I’d like to tell you when something is nothing, where “something” refers to the bigraded homotopy sheaves of the motivic sphere spectrum, “nothing” means that this sheaf is 0, and “when” means a specific range of bigradings.  These results are recorded formally in a recent preprint I coauthored with Oliver Röndigs and Paul Arne Østvær:

The paper breaks down into two parts.  First, we determine a vanishing range in the bigraded stable stems of the η-complete motivic sphere spectrum.  We then use a sequence of fracture squares and Bachmann’s theorem identifying ρ-periodic motivic spectra with sheaves of spectra on the real étale site to “uncomplete” our vanishing region.

In order to frame this theorem, let’s recall a few well-known characteristics of motivic stable stems.  I’ll use the “m+nα” grading in which m tells you the number of simplicial circles and n is the number of geometric circles, A10, and I’ll write ∏(m+nα) for the (m+nα)-th homotopy sheaf of the sphere spectrum (primarily because I don’t want to fiddle with too much fancy formatting in WordPress).  Morel’s connectivity theorem tells us that ∏(m+nα) = 0 for m < 0, so we have an entire half plane vanishing region already.  Morel has also computed ∏(0+nα) in terms of Milnor-Witt K-theory, which is nonzero for every n, so the Z-graded sheaf ∏(m+∗α) is not bounded in general!

Inspired by Morel’s theorem, let’s call ∏(m+∗α) the m-th Milnor-Witt stem (of the motivic sphere).  Work of Andrews and Miller [pdf] on the η-periodic motivic sphere tells us that the m-th Milnor-Witt stem over over a characteristic 0 field is in fact not bounded whenever m is nonnegative and congruent to 0 or 3 mod 4.  Perhaps the Milnor-Witt stems are generically unbounded?

This turns out to not be the case, and we prove it in our paper.  The η-complete variant is especially clean:

Theorem. The (m+nα)-th homotopy sheaf of the η-complete sphere spectrum is 0 whenever m > 0, is congruent to 1 or 2 mod 4, and 2n > max{3m+5, 4m}.

The proof of this theorem is a “simple” application of the slice spectral sequence, at least after the pioneering work of Röndigs-Spitzweck-Østvær.  This spectral sequence converges to the η-complete motivic stable stems, and its E1-page is given by applying a shift of the motivic Eilenberg-MacLane functor to the E2-page of the MU-Adams spectral sequence (i.e., the Novikov spectral sequence).  By Andrews-Miller, this E2-page only consists of η-towers above a piecewise-linear curve.  These towers of F2‘s transform into towers of motivic HF2‘s in the slice spectral sequence. The bigraded homotopy sheaf of HF2 is a polynomial algebra in a single variable τ over mod 2 Milnor K-theory.  It turns out that there is a d1 differential in the slice spectral sequence linking these towers by τ-multiplication.  When you pass to the E2-page, you’re just left with a bunch of mod 2 Milnor K-theory groups above the piecewise-linear curve when m is nonnegative and  congruent to 0 or 3 mod 4; when m is positive and congruent to 1 or 2 mod 4 and you’re above the curve, there are only 0’s:  that’s the vanishing range!

When the base field is nonreal (so -1 is a sum of squares), you actually get vanishing for the integral motivic stable stems in the same range (up to a cohomological dimension condition in the positive characteristic case).  This follows from Levine’s theorem identifying the motivic sphere and η-complete motivic sphere when the base field has finite cohomological dimension.

When the base field is formally real (so -1 is not a sum of squares), the integral vanishing range becomes complicated by topological information:

Theorem. If the base field is formally real, then ∏(m+nα) = 0 if the corresponding η-complete motivic stable stem is 0 and the m-th topological stable stem contains no odd torsion.

We deduce this by analyzing both the η-primary fracture square and the η-inverted 2-primary fracture square.  In the latter square, the (2,η)-periodic motivic sphere spectrum appears.  But, by the relation (2+ρη)η = 0 in the 0-th Milnor-Witt stem, this object is the same as the (2,ρ)-periodic motivic sphere spectrum.  By Bachmann’s theorem, the (2,ρ)-periodic stable motivic homotopy category is equivalent to the homotopy category of 2-periodic sheaves of spectra on the real étale site of the base field.  This category is in turn equivalent to 2-periodic sheaves of spectra on the Harrison space X of orderings of the base field.  Passing to a real closure of the base field turns X into a point, in which case we are working with the 2-periodic Spanier-Whitehead category, and this is how the surprising topological condition in the theorem appears.

There’s plenty left to study in this area.  For instance:

1. Our bounds are not optimal (and are likely far from optimal).  I doubt the slice spectral sequence is the right tool for producing optimal bounds, but you could make progress that way.
2. The first two “top” motivic stable stems in bounded Milnor-Witt stems correspond to topological stable stems:  ∏(1+2α) = Z/24 = π3 and ∏(2+4α) = Z/2 = π6.  That’s a bit thin to make a conjecture, but perhaps it’s the start of a pattern?
3. When a Milnor-Witt stem is bounded above, its structure is constrained by Morel’s contraction construction.  The contraction ωG of a sheaf is what you get when you take the kernel of the natural map 1 : Spec(F) → A1 – 0.  This corresponds to taking (A1 – 0)-loops, so on motivic stable stems we have the isomorphism ω∏(m+nα) = ∏(m+(n+1)α).  Tom Bachmann pointed out to us that ωG = 0 is equivalent to G being birational. It follows that in bounded Milnor-Witt stems, the top motivic stable stem is birational.  Can we say something structural about the lower motivic stable stems?  And can we leverage this structure in calculations?

Maybe you have further questions (or answers!).  If so, let me know.  For further technical details, remember to check out the paper.

