Inspired by Feynman integral computations in quantum field theory, Kontsevich conjectured in 1997 that the number of points of graph hypersurfaces over a finite field ${\mathbb{F}}_q$ is a (quasi-) polynomial in $q$. Stembridge verified this for all graphs with at most twelve edges, but in 2003 Belkale and Brosnan showed that the counting functions are of general type for large graphs. In this paper we give a sufficient combinatorial criterion for a graph to have polynomial point-counts and construct some explicit counterexamples to Kontsevich’s conjecture which are in ${\varphi }^4$ theory. Their counting functions are given modulo $p{q}^2$ ($q=p^n$) by a modular form arising from a certain singular K3 surface.

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analogue of the Hirzebruch–Riemann–Roch formula for the Euler characteristic of the $Hom$-space between a pair of matrix factorizations. We also establish $G$-equivariant versions of these results.

We consider the corresponding Christoffel–Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial a priori $C^2$-estimate for the corresponding curvature equation on ${\mathbb{S}}^n$.

By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion-free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously proved that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.

We establish an asymptotic formula with a power savings in the error term for traces of cycle integrals of the classical modular $j$-function

$$j\left(z\right)=q^{-1}+744+196884q+21493760q^2+\cdots .$$

This implies a conjecture of Duke, Imamoḡlu, and Tóth.

We show that if a big set of integer points $S\subseteq [0,N{]}^d$, $d>1$, occupies few residue classes mod $p$ for many primes $p$, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of Helfgott and Venkatesh.