Separability within alternating groups and randomness

This thesis promotes known residual properties of free groups, surface groups, right angled Coxeter groups and right angled Artin groups to the situation where the quotient is only allowed to be an alternating group. The proofs follow two related threads of ideas. The first thread leads to `alternating' analogues of extended residual finiteness in surface groups \cite{scott1978subgroups}, right angled Artin groups and right angled Coxeter groups \cite{haglund2008finite}. Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index convex-cocompact subgroup $H$ of $W$ and any finite subset $\{ \gamma_1, \ldots , \gamma_n \} \subset W \setminus H$ there is a surjective homomorphism $f$ from $W$ to a finite alternating group such that $f (\gamma_i) \notin f (H)$ . A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense. Similarly, finitely generated subgroups of closed, orientable, hyperbolic surface groups can be separated from finitely many elements in an alternating quotient, answering positively a conjecture of Wilton \cite{wilton2012alternating}. The second thread uses probabilistic methods to provide `alternating' analogues of subgroup conjugacy separability and subgroup into-conjugacy separability in free groups \cite{bogopolski2010subgroup}. Suppose $H_1, \ldots H_k$ are infinite index, finitely generated subgroups of a non-abelian free group $F$. Then there exists a surjective homomorphism $f:F \longrightarrow A_m$ such that if $H_i$ is not conjugate into $H_j$, then $f(H_i)$ is not conjugate into $f(H_j)$.EPSRC International Doctoral Scholar schem

One or Nothing: Anti-unification over the Simply-Typed Lambda Calculus

Generalization techniques have many applications, such as template construction, argument generalization, and indexing. Modern interactive provers can exploit advancement in generalization methods over expressive-type theories to further develop proof generalization techniques and other transformations. So far, investigations concerned with anti-unification (AU) over lambda terms and similar type theories have focused on developing algorithms for well-studied variants. These variants forbid the nesting of generalization variables, restrict the structure of their arguments, and are unitary. Extending these methods to more expressive variants is important to applications. We consider the case of nested generalization variables and show that the AU problem is nullary (using capture-avoiding substitutions), even when the arguments to free variables are severely restricted.Comment: 12 pages, submitted to Journal (a revised version of the previous submission with more precise proofs

Hunt for new phenomena using large jet multiplicities and missing transverse momentum with ATLAS in 4.7 fb−1 of s√=7TeV proton-proton collisions

Results are presented of a search for new particles decaying to large numbers of jets in association with missing transverse momentum, using 4.7 fb−1 of pp collision data at s√=7TeV collected by the ATLAS experiment at the Large Hadron Collider in 2011. The event selection requires missing transverse momentum, no isolated electrons or muons, and from ≥6 to ≥9 jets. No evidence is found for physics beyond the Standard Model. The results are interpreted in the context of a MSUGRA/CMSSM supersymmetric model, where, for large universal scalar mass m 0, gluino masses smaller than 840 GeV are excluded at the 95% confidence level, extending previously published limits. Within a simplified model containing only a gluino octet and a neutralino, gluino masses smaller than 870 GeV are similarly excluded for neutralino masses below 100 GeV