If one is resting at A, he explains, and desires to rest in a distant place B, one can only do so by resting for infinitely brief intervals in innumerable intermediate places.  Thus there is no difference essentially between what happens when one is resting at A before the start of the ‘journey’ and what happens when one is ‘en route’, i.e., resting in one or other of the intermediate places.  (Flann O’Brien, The Third Policeman)

4:45pm, the day before:  David Ayala and I pore over an IMTUF 100 course map and elevation profile, debating target times and starting pace.  I advocate for nine-minute miles, David makes the case for ten.  We are near the north end of Payette National Forest, leaning against a crude counter while an ancient barn looms over us.  The barn and counter belong to Burgdorf Hot Springs.  A woman from Silverton, Colorado walks by and introduces us to her three-month-old pig, Simba, who just made a 14-hour car ride.

Course profile.  106 miles, 22,000 vertical feet.

Lodging at Burgdorf Springs.  Photo: Mike Ormsby.

6:00am, mile 0:  Sunrise is an hour away and it’s 28ºF at Burgdorf.  I am shoulder-to-shoulder with 122 other racers.  Jer Humphrey lifts an elk bugle and signals the start with the sound of 30 toddlers screaming through a length of flexible tubing.   Everyone jogs and chats and headlamp beams bob across the dirt road.

With David at the starting line.  Photo:  Sarah Biber Ormsby

Mile 0.  Photo:  Sarah Biber Ormsby

7:40am, mile 11:  The lead pack cruises in to Loon Lake.  David and I are at the front.  A photographer snaps photos and I try to look casual and relaxed.  I am casual and relaxed.  Why am I trying to look that way as well?  Marshaling an unexpected amount of effort, I pry the cap off a green chisel tip Sharpie marker and scrawl a small X between the 9 and 7 on the race bib pinned to my shorts, signifying that I have not skipped the half mile excursion to the lake shore.  I share encouragement and smiles with the dozen other racers I see on my way back to the Loon Creek Trail loop.

Arriving at Loon Lake.  Photo: Howie Stern.

10:25am, mile 27:  David, Jesse Langner, and I crest Diamond Ridge, the first notable climb on the course.  The air is thin and the sky is slightly overcast.  The wind whistles through the trees.  Distant crowds cheering on our run?  In pure Ayalaian fashion, David plummets down the descent trail, propelled by gravity and a childhood of competitive skiing and Zion slickrock hopping.  My own descending chops have improved over the past year, but they’re no match.

11:20am, mile 33:  I cruise into Upper Payette, handing off my race vest to my crew: wife Sarah, mom Cindy, and dad Mike.  They inform me that David is five minutes ahead.  Gels and water are added to the pack.  I wipe my face with a washcloth.  Sarah applies spray-on sunscreen to any exposed skin.  I get a summary of the upcoming chunk of course:  a short flat followed by a gradual 1,200-footer and quick downhill to Snowslide Trailhead, about 15 miles total.

11:40am, mile 35:  I see David ahead on the trail and announce my presence with a quick turkey call.  (I learned how to do this from Sarah, and got a lot of practice with the flock in Berkeley.)  We run together.

1:10pm, mile 43:  The ecstasy of pear chunks, perfectly ripe, in a tupperware on a plastic picnic table decorated with rubber duckies.

Nearing Duck Lake.

David and Ana ascend Snowslide.

2:40pm, mile 50:  I crest Snowslide Summit with Bow Angemi, my pacer, after 47 minutes of hard power hiking.  It’s been raining and the rocks are slick.  We followed David and his pacer/girlfriend, Ana, up the steep, 2,000′ climb, but David has now bombed off the backside of the mountain, careening through primitive switchbacks in an avalanche chute.  I focus my thoughts on efficient, smart running, but feel my energy lag as I approach Lake Fork.  David accumulates a 15-minute lead on the descent and I speculate that it will only grow.

Ascending Fall Creek to Crestline.  Photo: Rick Valentine.

6:00pm, mile 68:  I’ve been running for 12 hours and am 18 miles past the furthest I’ve ever run before, trotting down the backside of Fall Creek Summit, the most difficult and penultimate major climb in the race.  David is an unknown distance ahead of me, but I am in second, slightly euphoric and slightly clumsy, reflecting on the impossibility of time as I skip down the Crestline Trail in Payette National Forest.  Rick Valentine, my pacer, trots behind me, intermittently stopping to snap photos and text them to my nervous crew.

6:10pm, mile 69:  Rick is somewhere, peeing or taking pictures.  I’m navigating an ATV track through rolling hills, high above McCall and nearly everything else in the area.  The rain has knocked the dust down and this section is reasonably runnable.  I’m gamely jogging, but my stride is far from energetic.  A runner comes up on my shoulder.  “How’s it going, Rick?”  Silence.  “Is that Rick?”  “This is Elliott.”  “Oh….  Hi, Elliott.  How’s it going?”  “Pretty well.”  “These are beautiful trails.”  “They are.”  “Good luck out there.”  “Good luck.”  I am in third place.

Crestline at dusk.  Photo: Rick Valentine.

8:15pm, mile 74:  An insanitary condition of the atmosphere due to accretions of black air descends on Crestline Trail.  Rick and I force our way to the Box Creek aid station before rummaging for headlamps in race vests.  At the aid station, we find six goats: four large white goats, and two brown kids who, with the assistance of several humans, have established this makeshift backcountry camp.  There is a fire and there is hot soup and there is water.  Rick snaps my photo with one of the goats, and we head back out to the wet trail just as Sam Ritchie enters Box Creek with his pacer.

Receiving some love from one of Box Creek’s cheer goats, Buzz.  Photo: Rick Valentine.

9:15pm, mile 78:  Sam double poles past me with authority on the dark, misty climb to North Crestline.  “Where are you from?”  “Boulder.”  “I’ve heard of it.  Do they have mountains there?”  “They do.”

11:45pm, mile 89:  I shuffle into Upper Payette for the second time, a hypothermic, despondent zombie who has just had treefuls of cold water poured under his rain jacket by the foliage choking the 1.5 mile sheep path known as Terrible Terrence.  I have finished the crux Crestline section of the course and only have 15 miles to the finish: a 2,500′ climb up Bear Pete Ridge to finally break 8,000′ feet and then a screaming descent back to Burgdorf.

