The humble interval is a gateway to both homotopy theory and higher-dimensional geometry. The interval allows us to define homotopies between continuous functions, while taking products of the interval with itself gives rise to the square, cube, tesseract and beyond. Abstracting away from topological spaces, one may speak of interval objects, cylinder objects or even cylinder functors on other categories, and use these to define homotopies there. However, the connection to higher-dimensional cubes is lost in the process. In this talk, I will show that in categories with algebraic weak factorization systems (AWFS) -- which provide a conducive setting for abstract homotopy theory -- we can recover cylinder functors that share both the homotopical and "cubical" aspects of the interval. More precisely, for any object $X$ in a category $\mathcal{C}$ equipped with coproducts and an AWFS, there is a functor from the category of cubes-with-connections to $\mathcal{C}$ that sends the 0-cube to $X$, the 1-cube to the cylinder on $X$, and so on. As a corollary, any such category is enriched in cubical sets with connections. (Joint work with Chris Kapulkin.)

Darboux's theorem states that all symplectic manifolds locally look alike. Consequently, there are no local invariants in symplectic geometry, and one must look for global invariants to probe symplectic manifolds. Such invariants can be obtained by investigating the homotopy type of mapping spaces related to the symplectic structure. In this talk, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once under the presence of hamiltonian group actions of either $S^1$ or finite cyclic groups.