Diversity, identity and belonging in e-learning communities: some theories and paradoxes Teaching in Higher Education

It is often assumed that online collaborative learning is inclusive of diversity. In this exploratory paper I challenge this notion by developing a theory which proposes that inclusion occurs through congruence between learners’ social identities and the identities implicitly supported through the interactions in a particular community. To build identity congruence, e-learning communities need spaces for both commonality and diversity and I present three paradoxes which underlie the aims of online learners and teachers to embrace diversity online. I illustrate these with some examples from online learning and teaching. The ability to ‘listen’ to each other online offers a way forward and the paper ends with some future possibilities about how we can ensure that e-learning communities benefit from diversity

Spectra of lens spaces from 1-norm spectra of congruence lattices

To every $n$-dimensional lens space $L$, we associate a congruence lattice $\mathcal L$ in $\mathbb Z^m$, with $n=2m-1$ and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on $L$ with the number of lattice elements of a given $\|\cdot\|_1$-length in $\mathcal L$. As a consequence, we show that two lens spaces are isospectral on functions (resp.\ isospectral on $p$-forms for every $p$) if and only if the associated congruence lattices are $\|\cdot\|_1$-isospectral (resp.\ $\|\cdot\|_1$-isospectral plus a geometric condition). Using this fact, we give, for every dimension $n\ge 5$, infinitely many examples of Riemannian manifolds that are isospectral on every level $p$ and are not strongly isospectral.Comment: Accepted for publication in IMR

Dimension formulas for Siegel modular forms of level 44

We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree $2$ with respect to certain congruence subgroups of level $4$. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A})$ whose local component at $p=2$ admits non-zero fixed vectors under the principal congruence subgroup of level $2$. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at $p=2$, we obtain formulas for the number of relevant automorphic representations. These in turn lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level $4$.Comment: 48 pages. Fixed some typographical errors and improved exposition. Final version which has been published in Mathematik