A Combinatorial Tale of Two Scattering Amplitudes: See Two Bijections

In this thesis, we take a journey through two different but not dissimilar stories with an underlying theme of combinatorics emerging from scattering amplitudes in quantum field theories. The first part tells the tale of the $c_2$-invariant, an arithmetic invariant related to the Feynman integral in $\phi^4$-theory, which studies the zeros of the Kirchoff polynomial and related graph polynomials. Through reformulating the $c_2$-invariant as a purely combinatorial problem, we show how enumerating certain edge bipartitions through fixed-point free involutions can complete a special case of the long sought after $c_2$ completion conjecture. The second part tells the tale of the positive Grassmannian and a combinatorial T-duality map on its cells, as related to scattering amplitudes in planar $\mathcal{N} = 4$ SYM theory. In particular, T-duality is a bridge between triangulations of the hypersimplex and triangulations of the amplituhedron, two objects that appear as images of the positive Grassmannian. We give an algorithm for viewing T-duality as a map on Le diagrams and characterize a nice structure to the Le diagrams (which can then be used in lieu of the algorithm). Through this Le diagram perspective on T-duality, we show how the dimensional relationship between the positroid cells on either side of the map can be directly explained.