My crew welcomes me and I demand warmth.  I am settled into a chair and fed hot broth and coffee and begin the slow process of changing into dry clothes.  I am ready to quit.  Why would I continue?  I have nothing to prove.  I am cold and exhausted and have run further and harder than I ever have before and finish lines are arbitrary human constructs that signify nothing and I should quit.  Despite the absolute necessity of me quitting, I am given ibuprofen and electrolyte pills and am told that I will finish.  My crew swarms around my peripheral, full of care and anxiety.  “Keep going.”  “You can do it.”  Bow stands ready to shepherd me over Bear Pete.  “You’ll turn the corner.”  “You didn’t say you could quit because you’re cold and tired.”  The race director appears, eyes shining, and encourages me onwards.  “Has he had caffeine?  Someone find him a caffeine pill.”

Finally, a bargain is offered:  Just continue to Cloochman Saddle.  The crew will be there.  It’s only five miles away and I can quit there if I still want to.  Grudgingly, I agree.  Sarah delivers a 200mg caffeine pill as I pull on my warmest mittens and trot out of camp with Bow, the trail briefly marked by glowing Chinese lanterns.  The 25-minute rest has reinvigorated my legs and my core temperature is back up.  I’m not setting any speed records, but I’m on my way running again.

Bargaining at Upper Payette.  Photo: Rick Valentine.

2:50am, mile 99:  I am swimming through pea soup at 8,000′.  I should have seen the aid station a mile ago, and it’s impossible to make out the topography of this place for all the fog.  Bow and I keep picking out pink streamers marking the course, but I begin to wonder if we are retracing our steps.  Maybe we’ve made a loop on Bear Pete ridge and we’re going to repeat it all night long.

3:10am, mile 100:  The Bear Pete aid station.  An offer of whiskey from strangers in sweatshirts by a roaring fire.  The HAM radio operator is chipper and caffeinated, alternately wearing a reflective vest and a huge blanket.  A few moments of warmth and then off, to the finish.

4:10am, mile 105:  The jarring descent trail gives way to gravel road and a woman in a white pickup truck parked at the trailhead hollers encouragement.  “One more mile to go!”  I make out headlamps in the dark.  First my dad, and then Sarah and Rick join me and Bow.  This may be the longest Sarah and I have run together since she started her third trimester.  There is chatter and people seem excited.  My face is a mask and my eyes must be a half inch deeper in their sockets.

We cross the finish line at 4:20am to the applause of my mom, race directors Jer and Brandi, and David and Ana.  There is also the enthusiastic wriggling of my dog, Aesop.  I collapse into Sarah’s arms, heave-sobbing, and then we hobble — could this body ever run? — to the Burgdorf office.  A runner who has dropped is curled asleep on a sofa by the wood stove, and the lead HAM operator sits at a table recording splits from across the course.

Belt buckle AND bottle opener.

6:30am:  A bowl of minestrone soup sits on the kitchen table of our rented cabin in McCall, Idaho.  I consider the impossibility of excavating its Martian surface with my spoon.  The hard boiled eggs to the side of the bowl are too foreign to contemplate, but I manage to consume one in four fearful bites.  My forebrain flickers with awareness of my 11,000 Calorie deficit and then sputters into reptile acquiescence.  I shuffle to bed and join my wife and her enormous belly.

Acknowledgments:  Sarah for her patience, mom and dad for their support, Bow and Rick for their audacity, Joelle and Trail Factor for their encouragement, David for conspiring, Jer and Brandi for their care, all the volunteers for all the million things, Ben Nelson for his craft, and de Selby for the title of this post.

Other drama:  David won the race in 19:52, breaking Seth Swanson’s 2012 course record by 1 hour 14 minutes.  Sam passed Elliott in the final two miles.  Both of them were under the old course record and they finished within four minutes of each other.  The top four women were under the old course record as well, led by Darla Askew in a time of 26:42.  Of the 123 starters, 78 finished the race under the 36-hour cutoff; Julie Seydel received a finishing time of 36:01:19.

Supplementary material:  Slightly erratic GPS track (watch was off for 3.5 miles headed out of South Crestline; satellites lost in the fog on Bear Pete), race website, results.

Oregon is renowned for many things: the soaring Cascade mountains whose glaciers feed our streams and rivers; lush forests hosting woodland creatures and mycological delights; the Columbia River Gorge and its impeccable waterfalls; the bookstores and bike lanes of Portland; the picturesque Pacific coastline, jagged with rock formations and pocketed by tide pools.

And Oregon is not bereft of culinary delights: its urban hubs host a staggering number of food carts per capita; there are 58 breweries in the city of Portland, 234 in the state; Willamette Valley pinot noir commands the respect (and checkbook) of many wine connoisseurs.

But there is one thing Oregon does not have: burritos.  Walk through any PDX neighborhood and it’s more likely that you’ll find a crème brûlée donut equipped with an eyedropper of triple sec (intended for injection immediately prior to ingestion) than a quality burrito served with passable guacamole.

Oregonians, search no more.  After an intense journey of personal discovery, I have found the best – perhaps the only – way to create and consume the perfect burrito in the Beaver State.  The recipe is more elaborate than most, but well worth the time and effort, and if you survive its creation you will doubtless return for second and third helpings.

Ingredients

Four months of trail ultrarunning training, the taste of defeat from the Peterson Ridge Rumble 40-miler, excitement at the prospect of a rematch, soaring temperatures and full sun forecast in Terrebonne, perfect race organization for the Smith Rock Ascent 50k by GoBeyond Racing, supportive friends and loved ones on the course, a huge chunk of the amazing Portland trail running family participating in the race, a finish area catered by Longboard Louie’s with their famous build-your-own burritos.

