homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Differential Galois theory is an analogue of Galois theory where fields are generalized to differential fields, hence is a theory for differential equations rather than just algebraic equations.
Pierre Deligne, Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp.111-195.
Andy Magid, Differential Galois theory, Notices of the AMS (1999) pdf
Teresa Crespo, The origins of differential Galois theory (pdf slides)
Andy R. Magid, Universal covers and category theory in polynomial and differential Galois theory, pdf
Textbook accounts include
Marius van der Put, Michael Singer, Galois theory of linear differential equations, Springer, Berlin (2003)
M. Singer, Direct and inverse problems in differential Galois theory, in H. Bass, et al. (eds.) Selected works of Ellis Kolchin with commentary , Amer. Math. Soc. 1999, 527–554.
Discussion in terms of D-geometry is in
João Pedro P. dos Santos, The behaviour of the differential Galois group on the generic and special fibres: A Tannakian approach, J. reine angew. Math. 637 (2009), 63-98 (pdf)
João Pedro P. dos Santos, Lifting D-modules from positive to zero characteristic (pdf)
based on
A connection to geometric invariant theory is proposed in
Model theoretic aspects are discussed in