7 ms·

Whitney Stratifications and Intersection Homology (2023) [pdf]

17 points by aebtebeten 2y ago

esafak 2y ago
In layman's terms, what is this about and why is it interesting?
- 082349872349872 2y ago
  For my part, it's a move in an HN-based glass bead game*, so I was completely surprised that anyone else bothered to upvote. (we normally play well away from the front page) Hassler Whitney I know from matroid theory (in particular, antimatroids for the mathematical psychology of learning), so stratification theory might be interesting because that also was. René Thom I know from catastrophe theory, so stratification theory might be interesting because that also was. Boundaries of boundaries are interesting, as are their arrow-reversed variants, but that's likely 1*1=1 compared with this material? (which I take to have something to do with finding nice boundaries along which to slice wholes up to better study their parts?) For ML in particular, maybe it could provide a reasonable way to divide-and-conquer manifold learning? In fact, as of this comment, although I'm pretty sure the vertices of my play are interesting, I have yet to learn whether its convex hull will also turn out to be interesting — if someone else can provide the ELI5, that'd be great! * https://en.wikipedia.org/wiki/The_Glass_Bead_Game https://en.wikipedia.org/wiki/The_Glass_Bead_Game
  - deleted 2y ago
    [deleted]
  - vinnyvichy 2y ago
    Retro reto retro hugos lolo yolos >In 2019, the novel was nominated for the 1944 Retrospective Hugo Award for Best Novel. Apologies for the rest, but in return for not not not expressing incuriosity* to your questions, may I suggest the use of poisk-fu to uncover the (anholocratic) Bourbaki "leader"? https://news.ycombinator.com/item?id=40595575 https://news.ycombinator.com/item?id=40595575 https://en.wikipedia.org/wiki/Ultrafiltration_(kidney) https://en.wikipedia.org/wiki/Ultrafiltration_(kidney) *second take at the Gr ineq?
    - 082349872349872 2y ago
      Sorry I didn't —to my great chagrin— spot the incarnate filtration! (I come from the cryppie branch, neither hydro nor celmech, so my only excuse is that I normally contemplate topless models and their bottoms, not any of their Manichaean relations) I'll just have to cool me down to take another round before I'm back in the ring (assuming XOR distributes under AND) to take another swing, so please bear with me. Besides, I'm guessing this'll be a job that, rather than requiring poisk-fu, instead needs a fast machine, one that keeps the research motor clean. > All Gaul is divided into three parts: the part that cooks with lard and goose fat, the part that cooks with olive oil, and the part that cooks with butter. —DC
      - 082349872349872 2y ago
        > to uncover the (anholocratic) Bourbaki "leader"? Holomorphic if not holocratic, how about HPC? One might lexically (or even temporally) order https://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image https://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image before https://www.sciencedirect.com/science/article/abs/pii/0003491681901998?via%3Dihub https://www.sciencedirect.com/science/article/abs/pii/000349... ... at the risk of putting the Cartan before the Horwitz. From https://fr.wikipedia.org/wiki/Henri_Cartan#Carrière https://fr.wikipedia.org/wiki/Henri_Cartan#Carrière > Continuant à garder des contacts en Allemagne durant [la guerre], il