Modern research into Grassman's "theory of forms"?

Solution 1:

Universal Algebra: Broadly speaking, Whitehead in his Treatise on Universal Algebra was motivated by the work of Boole, Hamilton and Grassmann, and eventually Maltsev and Polish mathematicians developed the general theory. However, Grassmann himself was mostly focused on binary associative operations.

Hypercomplex numbers: From abstract algebra point of view Grassmann studied some finitely generated associative algebras over the field of real numbers. In 19th century this area came to be known as linear associative algebra, and was further developed by Peirce and Wedderburn among others. The modern name is hypercomplex numbers and associativity is no longer required, examples are dual and split-complex numbers, quaternions, octonions, etc. Although purely algebraic approach is more in the spirit of Hamilton than Grassmann some hypercomplex numbers are closely related to non-Euclidean geometries and appear in mechanics, relativity and quantum theory. New interesting hypercomplex numbers were introduced by Musès in 1980s, although most of his work is more in the realm of numerology than mathematics.

Multilinear Algebra: Grassmann's own work was more specific than universal algebra and more intuitive than abstract algebra, in modern terms he was interested in algebras generated by a vector space, like exterior or Clifford algebras, with emphasis on geometric interpretations. And it's not just the scope but also the viewpoint, he was interested in geometric content of the operations (wedge product, Clifford product, etc.) more than their algebraic properties. Formally, the closest modern analog would be multilinear algebra if not for the prevailing tensor approach that reduces everything to index manipulations and assigns geometric interpretations after the fact if at all.

Geometric Algebra: Gian-Carlo Rota is the most prominent recent proponent of Grassmann's original point of view, he and his students introduced new geometric operations in his spirit, and developed their theory. Rota's motivation came from the classical invariant theory and he injected some combinatorial flavor into Grassmann's constructions. "Geometric algebra" is a term occasionally used to describe this circle of ideas, but most authors treat it as a synonym of Clifford algebra, which is just one example albeit universal in some sense (not to be confused with geometric algebra of ancient greeks, which is something else). And Rota's approach is an exception rather than a rule in multilinear algebra.

Differential geometry: Where the spirit of Grassman is preserved more it's differential geometry. After Elie Cartan's introduction of differential forms there was a tendency to abolish tensor algebra's "debauchery of indices" in favor of coordinate invariant notation because there is no global choice of coordinates on manifolds. Differential forms are Grassman's exterior forms that vary over a manifold, i.e. one generates exterior algebras over tangent spaces and takes smooth cross sections across the manifold. Of course, other Grassman type algebras can and are used instead. The advantage is that Grassman's definitions of operations are intrinsic, and automatically extend to bundles once some underlying structure is fixed (e.g. Riemannian metric for Clifford algebras). They can then be used to define differential operators that are explicitly invariant under transformations that preserve the underlying structure (e.g. defining codifferential using Hodge star). Physicists working with gauge theories on manifolds sometimes favor this approach, but even in differential geometry tensors dominate because they can be manipulated with much less geometric insight.

Sources: Yaglom’s book has an excellent historical exposition of Grassman’s ideas and their relation to Hamilton’s quaternions and hypercomplex numbers in general. This is a more recent historical survey of the subsequent work of Clifford, Lipschitz, Study and Cartan with emphasis on applications to physics. Wikipidea article on hypercomplex numbers gives a good overview with many historical and modern references, for Musèan hypernumbers see here. Rota’s programmatic paper reviving Grassman’s ideas and applying them to the classical invariant theory is quite readable, its short and elementary review is here. Applications to spinors and representation theory can be found in this book. Warner’s text systematically applies invariant approach in differential geometry.

Solution 2:

I think this passage is referring to the exterior algebra of a vector space. It's designed to have properties suitable for Grassmann's approach, and indeed can be viewed as a universal arrow in a certain category, but it does not really have anything to do with universal algebra, per se.

Grassmann's work later was applied in multilinear algebra and seeded the study of differential forms. Other than this, Grassmann's ideas were also built upon by Clifford. This eventually lead to the study of Clifford algebras and so-called "geometric algebras," which are special cases of Clifford algebras. This direction has a slightly different flavor than that of differential forms, but that's not to say they don't overlap and work together.

A great thing to read to get a better feel for the role of differential forms and Grassmann's and Clifford's legacies is Roger Penrose's Road to reality chapters 6 and 11.