Method

1. On Friday evening, drive to the Smith Rock bivy and park illegally on the grass knowing you’ll be able to sneak into a legal spot before they give out tickets when some climbers undertake an alpine start the next morning.  Set up camp, have a beer, and pray that you sleep well, but don’t worry too much if your fiancée’s sleeping pad deflates unexpectedly and a search and rescue operation keeps you awake from 11pm-1am.  (Be thankful that said S&R activity is successful.)
2. Wake up at 5am full of anticipation.  Do not confuse your anticipation with fear.  It’s definitely not fear, right?  Fire up the camp stove to make a moka pot of coffee and a big helping of oatmeal.  Pack the car, get changed, fill your handheld water bottle, and head to the starting line.
3. Engage in a number of stimulating conversations while waiting in the porta-potty line.  Collect your bib number and attach it to your singlet.  Wonder if your outfit clashes too much but decide that it looks awesome.  Do some light dynamic stretching and try not to look anxious.  Stare in awe at the main Smith Rock formation.
4. On cue, take off down the southern embankment of the Crooked River, over the footbridge, and then west along the northern bank trail.  Settle into a quick-but-comfortable rhythm just off of Jacob Phillips’s left shoulder.  Strike up a conversation with Rick Stilson who is just behind you.  Wonder who that guy in the green shirt who is not quite keeping pace is, writing him off as someone who will get caught in no-man’s land and struggle through the rest of the race in solitude.
5. Observe as the first climb strings out the clique of frontrunners.  Assume second position but notice that Rick is catching up as you near the summit and a commanding view of South Sister and the rest of the Oregon Cascade range.
6. Wonder at Jacob’s audacity as he blitzes the first aid station and opens up a not-insignificant gap as you refill your water bottle.
7. Downhill.  Sweet, merciful, quad-pounding downhill.  Revel in your gravity-powered fluid speed.  Let your feet dance across the tops of rocks.  Feel utterly confident that you are relaxed and this blistering pace (5:51 for the 11th mile!) will not affect your climbing ability later.  Reel in Jacob and take the lead, but notice that Jacob sticks to your shoulder, not letting you slip alone into the central Oregon desert.
8. Hit aid station 2 at the bottom of Cole Loop Trail.  Down a salt pill and a glass of water.  Accidentally drink coke out of the glass Jacob started sipping from.  Refill your water bottle and watch Jacob speed out and up a hill as your screw the lid on.  Think “patience.”  Decide not to chase.  Decide to let the hill dictate your pace until your climbing legs come back.  Mentally curse the horses that have trampled the trail into loose sand that gives away with each step.
9. Notice that Rick is gaining on you.  Exchange pleasantries as he passes you on the uphill, and shortly thereafter feel comforted that he too is hiking the wickedly steep section around mile 15.5.  He is not far ahead when you crest the hill and you quickly re-pass him.
10. Another long downhill.  Gravity courses through your legs again and you start to feel better.  You’re in second, and Jacob is slowly but surely coming back.  And the smell!  It smells incredible.  The warm sun is heating the scrub brush and juniper bushes and the air is redolent with their odor.  It’s like you’re running through the steam wafting off a cup of rooibos tea.  Between the gravity and the smell your senses are overloaded by how wonderful this place is.
11. But then a distraction.  What’s that noise over your shoulder?  Is Rick making a comeback?  He’s found his downhill legs is making a move?  No!  It’s not Rick at all.  It’s green shirt from the first mile!  (Aka Rob Russell.)  Where did he come from?!  He’s moving fast and your legs have no response as he charges past on your left.  You try to defuse the situation with a joke:  “Uh-oh.  Here comes the guy with the race plan!” you holler.  This sounds stupid as you say it, but Rob is nice enough to chuckle.  This is, apparently, a race.
12. Round a corner and hallucinate the vision of aid station 3, stocked with water, gels, electrolyte pills, and other goodies.  Hallucinate that Jacob is still at the aid station, slowly gathering his things.  Hallucinate that your fiancée, Sarah, has driven your Mazda3 over treacherous high-clearance roads to greet you at this aid station and offer encouragement and cantaloupe.  (You would never accept this hallucinatory cantaloupe as there is no crewing at this race and you don’t want to be disqualified.)  Realize that this is not a hallucination.  This is real and you are now tied for second.  Declare that you are tired, this is hard, and you would like to take a nap as you down another salt pill, collect two gels, refill your bottle, down a glass of coke and a glass of water, kiss Sarah (who pretends to not mind the sweat), and take off down the trail.
13. Jacob follows and you are running together now, chatting a bit with him on your shoulder.  Suddenly, an eruption of sound from the left of the trail.  A dark, hulking mass shifts its weight.  A dinosaur.  No, Sasquatch!  Neither.  Cattle.  A herd of loud, disgruntled cattle.
14. Doubt creeps in as you ask yourself a simple question:  Where are the orange cones and streamers?  There have been so many thus far, constantly reminding you that you are on course.  You look down at your watch and note that you are 22.4 miles into the race.  You ask Jacob if he’s seen any course markings.  He hasn’t for a while.  If you don’t see a marker by 22.9, you’ll start to worry.  23.  23.1.  Where are the markers?!  Did you miss a turn off?  If you turn around now you’ll either get caught by Rick and know (more or less) that you’re on course, or you’ll find the turn off, or you’ll bumble by the turn off again and end up back at the aid station asking idiotic questions.  Faith.  Hold on.  23.3.  A marker!  An orange streamer on a pine tree that looks absolutely beautiful.
15. You follow a few more cones and burst into Skull Hollow campground.  A green-shirted figure flits through a gate and up the trail.  Up the trail.  Uphill.  Again.  You try to find your rhythm.  Mercifully, this is an easier climb.  Maybe 5 or 7% grade on groomed trails that you should charge up.  But the best you can do is hold yourself together.  10-minute pace does not feel proud, but it will have to do.  Jacob has disappeared, and quickly.  Rob appears behind you, moving quite well.  As he passes, he tells you not to worry, you’ll get him on the downhill.  “We’ll see,” you say.
16. When you crest the hill you gain a ridge that you’ll follow to aid station 4 (née 1).  South Sister greets you again and you have an unobstructed view of the course ahead.  Rob must have two minutes on you, while Rick appears to be gaining steadily, eating into the hefty buffer Rob has built.  This looks like a good show!  Maybe they’ll have popcorn at the final aid station and you can watch this duel from the best seat in the house!
17. Your cadence is back up as you reach the final aid station.  One of the volunteers knows exactly what you need (a gel, electrolytes, more water) and scurries to help out.  You are five miles from the finish and a glance at your watch reveals that a sub-4 hour time is still in reach.  That’s the goal now.  You are tired and your head is swimming but if you hold it together on the Burma Road descent and put in some solid miles, you can break four hours, your tentative time goal for the race.  And is that Rick catching Rob up ahead?  What a finish they’re going to have!
18. But then all of a sudden Rick is coming back, and coming back fast.  Just a mile from the finish, as you hop from Burma Road to Wolf Trail on one of the few techy parts of the course, Rick stops suddenly and howls in pain.  You flow by, momentum too much to divert at this point.  You holler back to ask if he is all right and he grimaces a “Yeah” in return.  You are worried he has twisted an ankle, but Rick is in fact struggling with severe calf cramps.
19. You are in the trees along the river and Rob’s green shirt is not in sight.  A flat mile remains ahead followed by the short, brutal 200 foot climb to the finish line.  The Smith Rock formation looms to your right, a perfect backdrop for photos.  Speak of the devil!  You pass the photographer, Paul Nelson, who claims you’ll catch the leader.  This does not compute as a meaningful statement.  What, possibly, could that mean?  Will you look perplexed in your race photos?
20. Crossing the footbridge you spot Rob’s green shirt amongst the 15-mile racers who are finishing on the same stretch.  While you can get closer to him, it never feels like you’re in the hunt for first.  On the final climb you pass your colleagues Jim and Jamie who are dabbling in trail racing for the first time, running gamely towards the finish of an hours-long battle with Gray Butte.  You crest the hill, turn right, and feel a wave of satisfaction, followed by a wave of heat exhaustion, spill over you as you cross the finish line.  Rob congratulates your strong finish; you were only 21 seconds behind him.  Sarah grabs your finisher’s pint glass and asks what you need.  Ice.  You need ice.  And water.  And a nap.
21. Cheer as Rick crosses the finish line just 31 seconds behind you, making the top-three spread a mere 52 seconds – a photo finish in the ultrarunning world.
22. Bumble around the finish area in a stupor, cheering for Jamie and then Jim as they cross the finish line.  Congratulate your colleague Alison who was first woman in the 15-miler.  Congratulate Sarah because she was second woman in the 4-mile race.  Cheer for Jacob as he finishes within eight minutes of the winning time.  Cheer for Joelle Vaught as she crushes the women’s course record, spending a scant 4:16:05 on the trails.
23. Work up your courage and approach the burrito table.  Timorously lift a flour tortilla, scoops of rice and beans, a handful of spinach, guac, and a healthy portion of Pacific salmon onto your plate.  Consume over the next hour in between sips of Lucky Lab Superdog IPA while trading stories with your compatriots and soaking in the post-race vibe.

