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A question for the more informed: "What are recent developments in linear algebra?" Perhaps, we can broaden the scope and include numerical/randomized linear

A question for the more informed: "What are recent developments in linear algebra?" Perhaps, we can broaden the scope and include numerical/randomized linear algebra. I come from a machine learning background, and the most recent directions I am aware of are in applications of Krylov subspace methods to very large number of data points and low-precision training (both in the context of Gaussian processes).

WoahNoun 5y ago
Depends on how you define "linear algebra." Representation theory, lie groups, and grassmannians can all be considered "linear algebra" and are active areas of math research, but it probably isn't what you are looking for.
- activatedgeek 5y ago
  Fair enough. I don't quite care about obscure and beautiful theoretical problems. I am looking for new developments that have shown promise to make big practical impact. For what it's worth, I am no linear algebra definition police so feel free to list what you think is reasonably relevant.
Syzygies 5y ago
I teach linear algebra in a pure math department. We like to think that freeing people to be able to daydream about linear algebra is our goal, that there's more to it than what function calls to make in MatLab. Still, you could throw out our course and spend a semester teaching people how to really think about singular value decomposition. It's everywhere, critical for example in machine learning, yet pure math types think it's "applied". Riemannian geometry begins with an inner product at each point (though what's amazing about O'Neill's text is that he slips by you a categorical definition first at every step). As another route up the mountain, one could now understand parallel transport as a local version of the singular value decomposition, and work out the whole theory that way. This requires a fluid, intuitive grasp of singular values, rather than the textbook definition. We should teach that understanding.
dellamonica 5y ago
Linear Programming methods (barrier/simplex) have been able to run faster and solve larger instances in great part due to algorithmic advances (the other contribution being faster hardware, more memory etc). Considering that LP is a fundamental tool for mixed/integer programming, these improvements are incredibly valuable. Gurobi had a white paper mentioning how much their solvers' performance was improved by which techniques (I'll link it here if I can find it).
bmc7505 5y ago
> "What are recent developments in linear algebra?" Have you heard of geometric algebra? It takes a while to wrap your head around, but unifies a lot of mathematics in a very elegant way. For instance, you can view Euclidean geometry as a special case, and concepts like quaternions and rotations can be explained in a very natural way. There are a few good resources below if you're interested. [0]: http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-and-geometric-algebra/ http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-an... [1]: https://www.youtube.com/watch?v=ipqKvNHnABs https://www.youtube.com/watch?v=ipqKvNHnABs [2]: https://alexkritchevsky.com/2020/10/15/ea-operations.html https://alexkritchevsky.com/2020/10/15/ea-operations.html [3]: https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ... [4]: https://crypto.stanford.edu/~blynn/haskell/ga.html https://crypto.stanford.edu/~blynn/haskell/ga.html [5]: https://bivector.net/ https://bivector.net/

WoahNoun 5y ago

Depends on how you define "linear algebra." Representation theory, lie groups, and grassmannians can all be considered "linear algebra" and are active areas of math research, but it probably isn't what you are looking for.

activatedgeek 5y ago
Fair enough. I don't quite care about obscure and beautiful theoretical problems. I am looking for new developments that have shown promise to make big practical impact. For what it's worth, I am no linear algebra definition police so feel free to list what you think is reasonably relevant.

Syzygies 5y ago

I teach linear algebra in a pure math department. We like to think that freeing people to be able to daydream about linear algebra is our goal, that there's more to it than what function calls to make in MatLab. Still, you could throw out our course and spend a semester teaching people how to really think about singular value decomposition. It's everywhere, critical for example in machine learning, yet pure math types think it's "applied". Riemannian geometry begins with an inner product at each point (though what's amazing about O'Neill's text is that he slips by you a categorical definition first at every step). As another route up the mountain, one could now understand parallel transport as a local version of the singular value decomposition, and work out the whole theory that way. This requires a fluid, intuitive grasp of singular values, rather than the textbook definition. We should teach that understanding.

dellamonica 5y ago

Linear Programming methods (barrier/simplex) have been able to run faster and solve larger instances in great part due to algorithmic advances (the other contribution being faster hardware, more memory etc). Considering that LP is a fundamental tool for mixed/integer programming, these improvements are incredibly valuable. Gurobi had a white paper mentioning how much their solvers' performance was improved by which techniques (I'll link it here if I can find it).

bmc7505 5y ago

> "What are recent developments in linear algebra?" Have you heard of geometric algebra? It takes a while to wrap your head around, but unifies a lot of mathematics in a very elegant way. For instance, you can view Euclidean geometry as a special case, and concepts like quaternions and rotations can be explained in a very natural way. There are a few good resources below if you're interested. [0]: http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-and-geometric-algebra/ http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-an... [1]: https://www.youtube.com/watch?v=ipqKvNHnABs https://www.youtube.com/watch?v=ipqKvNHnABs [2]: https://alexkritchevsky.com/2020/10/15/ea-operations.html https://alexkritchevsky.com/2020/10/15/ea-operations.html [3]: https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ7U8KFvtiJY4K https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ... [4]: https://crypto.stanford.edu/~blynn/haskell/ga.html https://crypto.stanford.edu/~blynn/haskell/ga.html [5]: https://bivector.net/ https://bivector.net/