Inquiry: Algebraic Geometry and Topology
Speaking of money and such matters, a call for assistance from Quantivity:
Algebraic geometry and topology traditionally focused on fairly pure math considerations. With the rise of high-dimensional machine learning, these fields are increasing being pulled into interesting computational applications such as manifold learning. Algebraic statistics and information geometry offer potential to help bridge these fields with modern statistics, especially time-series and random matrices.
Early evidence suggests potential for significant intellectual cross-fertilization with finance, both mathematical and computational. Geometrically, richer modeling and analysis of latent geometric structure than available from classic linear algebraic decomposition (e.g. PCA, one of the main workhorses of modern $mathbb{P}$ finance); for example, cumulant component analysis. Topologically, more effective qualitative analysis of data sampled from manifolds or singular algebraic varieties; for example, persistent homology (see CompTop).
As evidence by Twitter followers, numerous Quantivity readers are experts in these fields. Thus, perhaps the best way to explore is to seek insight from readers.
Readers: please use comments to suggest applied literature from these fields; ideally, although not required, that of potential relevance to finance modeling. All types of literature are requested, from intro texts to survey articles to preprint working papers on specific applications.
These suggestions will be synthesized into one or more subsequent posts, along with appropriate additions to People of Quant Research.
If you or a member of your family knows of any relevant resources, please go to: Inquiry: Algebraic Geometry and Topology and volunteer those resources as comments. You might even make the People of Quant Research list!
I wonder if there would be any interest in tracking bundled instruments using topic maps? That could be an interesting question.