Us with I-III

Us without

There’s no better time than

When you’re awake to rest

And permit expectations

Of tomorrow to slide,

Watch-hammer like, across

The land until it catches

On the horizon line.

A division lasts longer

Every day as the earth

Conquers more but inherits

Less; rockpools fill with dust

As pillars return to sand

From contexts that pinned them

Skyward. To walk through it

Is a peaceful havoc,

A resurgence of green

In which footpaths occupy

The role of lean highways

Rendered trustworthy by

The shallowness of their

Camber; they’re now regarded

In the right light, or just

Through the wrong sight before.

 

Us withheld

These questions inhabit

The air cloistered over

Paths intended for feet

But tread by them instead,

A gallery of earth conjoined to

A mausoleum for gratitude,

Spoken sotto voce

And blending into the

Clay haze of the foreground.

The moorland is quilted

By bridges suspended

Because of rivets replaced

By rivulets; useless

Opinions wandering

Overused land cross them

Like so many shadows

Straining the girders in

The shade, still forms astride

Still water, but red fruit

Bobbing components of

A green canal lying

Above the riverbank.

 

Us withdrawn

There are steps the river

Flows down that, despite its

Many cycles, remain

Unanticipated,

Unclued within the path’s

Prequel, but a fixture

Of its rotation in

Any case.

Either the

Steps or the river leads

Down towards settlement,

Are persuaded into

Dividing their forces

To skirt streets and dabble

In a more radical

Interface.

Time proves that

By night it’s a different

Empire: low-lying clouds,

That suppress the light by

Day, colonise the night

With an urban orange,

Convert constellations

Into suburbs laced with

Halogen; ungated

But framed as if a memory

Favoured by the earth.

 

It shares no nostalgia

For the dust, nor sunny

Rocks widely bearing

Attributes intrinsic

To their position and

Yet matching the posture

Of the soil.

One is led

To mistake walkways

For solitude by the

Soliloquy of each

Wayward adjective that

Passes, though their epitaphs

Evoke a nativity

Of expression

That begins: us withdrawn,

You and I to live.

A quiet name

The clouds mass in indecision, their grey

Partly silhouettes the features of the land

Offset by the half-dry, half-wet haziness

Found there. Every so often there’s

A jet of blue, lounging lighthouse-like

In the space its travels have earned.

 

The crops identify the soil, ascribe a

Quiet name that must be sought to

Be heard, and hence nudge the

Listener towards remembrance.

 

Other spaces are tilled by footfall

And have waived their right to an

Unobstructed view of the sky.

 

(I usually try not to comment on my own writings, but some I think that some context might be constructive (and fun!) here. I mostly wrote this on a train journey from Edinburgh to London when visiting the UK in April and it lay around in a folder for a while – which I accidentally left in Kyoto after a conference – until I eventually got around to writing it up and editing it slightly. I originally intended to extend it to something more coherent but, after rereading it a few times, I quite like it as it is.)

Some comments on colouring

A friend of mine posted a neat colouring problem to stack.exchange this week that implicitly contained the fact that any graph embedded on a sphere can be 4-coloured, which follows from the fact that a graph is planar iff it can be embedded on a sphere. This follows immediately from sensible stereographic projection, but it raises the more interesting question of how colourings behave for graphs embedded on more general surfaces. Perhaps there’s my interest in dimer models ticking away subconsciously too…

It turns out that there is a beautifully uniform treatment of the chromatic number for even not necessarily orientable surfaces due to Ringel and Youngs in the 1960s, completing work of Heawood in the 1890s. Sadly, Heawood believed that he had proved the strong version that the other two authors later corrected using more sophisticated techniques. This is also an instance of a single exceptionally tricky case of a problem resisting proof, since Ringel and Youngs prove Heawood’s conjecture for all surfaces apart from the sphere (or the plane), which was the subject of the four colour theorem proved a decade later.

I say ‘an instance’ as this is quite analogous to the Poincaré conjecture, where it was proved that a [topological] n-manifold homotopic to a n-sphere is actually homeomorphic to the sphere when n\geq 5 by Smale also in the 1960s. There’s an amusing and slightly sorrowful anecdote I was told during my time at Warwick about the late Sir Christopher Zeeman who proved the Poincaré conjecture in dimension 5 shortly before Smale obliterated the problem in all dimensions except 3 and 4. Freedman proved the conjecture in dimension 4 in 1982, before Perelman’s proof for dimension 3 in the early 2000s.

This area also exhibits pleasing applications of logic to everyday mathematical problems: one can extend colouring theorems such as these from finite graphs to infinite graphs immediately by appealing to Gödel’s compactness theorem. Here are my notes on the proof of Heawood’s conjecture for nonspheres, which should be moderately accessible to nonmathematicians who care more about my poetry!