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As a one-time mathematician, this was a really fascinating article. The similarities seem to be entirely coincidental, but what would have been my doctoral diss

As a one-time mathematician, this was a really fascinating article. The similarities seem to be entirely coincidental, but what would have been my doctoral dissertation was also about generalizing some concepts from smooth manifolds to a "non-smooth" setting, and the crux of my work also hinged on optimal transport. Actually I feel optimal transport is a pretty underrated concept in both pure and applied math, and I would have loved to explore it had I continued in academia. But oh well, one must make choices in life...

xelxebar 1y ago
Small world. My graduate research was precisely on this topic as well. I was going in a more algebraic direction, though. My master's thesis was essentially about different discrete analogues of curvature using cooked-up cohomological constructions. I really wish academia consistently provided as much security as industry. Would have loved to continue this line of research.
pkoird 1y ago
I can't simply help but think that optimal transport is intricately linked to the principle of least action (and as we know POLA is everywhere in nature). At the end, natural interactions seem to be one big optimization problem.
bmacho 1y ago
Do I understand correctly that with sectional curvature/triangle comparison methods you can do differential geometry on non-smooth manifolds (e.g. on a cube)? If so, I've completely missed this fact before.
- youcandoittoo 1y ago
  Sure, see (2010) A curved Brunn-Minkowski inequality on the discrete hypercube, Or: What is the Ricci curvature of the discrete hypercube? http://www.yann-ollivier.org/rech/publs/cube.pdf http://www.yann-ollivier.org/rech/publs/cube.pdf Or this: (2011) A visual introduction to Riemannian curvatures and some discrete generalizations http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf Taken from the site of Yann Ollivier http://www.yann-ollivier.org/rech/index http://www.yann-ollivier.org/rech/index

xelxebar 1y ago

Small world. My graduate research was precisely on this topic as well. I was going in a more algebraic direction, though. My master's thesis was essentially about different discrete analogues of curvature using cooked-up cohomological constructions. I really wish academia consistently provided as much security as industry. Would have loved to continue this line of research.

pkoird 1y ago

I can't simply help but think that optimal transport is intricately linked to the principle of least action (and as we know POLA is everywhere in nature). At the end, natural interactions seem to be one big optimization problem.

bmacho 1y ago

Do I understand correctly that with sectional curvature/triangle comparison methods you can do differential geometry on non-smooth manifolds (e.g. on a cube)? If so, I've completely missed this fact before.

youcandoittoo 1y ago
Sure, see (2010) A curved Brunn-Minkowski inequality on the discrete hypercube, Or: What is the Ricci curvature of the discrete hypercube? http://www.yann-ollivier.org/rech/publs/cube.pdf http://www.yann-ollivier.org/rech/publs/cube.pdf Or this: (2011) A visual introduction to Riemannian curvatures and some discrete generalizations http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf Taken from the site of Yann Ollivier http://www.yann-ollivier.org/rech/index http://www.yann-ollivier.org/rech/index