### Homological coherence of one-relator groups

Seems to be an interesting time right now to be working on coherent groups (previous blog posts: link and link). Recall that coherent groups are those whose finitely generated subgroups are finitely presented, and that coherence of one-relator groups is one of the main open problems. It is known that one-relator groups with torsion elements … Continue reading "Homological coherence of one-relator groups"

### Virtually free-by-cyclic groups

Last month I blogged about coherent groups (i.e. groups whose finitely generated subgroups are finitely presented). There I also referred to an article of Daniel Wise about coherent groups that contains many open problems at the end. One of these problems is whether every one-relator group is coherent (a question posed by Baumslag in the … Continue reading "Virtually free-by-cyclic groups"

### Differentials in the Adams spectral sequence

In this post we consider the Adams spectral sequence which computes the stable homotopy groups of spheres at the prime 2. The classes $$\{h_j\}_{j\ge 0}$$ on the 1-line of the second page are called Hopf classes since they are related to the Hopf invariant: Adams proved that Hopf invariant one classes can only exist in … Continue reading "Differentials in the Adams spectral sequence"

### Coherent groups

Recall that a group G is called coherent if every finitely generated subgroup of G is finitely presented. Main examples are 3-manifold groups and (virtually) polycyclic groups. Instead of trying to motivate the study of coherent groups in my own words, let me instead just refer to Section 2 of the article `An Invitation to … Continue reading "Coherent groups"

### Regularity of minimizing hypersurfaces

Let us consider the following classical problem from geometry (the case $$n=3$$ is basically Plateau’s problem): Let $$\Gamma$$ be a smooth, closed, oriented, $$(n−1$$)-dimensional submanifold of $$\mathbb{R}^{n+1}$$. If we consider all the smooth, compact, oriented hypersurfaces $$M \subset \mathbb{R}^{n+1}$$ with $$\partial M = \Gamma$$, does there exist one with least area among them? In the … Continue reading "Regularity of minimizing hypersurfaces"

### Illustrating the Impact of the Mathematical Sciences

A series of posters and some other related media were produced by the National Academy of Sciences of the USA to showcase mathematics of the twenty-first century and its applications in the real world: link. If you still don’t know what to put on your office walls, have a look at those posters!

### Double Soul Conjecture

Recall the Soul Theorem of Cheeger and Gromoll: If $$(M,g)$$ is a complete and connected Riemannian manifold of non-negative sectional curvature, then there exists a closed, totally convex and totally geodesic embedded submanifold whose normal bundle is diffeomorphic to $$M$$. Such a submanifold is called a soul of $$(M,g)$$ and the Riemannian metric induced on … Continue reading "Double Soul Conjecture"

### Progress on the union-closed sets conjecture

The union-closed sets conjecture is the following extremely easy to state conjecture about subsets of finite sets: Assume that $$\mathcal{F}$$ is a family of subsets of $$\{1, 2, \ldots, n\}$$ which is union-closed; this means that for any two sets $$A,B$$ in $$\mathcal{F}$$ their union $$A \cup B$$ is also a member of $$\mathcal{F}$$. Then … Continue reading "Progress on the union-closed sets conjecture"