I can say a little about the work of Brown and Schretz, since Brown gave a talk at BIRS last month.

If you take a graph with $N$ edges and some restriction on valence (called a Feynman graph), there is a certain integral on an $N-1$-dimensional domain with boundary that in small cases appeared to yield linear combinations of multiple zeta values. This was a discovery of Broadhurst and Kreimer, by numerical brute force.

This raised the question of which graphs yield combinations of multiple zeta values. This is interesting in part because when multiple zeta values show up in an integral, it is a sign that mixed Tate motives are lurking somewhere. In fact, Brown showed last year that $MT(\mathbb{Z}) = \pi_1^{mot}(\mathbb{P}^1 \setminus \{0,1,\infty \})$, i.e., every mixed Tate motive over $\mathbb{Z}$ has periods in $\mathbb{Q}[(2 i \pi)^{-1}, MZV]$. The connection between graphs and motives comes through work of Bloch, Esnault and Kreimer, where they defined motives of graphs $G$ by making a variety $X_G$ by some blow-up construction and taking $H^{N-1}$. You can ask if these motives are mixed Tate, but this is apparently way too hard. An easier question is computing cohomology, and easier still is counting points over finite fields.

Kontsevich conjectured that the function that takes a prime power $q$ and returns the number of $\mathbb{F}_q$-points in $X_G$ is a polynomial, and it was true for small $G$, but this recent work gives a construction of counterexamples. Some of the functions are (or appear to be) $q$-expansions of modular forms of small level and weight, but that seems to be a transition zone between polynomials and wilder functions that no one has been able to identify.