Probably
one of the most underrated and exciting developments in mathematics currently is the expansion of geometric algebra. One of the most amazing realizations in Math is the fact that most fields are actually connected. There's a branch of math that seeks these analogies: Representation theory and its generalization Category theory. They require an advanced stage of mathematical education...but
What if you could do Vector calculus mentally thanks to the addition of new primitives beyond points and lines? A math closer to perception.
"Clifford Algebra, a.k.a. Geometric Algebra, is a most extraordinary synergistic confluence of a diverse range of specialized mathematical fields, each with its own methods and formalisms, all of which find a single unified formalism under Clifford Algebra. It is a unifying language for mathematics, and a revealing language for physics."
Physicists just love it.
An introduction: https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
Where to study from: Lectures on Exceptional Lie Groups - J. F. Adams
Books: Alan Macdonald's two books Geometric algebra; Geometric calculus
A great introductory book is Geometric Algebra for Computer Science:
http://www.geometricalgebra.net/index.html
This book includes a lot of the geometry side of Geometric Algebra.
For a more axiomatic treatment, see Doran & Lasenby's "Geometric Algebra for Physicists":
http://www.mrao.cam.ac.uk/~cjld1/pages/book.htm
Note that the applications in this book require a great deal of physics background, and aren't for the faint of heart.
I'm continually impressed how many different aspects of mathematics can be brought under the umbrella of GA. For example, all of Stokes theorem, Green's theorem, Cauchy integral equations, and Divergence theorems are all specific cases/applications of the Fundamental theorem of geometric calculus. There are a number of other algebraic structures that have GA representations. Examples include: Cramer's rule, differential forms, complex numbers, quaternions, and Pauli and Dirac algebras.
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