Tangent cones and C1 regularity of definable sets

Let $X?{\mathbb{R}}^n$ be a connected locally closed set which is definable in an o-minimal structure. We prove that the following three statements are equivalent: (i) X is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of X coincide at every point in X, (iii) for every $x\in X$, the tangent cone of X at the point x is a k-dimensional linear subspace of ${\mathbb{R}}^n$ (k does not depend on x) varies continuously in x, and the density $\theta (X,x)<3/2$.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

Classification of some 3-subgroups of the finite groups of Lie type E6

We consider the finite exceptional group of Lie type $G=E_6^{\epsilon }(q)$ (universal version) with $3|q?\epsilon$, where $E_6^{+1}(q)=E_6(q)$ and $E_6^{?1}(q){=}^2{E}_6(q)$. We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local $M<G$ which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing $Z(G)$, all extraspecial subgroups containing $Z(G)$, and all cyclic groups of order 9 containing $Z(G)$. These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.     

Partial cohomology of groups and extensions of semilattices of abelian groups

We extend the notion of a partial cohomology group $H^n(G,A)$ to the case of non-unital A and find interpretations of $H^1(G,A)$ and $H^2(G,A)$ in the theory of extensions of semilattices of abelian groups by groups.     

On the stability of sparse convolutions

We give a stability result for sparse convolutions on ${?}^2(G)\times {?}^1(G)$ for torsion-free discrete Abelian groups G such as $\mathbb{Z}$. It turns out, that the torsion-free property prevents full cancellation in the convolution of sparse sequences and hence allows to establish stability, that is, injectivity with an universal lower norm bound, which only depends on the support cardinalities of the sequences. This can be seen as a reverse statement of the Young inequality for sparse convolutions. Our result hinges on a compression argument in additive set theory.     

An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by $X^{(n)}$ the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on $H^1(X^{(n)})$ to define an integrable connection on the trivial vector bundle on $X^{(n)}$ with fiber the universal algebra of the Lie algebra associated with the descending central series of ${\pi }_1$ of $X^{(n)}$. The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.     

Extension des scalaires par le morphisme de Frobenius,pour les groupes réductifs

Let G be a reductive group over a field k of characteristic $p>0$. For $n?0$ and $q:=p^n$, let $G^{\{n\}}$ be deduced from G by the extension of scalars $x?x^q:k?k$. If k is perfect, this keeps making sense for $n?\mathbb{Z}$. We show that, if k is perfect, there exists $m>0$ such that the algebraic groups G and $G^{\{m\}}$ over k are isomorphic. The isomorphism class of $G^{\{n\}}$, as a reductive group over k, then depends only on n modulo m. For k not necessarily perfect, we show that such a periodicity remains true for n large enough.     

Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of a graph $G$, respectively. The matrix $D(G)+A(G)$ is called the signless Laplacian matrix of $G$. The spectrum of the matrix $D(G)+A(G)$ is called the Q-spectrum of $G$. A graph is said to be determined by its Q-spectrum if there is no other non-isomorphic graph with the same Q-spectrum. In this paper, we prove that all starlike trees whose maximum degree exceed $4$ are determined by their Q-spectra.