Related papers: Super approximation for $\text{SL}_2\times \text{S…

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple…

We derive a modular anomaly equation satisfied by the prepotential of the N=2* supersymmetric theories with non-simply laced gauge algebras, including the classical B and C infinite series and the exceptional F4 and G2 cases. This equation…

We propose a superspace formulation for conformal $(p,q)$ supergravity in two dimensions as a gauge theory of the superconformal group $\mathsf{OSp}_0 (p|2; {\mathbb R} ) \times \mathsf{OSp}_0 (q|2; {\mathbb R} )$ with a flat connection.…

A consequence of Rapinchuk et al. is that for $S$ a closed surface of genus $g\geq 2$, the set of Hitchin representations of $\pi_1(S)$ with image in $\mathrm{SL}(n,\mathbb{Q})$ is dense in the Hitchin component. We give a dynamical proof…

We construct an explicit family of arithmetic Teichm\"uller curves $\mathcal{C}_{2k}$, $k\in\mathbb{N}$, supporting $\textrm{SL}(2,\mathbb{R})$-invariant probabilities $\mu_{2k}$ such that the associated…

The present paper studies the homology of the groups $SL_2(k[C])$ and $GL_2(k[C])$ where $C=\overline{C}\setminus\{P_1,\dots,P_s\}$ is a smooth affine curve over an algebraically closed field $k$. It is well-known that these groups act on a…

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry…

The goal of this paper is to demonstrate the use of techniques from hyperbolic geometry to compute generating sets of certain subgroups of $SL^+(2,\mathbb{C})$; specifically, $SO^+(Q,\mathbb{Z})$ for $Q$ some integral quadratic form of…

The Seiberg-Witten curves and differentials for $\N=2$ supersymmetric Yang-Mills theories with one hypermultiplet of mass $m$ in the adjoint representation of the gauge algebra $\G$, are constructed for arbitrary classical or exceptional…

We review recent work on the study of N=2 super Yang-Mills theory with gauge group SU(N) from the point of view of the Whitham hierarchy, mainly focusing on three main results: (i) We develop a new recursive method to compute the whole…

We give an example of an infinite family of finite groups $G_n$ such that each $G_n$ can be generated by 2 elements and the diameter of every Cayley graph of $G_n$ is $O(\log (| G_{n}|))$. This answers a question of Lubotzky.

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is…

For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long…

Let $S$ be a set of transpositions that generates the symmetric group $S_n$, where $n \ge 3$. The transposition graph $T(S)$ is defined to be the graph with vertex set $\{1,\ldots,n\}$ and with vertices $i$ and $j$ being adjacent in $T(S)$…

Multi-parametric families of AdS$_{4}$ vacua with various amounts of supersymmetry and residual gauge symmetry are found in the $\,[\textrm{SO}(1,1) \times \textrm{SO}(6)] \ltimes \mathbb{R}^{12}\,$ maximal supergravity that arises from the…

Let $X$ be a smooth, geometrically connected curve over a perfect field $k$. Given a connected, reductive group $G$, we prove that central extensions of $G$ by the sheaf $\mathbf K_2$ on the big Zariski site of $X$, studied by J.-L.…

We construct 4D $\mathcal{N}=2$ theories on an infinite family of 4D toric manifolds with the topology of connected sums of $S^2 \times S^2$. These theories are constructed through the dimensional reduction along a non-trivial $U(1)$-fiber…

Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. We continue our investigation from arXiv:2505.10683 [math.RT] into when the resulting skew-group algebra $R \ast G$ is a…

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

In this paper, we construct, for some $2$-groups $G$, explicit Galois extensions $E/\mathbb{Q}(T)$ of group $G$ with $E\cap\overline{\mathbb{Q}}=\mathbb{Q}$. We also provide explicit arithmetic progressions of integers $t_0$ such that the…