And that’s it!  You’ve just created and consumed the best burrito in Oregon.  Bon appétit, and may all your culinary adventures be just as expansive, lengthy, and delicious!

Rick, Rob, and me, post-burritos.

The sun sets on this epicurean escapade.

Ricardo Rojas-Echenique has a new paper up on the arXiv comparing G-sets and quadratic forms via Dress’s Burnside-to-Grothendieck-Witt map:

Ricardo was one of my students in the K-group over the summer, and I’m very excited to see this research come to fruition.

Update (May 31, 2016): Ricardo’s paper is now published in JPAA!

The basic idea is as follows:  if L/k is a finite Galois extension of fields of characteristic not 2 with Galois group G, then there is a natural way to take G-orbits to quadratic forms.  One assigns to G/H the so-called trace form of the field extension LH/k.  This trace form takes x in LH to trLH/k(x2).  If you’re familiar with the Burnside ring A(G) and Grothendieck-Witt ring GW(k), it’s a very fun exercise to work out why this assignment induces a ring homomorphism hL/kA(G) → GW(k).

The study of analogies between the Burnside and Grothendieck-Witt rings goes back to Andreas Dress in Appendix B to his 1971 Bielefeld notes.  These notes are amazing, and also shockingly difficult to track down.  My collaborator Jeremiah Heller found them on microfiche(!) at the UIUC library and kindly scanned them to pdf.  If you’d like to take a look yourself, here they are:

For the uninitiated, allow me to at least briefly expound on the construction of these rings and why they’re so awesome.  In the case of the Burnside ring of a finite group G, one starts with the set of finite G-sets (up to equivariant isomorphism) along with the disjoint union and cartesian product operations.  These form a semi-ring, which we can complete to a ring via the Grothendieck construction.  This is the Burnside ring A(G).  As an abelian group, it is freely generated by orbits G/H where H runs through a set of conjugacy classes of subgroups of G.

For the Grothendieck-Witt ring of a field k of characteristic different from 2, you start with quadratic forms (up to isometry) along with direct sum and tensor product operations.  Again, you have a semi-ring, and the Grothendieck construction results in the Grothendieck-Witt ring GW(k).  As an abelian group, it is generated (but certainly not freely generated) by the one-dimensional quadratic forms 〈a〉which take x in k to ax.  (Here a is a unit in k.)

In addition to encoding fascinating representational and arithmetic information about groups and fields, respectively, these rings are also endomorphisms of unit objects in some fashionable stable homotopy categories.  The Burnside ring is isomorphic to the endomorphisms of the G-equivariant sphere spectrum, while the Grothendieck-Witt ring captures the endomorphisms of the motivic sphere spectrum over Spec(k).

A number of authors have studied links between G-equivariant and motivic stable homotopy theory, including myself and Jeremiah Heller in a paper that was recently accepted to the Transactions of the AMS, Galois equivariance and stable motivic homotopy theory, previously written about on this blog here.  We show that there is a functor SHG → SHk from the G-equivariant stable homotopy category to the motivic stable homotopy category over k (again where G = Gal(L/k)).  Moreover, when k is real closed and Lk(i) (so that G is cyclic of order 2), this functor is full and faithful (at least after completion with respect to the Hopf map).  But for all Galois extensions, this functor induces the Dress map hL/k on endomorphisms of the unit objects!  Hence the Dress map is the first obstruction to fullness and faithfulness of the functor.

