We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Čech’s closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the constructions of cell complexes by a finite sequence of closed cell attachments, which attach arbitrarily many cells at a time, agree. Likewise, the constructions of CW complexes relative to a compactly generated weak Hausdorff space that attach only finitely many cells, also agree. On the other hand, we give examples showing that the constructions of finite-dimensional CW complexes, CW complexes of finite type, and relative CW complexes that attach only finitely many cells, need not agree.

I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the spatial $L^2$-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into traveling combustion waves.

A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture [15], which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open.

A Steiner coset partition of G, with respect to distinct subgroups $H_1,\dots ,H_r$, is a coset partition of G that contains exactly one coset of each $H_i$. In the quest of a more structural version of the Herzog–Schönheim Conjecture, it was shown that there is no Steiner coset partition of G with respect to any $r\geq 2$ subgroups $H_i$ that mutually commute [1]. In this article, we show that this result holds for $r=4$ mutually commuting subgroups provided that G does not have $C_2\times C_2\times C_2$ as a quotient, where $C_2$ is the cyclic group of order $2$. We further give an explicit construction of Steiner coset partitions of the n-fold direct product $G^*=C_p\times \ldots \times C_p$ for p prime and $n\geq 3$. This construction lifts to every group extension of $G^*$.

In this article, we address the following question: Which hyperbolic or elliptic PDEs admit functional separable solutions. We shall focus on the study of a sinh-Gordon type equation. We construct solutions to this equation via the method of functional separation. We prove that these are the only families that have the property of functional separation and so we obtain a classification. To this end, we construct new families of solutions for the hyperbolic and elliptic versions of both sine and sinh-Gordon equations in a unified way.

If ${\mathcal {E}}, {\mathcal {F}}$ are vector bundles of ranks $r-1,r$ on a smooth fourfold X and $\mathop {\mathcal Hom}({\mathcal {E}},{\mathcal {F}})$ is globally generated, it is well known that the general map $\phi : {\mathcal {E}} \to {\mathcal {F}}$ is injective and drops rank along a smooth surface. Chang improved on this with a filtered Bertini theorem. We strengthen these results by proving variants in which (a) ${\mathcal {F}}$ is not a vector bundle and (b) $\mathop {\mathcal Hom}({\mathcal {E}},{\mathcal {F}})$ is not globally generated. As an application, we give examples of even linkage classes of surfaces on $\mathbb P^4$ in which all integral surfaces are smoothable, including the linkage classes associated with the Horrocks–Mumford surface.