facilita à la fin de la guerre le rapprochement entre mathématiciens français et allemands, communiquant en particulier avec Heinrich Behnke et Peter Thullen. Lagniappe: https://www.youtube.com/watch?v=p9g4Md_huII https://www.youtube.com/watch?v=p9g4Md_huII [note that Gen. CDSB was able to save over half his command (and 12'000 horses), despite the fact that near where he crossed the border it can get down to -40 (C or F, your pick) in winter; he ought to get more credit for an army half-full than he often does]
        vinnyvichy 2y ago
        Best lagniappe ever (SOMEHOW did not recognize FG in this one)
        2y ago
- vinnyvichy 2y ago
  >This has led to their use in engineering, control theory and robotics. https://en.wikipedia.org/wiki/Whitney_conditions#The_Whitney_conditions_in_Rn https://en.wikipedia.org/wiki/Whitney_conditions#The_Whitney...
  - vinnyvichy 2y ago
    Should we ask ourselves whether we should marvel that this stuff has (not just (,maybe-likely,)) ((non-software, maybe-likely,)) engineering applications X)
g15jv2dp 2y ago
What follows is not a jab at the author of the document. I think it's really nice for an undergrad to be interested in this topic. It's very close to my field of research, and not something that an undergrad would typically know about. But why would you post someone's draft of a bachelor thesis here? If someone is really interested in intersection homology (and perverse sheaves!), here is a set of lecture notes (not written by me): https://webusers.imj-prg.fr/~bruno.klingler/cours/courspervers.pdf https://webusers.imj-prg.fr/~bruno.klingler/cours/coursperve...
- vinnyvichy 2y ago
  Thanks for the link! (I'm afraid there's nothing for me to reciprocate with ATM) You don't have to defend my eccentricity, but I always pay attention to BS theses, even drafts (RIP Grothendieck) because, they almost always* teach me something deep, and more importantly, efficiently. YMMV, but I'd be curious about your counter-defence if so. *sometimes I do disagree with their headlines.
doormatt 2y ago
When there's a spelling mistake on the first page (signular), I have a hard time taking the rest of the paper seriously.
- rexpop 2y ago
  This is mountains/molehills or pennywise poundfoolishness if I've ever seen it. We are not typists—we have other, oblique strengths.
  - doormatt 2y ago
    Did they just ignore the spell-check then? If they can't be bothered to check the spelling, what else did they miss?
    - vinnyvichy 2y ago
      They might have missed that "signular" could be a meaningless typo (but "convinient" maybe not so much)
    - 082349872349872 2y ago
      > If they can't be bothered to check the spelling, what else did they miss? When needing to split a bill (an exercise in exact simple arithmetic), the tradition at my Uni included an absolute ban on asking a mathematician. Then there is the sort of applied physicist who is satisfied once they have made an even number of sign errors...
      - defrost 2y ago
        > When needing to split a bill (an exercise in exact simple arithmetic) Not so exact or simple when the diners and total (after tips, etc) are co-prime > an absolute ban on asking a mathematician you can see why, annoying theorists focusing on the general case are the last thing needed when Tony Soprano wants the money now. https://www.youtube.com/watch?v=96XJl7mxDtc https://www.youtube.com/watch?v=96XJl7mxDtc ?v=AxR2BOXln4w