Ricardo’s project, then, was to study when hL/k is either injective or surjective.  These are then necessary conditions for faithfulness and fullness of the functor SHG → SHk, respectively.  More generally, Ricardo’s work tells us something fascinating about when the representation theory of G and quadratic forms over k are nicely related.  His result completely classifies injectivity and surjectivity of hL/k in the following two theorems.

Theorem. For a finite nontrivial Galois extension L/khL/k is injective if and only if Lk(a1/2) where a ∈ k is not a sum of squares.

Theorem. For a finite Galois extension L/khL/k is surjective if and only if k is quadratically closed in L.

The proofs employ elementary but clever arguments, mixing together the expected ingredients: Galois theory, group theory, and the arithmetic of quadratic forms.  For more details, I’ll send you to the paper!

The week before spring term starts, Reed College hosts one of its most distinguishing events, Paideia.  Faculty, students, and alumni give mini-courses and workshops on topics they know or care about in a community-wide festival of spontaneous learning.  This year’s lineup included Dr. Demento, an introduction to Heidegger hosted by the college president, a chocolate tasting, an orienteering course, a talk on fractional calculus, an introduction to mountain trail running (led by yours truly), and dozens of other seminars and practical explorations.

Leading the intro to MTR gave me the opportunity to synthesize a lot of disperse thoughts on running, training, nutrition, and mountains into a slide presentation.  And then I got to decorate said slides with all of my favorite wilderness and mountain photos.  What a great excuse to reflect on some of my favorite things!

If you’re interested, you can access the slides by clicking here.

[Update as of 4.V.16: Riley’s work is now published in Journal of Algebra!]

[Update as of 9.VIII.16: Jeremiah Heller and I now have a paper on the arXiv detailing our results on Balmer’s comparison map from the tensor triangular spectrum of the stable motivic homotopy category to the homogeneous Zariski spectrum of Milnor-Witt $K$-theory.  I have updated the text below to reflect our current understanding of the subject.]

In this post I want to tell you about the wonderful world of tensor triangular geometry and what my student Riley Thornton’s work might tell us about stable motivic homotopy theory. For full details, check out Riley’s paper,

Suppose you have a triangulated category $\mathcal{C}$ with a compatible symmetric monoidal structure $\otimes$.  For instance, perhaps you’re a stable homotopy theorist studying topological spectra under smash product.  Or perhaps you’re a noncommutative geometer studying $C^*$-algebras via $KK$-theory. Or maybe you’re an algebraic geometer studying perfect complexes. Or a representation theorist studying stable $G$-modules. It’s a big umbrella.

One fruitful way to study your tensor triangular category is via its prime ideals: thick full subcategories $\mathcal{P}$ such that $c\otimes p\in\mathcal{P}$ whenever $p\in\mathcal{P}$ (so, ideals with respect to $\otimes$) satisfying a primality condition: $a\otimes b\in\mathcal{P}$ implies $a$ or $b\in \mathcal{P}$.

This is the approach of Paul Balmer‘s school, and it goes under the heading tensor triangular geometry. Without getting into the details, one builds a Zariski spectrum-esque topological space $\mathrm{Spc}(\mathcal{C})$ which, as a set, consists of the tensor triangular primes in $\mathcal{C}$.  Understanding $\mathrm{Spc}(\mathcal{C})$ sheds light on properties of $\mathcal{C}$ related to nilpotence and other cool stuff.  If you want to learn more about this perspective, you may as well start here [pdf].

But how can you get a handle on $\mathrm{Spc}(\mathcal{C})$? It turns out that the Zariski spectrum of the endomorphisms of the $\otimes$-unit object $1\in \mathcal{C}$ contains coarse but important information about $\mathrm{Spc}(\mathcal{C})$.  In particular, there is a continuous map $\rho:\mathrm{Spc}(\mathcal{C})\to \mathrm{Spec}(\mathrm{End}(1))$ which, under favorable circumstances, is surjective.  Given such a surjective map and knowledge of the structure of $\mathrm{Spec}(\mathrm{End}(1))$, we might hope to determine $\mathrm{Spc}(\mathcal{C})$ fiberwise.

But in certain contexts, $\mathrm{Spec}(\mathrm{End}(1))$ is a little too coarse. For instance, if $\mathcal{C} = \mathrm{SH}^{\mathbb{A}^1}(F)$ is the (for the experts: full subcategory of compact objects in) the stable motivic homotopy category of a field $F$, then $\mathrm{End}(1) = GW(F)$, the Grothendieck-Witt ring of quadratic forms over $F$, and this is a picture of $\mathrm{Spec}(GW(F))$:

Prime ideals in the Grothendieck-Witt ring.

So what exactly is going on here? We see a bunch of copies of $\mathrm{Spec}(\mathbb{Z})$ in which all of the points $(2)$ are glued together. There is a distinguished copy of $\mathrm{Spec}(\mathbb{Z})$ associated with the dimension homomorphism, and the rest are indexed by $X_F$, the space of orderings on $F$. (For these purposes, it’s best to think of an ordering as a group homomorphism $\alpha:F^\times\to \pm 1$ which also satisfies additivity: $\alpha(a+b)=1$ whenever $\alpha(a)=\alpha(b)=1$. We recover the positive cone of $\alpha$ via $P_\alpha = \alpha^{-1}(1)$.) The copy of $\mathrm{Spec}(\mathbb{Z})$ associated with $\alpha\in X_F$ arises via pullback along the signature homomorphism

$\mathrm{sgn}_\alpha:GW(F)\to GW(F_\alpha) \to W(F_\alpha) \cong \mathbb{Z}$.

Here $F_\alpha$ is the real closure of $F$ with respect to $\alpha$, and $W(E) = GW(E)/(h)$ is the Witt ring, given by modding out by the hyperbolic plane $h = x^2-y^2$. (If things like real closure are feeling hazy, go read your favorite algebra text’s treatment of the Artin-Schreier Theorem. Real closed fields are ordered fields which are maximal with respect to algebraic extensions which respect ordering.) The “dimension copy” of $\mathrm{Spec}(\mathbb{Z})$ arises via pullback along

$\mathrm{dim}:GW(F)\to GW(\overline{F})\cong \mathbb{Z}$

where $\overline{F}$ is the algebraic closure of $F$. All of this is essentially a classical result from Lorenz and Leicht’s 1970 Inventiones article.

