be warned, it’s really probably not the most coherent (ha ha) narrative.

For the sake of it, or in case you’re the sort of person who prefers to read instead of watch, I’ve included the transcript of the whole video below.

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Thesis online

03/07/20

Having finished my defence (entirely virtually, and hence, of course, plagued with many technical difficulties), I have now been able to share my thesis with the world. Most of the technical content can already be found in my two preprints (part I and part II), but I think the thesis is a much more self-contained and leisurely read, with a lot more examples and (hopefully) helpful appendices. It also has geese! You can find a copy of it on here on TEL.

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Some videos I recorded

02/06/20

Edit. I’m in the process of moving over from YouTube to Vimeo (no ads, less tracking, and no Google), so you can find my videos there now instead. In particular, the connections and curvature videos can be found here.

This is a very short post just to say that I’ve started uploading some maths-related videos to my YouTube page. At the moment there’s a talk I recently gave at a graduate seminar about connections and curvature from the point of view of somebody trying to avoid differential geometry, and a series I’ve been working on called Nice Analytic Sheaves For All which aims to discuss motivations for coherence conditions of complex-analytic sheaves from various points of view. Hopefully there will be more updates to the latter soon!

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Confusion about connections

29/05/20

I am a big fan of Stephen Bruce Sontz’s book Principal bundles: The Classical Case, and cannot recommend it enough, as somebody who usually finds differential geometry (a) dull, and (b) incomprehensible. Anyway, there’s a lovely quote from this book about how confusing the terminology surrounding connections can be, so let’s try to clear some of that up today.

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Simplicial Chern-Weil theory

29/03/20

This week just gone I uploaded two preprints to the arXiv:

  1. Simplicial Chern-Weil theory for coherent analytic sheaves, part I;
  2. Simplicial Chern-Weil theory for coherent analytic sheaves, part II.

Both have been extracted from my PhD thesis (which I’ve just sent off to the referees) and contain about 90% of the main mathematical content of my thesis, but with about 90% fewer inane footnotes and digressions. There are also a few appendices in my thesis which explain the background of some of the subjects in a bit more detail, which I think are quite nice, but I’ll probably turn the good ones into blog posts at some point in the coming months.

So what are these two papers about? And why is it split into two parts?

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The determinant bundle

08/12/19

As always, when I haven’t written anything on here in a while, I’m going to write about something entirely irrelevant to anything I’ve written about before (instead of continuing any of the many series of posts I’ve started or planned to start). Today we’re going to look at a ‘fun’ little calculation, which might be the most algebraic-geometry-like calculation I’ve done in a long time: we’re going to compute the determinant of the cotangent sheaf of the complex projective sphere. Note that even the statement of this problem makes me uneasy: I am nowhere near as comfortable with classical algebraic geometry as I should be!

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