17 points by aebtebeten 2y ago

esafak 2y ago

In layman's terms, what is this about and why is it interesting?

082349872349872 2y ago
For my part, it's a move in an HN-based glass bead game*, so I was completely surprised that anyone else bothered to upvote. (we normally play well away from the front page) Hassler Whitney I know from matroid theory (in particular, antimatroids for the mathematical psychology of learning), so stratification theory might be interesting because that also was. René Thom I know from catastrophe theory, so stratification theory might be interesting because that also was. Boundaries of boundaries are interesting, as are their arrow-reversed variants, but that's likely 1*1=1 compared with this material? (which I take to have something to do with finding nice boundaries along which to slice wholes up to better study their parts?) For ML in particular, maybe it could provide a reasonable way to divide-and-conquer manifold learning? In fact, as of this comment, although I'm pretty sure the vertices of my play are interesting, I have yet to learn whether its convex hull will also turn out to be interesting — if someone else can provide the ELI5, that'd be great! * https://en.wikipedia.org/wiki/The_Glass_Bead_Game https://en.wikipedia.org/wiki/The_Glass_Bead_Game
- deleted 2y ago
  [deleted]
- vinnyvichy 2y ago
  Retro reto retro hugos lolo yolos >In 2019, the novel was nominated for the 1944 Retrospective Hugo Award for Best Novel. Apologies for the rest, but in return for not not not expressing incuriosity* to your questions, may I suggest the use of poisk-fu to uncover the (anholocratic) Bourbaki "leader"? https://news.ycombinator.com/item?id=40595575 https://news.ycombinator.com/item?id=40595575 https://en.wikipedia.org/wiki/Ultrafiltration_(kidney) https://en.wikipedia.org/wiki/Ultrafiltration_(kidney) *second take at the Gr ineq?
  - 082349872349872 2y ago
    Sorry I didn't —to my great chagrin— spot the incarnate filtration! (I come from the cryppie branch, neither hydro nor celmech, so my only excuse is that I normally contemplate topless models and their bottoms, not any of their Manichaean relations) I'll just have to cool me down to take another round before I'm back in the ring (assuming XOR distributes under AND) to take another swing, so please bear with me. Besides, I'm guessing this'll be a job that, rather than requiring poisk-fu, instead needs a fast machine, one that keeps the research motor clean. > All Gaul is divided into three parts: the part that cooks with lard and goose fat, the part that cooks with olive oil, and the part that cooks with butter. —DC
    - 082349872349872 2y ago
      > to uncover the (anholocratic) Bourbaki "leader"? Holomorphic if not holocratic, how about HPC? One might lexically (or even temporally) order https://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image https://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image before https://www.sciencedirect.com/science/article/abs/pii/0003491681901998?via%3Dihub https://www.sciencedirect.com/science/article/abs/pii/000349... ... at the risk of putting the Cartan before the Horwitz. From https://fr.wikipedia.org/wiki/Henri_Cartan#Carrière https://fr.wikipedia.org/wiki/Henri_Cartan#Carrière > Continuant à garder des contacts en Allemagne durant [la guerre], il facilita à la fin de la guerre le rapprochement entre mathématiciens français et allemands, communiquant en particulier avec Heinrich Behnke et Peter Thullen. Lagniappe: https://www.youtube.com/watch?v=p9g4Md_huII https://www.youtube.com/watch?v=p9g4Md_huII [note that Gen. CDSB was able to save over half his command (and 12'000 horses), despite the fact that near where he crossed the border it can get down to -40 (C or F, your pick) in winter; he ought to get more credit for an army half-full than he often does]
      - vinnyvichy 2y ago
        Best lagniappe ever (SOMEHOW did not recognize FG in this one)
        2y ago
vinnyvichy 2y ago
>This has led to their use in engineering, control theory and robotics. https://en.wikipedia.org/wiki/Whitney_conditions#The_Whitney_conditions_in_Rn https://en.wikipedia.org/wiki/Whitney_conditions#The_Whitney...
- vinnyvichy 2y ago
  Should we ask ourselves whether we should marvel that this stuff has (not just (,maybe-likely,)) ((non-software, maybe-likely,)) engineering applications X)

g15jv2dp 2y ago

What follows is not a jab at the author of the document. I think it's really nice for an undergrad to be interested in this topic. It's very close to my field of research, and not something that an undergrad would typically know about. But why would you post someone's draft of a bachelor thesis here? If someone is really interested in intersection homology (and perverse sheaves!), here is a set of lecture notes (not written by me): https://webusers.imj-prg.fr/~bruno.klingler/cours/courspervers.pdf https://webusers.imj-prg.fr/~bruno.klingler/cours/coursperve...

vinnyvichy 2y ago
Thanks for the link! (I'm afraid there's nothing for me to reciprocate with ATM) You don't have to defend my eccentricity, but I always pay attention to BS theses, even drafts (RIP Grothendieck) because, they almost always* teach me something deep, and more importantly, efficiently. YMMV, but I'd be curious about your counter-defence if so. *sometimes I do disagree with their headlines.

doormatt 2y ago

When there's a spelling mistake on the first page (signular), I have a hard time taking the rest of the paper seriously.

rexpop 2y ago
This is mountains/molehills or pennywise poundfoolishness if I've ever seen it. We are not typists—we have other, oblique strengths.
- doormatt 2y ago
  Did they just ignore the spell-check then? If they can't be bothered to check the spelling, what else did they miss?
  - vinnyvichy 2y ago
    They might have missed that "signular" could be a meaningless typo (but "convinient" maybe not so much)
  - 082349872349872 2y ago
    > If they can't be bothered to check the spelling, what else did they miss? When needing to split a bill (an exercise in exact simple arithmetic), the tradition at my Uni included an absolute ban on asking a mathematician. Then there is the sort of applied physicist who is satisfied once they have made an even number of sign errors...
    - defrost 2y ago
      > When needing to split a bill (an exercise in exact simple arithmetic) Not so exact or simple when the diners and total (after tips, etc) are co-prime > an absolute ban on asking a mathematician you can see why, annoying theorists focusing on the general case are the last thing needed when Tony Soprano wants the money now. https://www.youtube.com/watch?v=96XJl7mxDtc https://www.youtube.com/watch?v=96XJl7mxDtc ?v=AxR2BOXln4w