Why might we be disappointed with $\mathrm{Spec}(GW(F))$ as the target of our comparison map $\mathrm{Spc}(\mathrm{SH}^{\mathbb{A}^1}(F))\to \mathrm{Spec}(GW(F))$? Well, doesn’t it feel a little unnatural that all the characteristic two primes are collapsed to a point? Do we really think that all the odd and zero characteristic triangular primes will know about the rich order structure on $F$, but that triangular primes which map to $(2)$ remain clueless? I mean, $(2)$ is the most interesting prime — if anything there should be more going on there!

And there is. In order to see this, we need to introduce a new character and prove two theorems. The character is Milnor-Witt $K$-theory, $K^{MW}_*(F)$. This is a $\mathbb{Z}$-graded ring defined as a quotient of the free associative algebra on symbols $[a]$ where $a\in F^\times$ (these are in degree $1$) and $\eta$ (in degree $-1$). I’ll send you to the paper for the explicit relations, but they include the Steinberg relation $[a][b] = 0$ for $a+b=1$, and $K^{MW}_*(F)$ is a sort of quadratic enhancement of Milnor $K$-theory. (In particular, $K^{MW}_0(F)\cong GW(F)$.)

A theorem of Morel tells us that $K^{MW}_*(F)$ is a graded ring of endomorphisms of the unit object $S_F$ in $\mathrm{SH}^{\mathbb{A}^1}(F)$. In particular,

$K^{MW}_n(F) \cong [S_F,\mathbb{G}_m^{\wedge n}]$,

the group of stable homotopy classes of maps from the motivic sphere spectrum to the $n$-fold smash product of $\mathbb{G}_m = \mathbb{A}^1\smallsetminus 0$. (The $\mathbb{Z}$-grading arises because $\mathbb{G}_m$ is a smash-invertible object in the stable motivic homotopy category.)

Let $\mathrm{Spec}^h(K^{MW}_*(F))$ denote the collection of homogeneous prime ideals in $K^{MW}_*(F)$; it has a natural Zariski topology. For any such homogeneous spectrum of a graded endomorphism ring, Balmer produces a continuous map from the tensor triangular spectrum to the homogeneous spectrum. In this case, it takes the form

$\rho^\bullet:\mathrm{Spc}(\mathrm{SH}^{\mathbb{A}^1}(F))\to \mathrm{Spec}^h(K^{MW}_*(F))$.

(Technical note:  we actually need to replace $\mathrm{SH}^{\mathbb{A}^1}(F)$ with its full subcategory of compact objects.  Later we will replace this category with compact cellular objects.  These are quite a bit simpler, but still very rich and the target of the map remains the same.)  We’ll be able to study $\mathrm{Spc}(\mathrm{SH}^{\mathbb{A}^1}(F)$ fiberwise via this map as long as

• we know the structure of $\mathrm{Spec}^h(K^{MW}_*(F))$, and
• we know that $\rho^\bullet$ is surjective.

This leads us to the main result of Riley’s paper (in cartoon form):

Theorem [Thornton]. If $F$ is a field of characteristic different from $2$, then the homogeneous prime ideals in $K^{MW}_*(F)$ take the form:

Homogeneous primes in Milnor-Witt K-theory.

If you want to know exactly which prime is what, I’ll send you to the paper: it’s quite readable if you have some basic background in Milnor-Witt $K$-theory. For our purposes, let’s simply observe that Milnor-Witt $K$-theory resolves the “problem” with the Grothendieck-Witt ring: we now have characteristic two primes indexed by $X_F\amalg \{\mathrm{dim}\}$. In fact, we even get a bonus characteristic two prime at the bottom of the diagram!

But all of this is for naught if $\rho^\bullet$ doesn’t hit these new primes. Balmer produces several criteria for surjectivity of $\rho$ and $\rho^\bullet$, and the connectivity of the stable motivic homotopy category guarantees that $\rho$ surjects onto $\mathrm{Spec}(GW(F))$. But none of Balmer’s criteria apply to $\rho^\bullet$ in this context. Nonetheless, we have the following result, which leverages Thornton’s computation to find explicit triangular primes in the stable motivic homotopy category living over each homogeneous Zariski prime.

In order to state it, a small bit of terminology: let $SH^{\mathbb{A}^1}(F)^c$ denote the full subcategory of compact motivic spectra over $F$.

Theorem [Heller-Ormsby]. Balmer’s map

$\rho^\bullet:\mathrm{Spc}(SH^{\mathbb{A}^1}(F)^c)\to \mathrm{Spec}^h(K^{MW}_*(F))$

is surjective.

Some brief notes on the proof/construction:

• It proceeds via explicit knowledge of the target (Thornton’s theorem) and topological arguments.
• Homogeneous Zariski primes not containing 2 are easy to hit since the map $(~)_0:\mathrm{Spec}^h(K^{MW}_*(F))\to \mathrm{Spec}(GW(F))$ is a homeomorphism away from these primes (and $\rho$ is surjective by connectivity).
• This means that characteristic 2 primes are the crux, and we rely on a topological argument to ensure they are hit by $\rho^\bullet$.  Note, though, that this does not produce any explicit tensor triangular primes over these ideals!  If we pass to the cellular motivic category, we can construct explicit tt-primes as subcategories of acyclics for novel cellular field spectra.

You can read a full account of these results here.

At this point, I think that more questions have been raised than answered. What else lives over Riley’s prime ideals? Do all of the triangular primes pull back from real and algebraic closures? How do the cellular primes compare to non-cellular primes?  Are the triangular primes in the above theorem maximal (or close to maximal)? (Note that $\rho^\bullet$ reverses inclusions.)  What about nilpotence in the stable motivic homotopy category? Certainly the exotic non-nilpotent elements of Andrews, Isaksen, et al will enter the story….  Jeremiah Heller and I are actively working on these questions and more, but everyone’s input is welcome!

After a decades-long relationship with road running, I have finally found the love and support I truly need — and deserve! — on the trails.

