In this paper, we are interested in proving that the infinitesimal variations of Hodge structure for hypersurfaces for hypersurfaces of high-enough degree lie in a proper subvariety of the variety of all integral elements of the Griffiths transversality system. That is, this proves that in this case, the geometric infinitesimal variations of Hodge structure satisfy further conditions rather than just being integral elements of the Griffiths system. This is proved using the Jacobian ring representation of the (primitive) cohomology of the hypersurfaces and a space of symmetrizers as defined by Donagi, but here, we use the Jacobian ring representation to identify a geometric structure carried by the variety of all integral elements.

We compute Grothendieck's torsion index of a compact Lie group for all simply connected groups and all groups of adjoint type. In particular, the torsion index of the group $E_8$ is $2^6·3^2·5$.

We compute Grothendieck's torsion index of the compact Lie group $\mathrm{Spin}(n)$ for any $n$. We explain the applications of the torsion index to topology and to the study of splitting fields for quadratic forms.

Let ${\stackrel{̲}{\mathcal{M}}}_{g,l}$ be the moduli space of stable algebraic curves of genus $g$ with $l$ marked points. With the operations that relate the different moduli spaces identifying marked points, the family ${({\stackrel{̲}{\mathcal{M}}}_{g,l})}_{g,l}$ is a modular operad of projective smooth Deligne-Mumford stacks $\stackrel{̲}{\mathcal{M}}$. In this paper, we prove that the modular operad of singular chains $S_{*}(\stackrel{̲}{\mathcal{M}};\mathbb{Q})$ is formal, so it is weakly equivalent to the modular operad of its homology $H_{*}(\stackrel{̲}{\mathcal{M}};\mathbb{Q})$. As a consequence, the up-to-homotopy algebras of these two operads are the same. To obtain this result, we prove a formality theorem for operads analogous to the Deligne-Griffiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field.

There is a well-known conjecture of Serre that any continuous, irreducible representation $\stackrel{̲}{\rho }:G_Q\to {\mathrm{GL}}_2({\stackrel{̲}{F}}_{\ell })$ which is odd arises from a newform. Here $G_Q$ is the absolute Galois group of $Q$, and ${\stackrel{̲}{F}}_{\ell }$ is an algebraic closure of the finite field $F_{\ell }$ of $\ell$ of ℓ elements. We formulate such a conjecture for $n$-dimensional mod ℓ representations of ${\pi }_1(X)$ for any positive integer $n$ and where $X$ is a geometrically irreducible, smooth curve over a finite field $k$ of characteristic $p$ ($p\ne \ell$), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for $\ell >2$ follows from a result announced by Gaitsgory in [G]. The methods are different.

We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of the Robbins conjecture on the degree of generalized Heron polynomials.

We correct a step in the published article ‘ElM’ by including additional cases.