This was a breakout season for me:  I hung up the road flats.  I embraced distance.  I took the mountain air.  I soaked my eyeballs in Western landscapes.  I practiced downhill running.  I pranced over boulder fields.  I capered through stream beds.  I surged up scree fields.  I raced for the W, and even got it — once.

As of yesterday, I had run three 50k trail races and won the Northwest Mountain Trail Series.

 Race Distance Elevation Place Time Points Mt Hood 50k 50 km 2,500′ 2nd 3:26 98.5 Volcanic 50 50+ km 7,000′ 2nd 5:27 98.4 Elk-King’s 50k 50 km 6,000′ 1st 4:24 100

Now were I a sponsored professional runner (cough hint cough, shoe and apparel companies), I might find the time to give each of these races the special attention they deserve.  As it stands, I am a professional mathematician who should be unpacking boxes from a move that happened three weeks ago.  So, citing a peculiar combination of laziness and busyness, I have opted to relay some vignettes and pictures from this season’s training and racing and call it a day.

Before I get to that, though, huge kudos to the folks who made this possible.  Go Beyond Racing put on all of these events and could not run them better if they tried (and yet still, they try); thank you Todd and Renee and the rest of the crew and the army of volunteers.  Trail Factor is full of wonderful people who have shared amazing advice and trail running wisdom; Jason Leman and Marta Fisher have been especially generous.  As friend and training-partner-when-colocated, David Ayala has been a huge inspiration as he burst onto the Montana ultra scene and took fourth in the Wasatch Front 100.  Ben Nelson at Lifestyles Physical Therapy has been a tremendous help in managing an achilles flare up.  And of course and most of all, Sarah has been hugely supportive of this odd running habit even when life is so busy we can’t see straight.  Thank you.

Chamonix

After stutter starting through the first four months of the year, my training really got going in May.  That gave me just enough buildup to feel fit when I spent 10 days in late June in Chamonix, the Disneyworld of French Alpine playgrounds.  [Mandatory pause to revel in my parents’ excellent choice of family vacation spots.  Thanks, Mom and Dad!]

Typical Chamonix trail view.

Running the Chamonix trails hit the reset button on my perception of hill steepness and length.  I got to spend the rest of the season thinking “This is hardly anything compared to the two vertical kilometers of gain I could get from the Chamonix valley!” Highlights of the trip included getting passed by a free soloing Kilian Jornet during a guided mixed climb of the Arête des Cosmiques.

Shortly after getting passed a fast-moving diminutive Basque man.

St Helens — the preview

My goal race for the season was the the Volcanic 50, so on August 1 (a day that would see temperatures soar to 97ºF), I set out to circumambulate Mt St Helens and learn the course.  Go Beyond Racing was hosting a group run, and I ran much of the loop with Don Gallogly.

Don climbs the ropes out of the Toutle River basin.

It was brutally hot, and I took a wrong turn approaching Windy Pass, sending me two miles out of my way.  I ran out of water traversing the endless reentrants (frustratingly dry) past the Plains of Abraham.  But I nonetheless made a nice 8.5-hour day out of it and got to take in some spectacular views.

Elk!

Mt Hood 50k

On July 12, I ran the Mt Hood 50k with one word on my mind:  systems.  If I was going to do well at trail ultras with four to eight miles of rugged terrain between aid stations, I needed to dial in my nutrition and hydration systems under those conditions.  So I undertook the Hood 50k — an honest but fast course on the Pacific Crest Trail — with the goal of honing those skills.

Elevation profile for the Mt Hood 50k.

Running fast and challenging for the win (only to be succinctly and convincingly outclassed by a very quick Patrick Reaves), were bonuses.  I was surprised and pleased by my speed on the course, averaging 6:39 miles despite a conservative start.

Charging downhill in second place during the Mt Hood 50k.

Teton Crest Trail

Despite being poorly positioned in both of our race schedules, David and I decided to sneak in some adventure running on August 20.  I wrote about our experience on the 41-mile long Teton Crest Trail here.

Volcanic 50

The Volcanic 50 was my goal race this season.  After climbing from the Marble Mountain Sno-Park, the course joins the Loowit Trail for a full clockwise circumnavigation of everyone’s favorite pyroclastic flow site, Mt St Helens.

Running the final descent in the Volcanic 50, about to take second place. Photo: Paul Nelson Photography.

Elevation profile for the Volcanic 50.

This race has everything: There are climbs on packed dirt.  There are boulder fields of volcanic tuff.  There are short sections of bushwhacking.  There is a river crossing.  There’s long sandy climb that will sap your strength.  There is a hard-charging Rod Bien who will be ever-so-friendly as he takes the lead from you around mile 18, encouraging you to work with him since this is just a training run for him and he sure would like the company.  (You will have the opportunity to replay this scene in your mind many times over.)  There is an oasis spring after the blast zone, followed by a steep climb to Windy Pass and scree slope descent to the Plains of Abraham from which you can see Rod Bien’s green shirt not getting closer despite the quick miles you are laying down on this mercifully flat terrain.  And then are the reentrants, gravelly gullies with washed out trails that you can stumble through while your right adductor cramps with astonishing ferocity.  And finally, there is one more boulder field, followed by one more wooded section, followed by a two-mile screaming descent into the parking lot where you are so so tired and just got second place by five minutes and are 22 minutes under the old course record and you are really happy to have done this and gosh should put on some warm clothes isn’t everyone freezing in this mist?

A case study in how to look tired when you cross the finish line.

Me and Aesop bundled up after the race. A far cry from the previous year’s 90+-degree temps.

Elk-King’s 50k

I was tired after the Volcanic 50, but also intrigued to learn that I was high in the standings of the NW Mountain Trail Series despite only having done two races.  There were three more scoring races left in the season, the Mountain Lakes 100 and Elk-King’s 25 and 50k races.  After an intense season, I had no intention of jumping into a 100-miler tired and unprepared, so had to choose between the two Elk-King’s races.  I ran the 25k last year, and knew that it was a brutal, quad-punishing romp over Elk and King’s Mountains on technical terrain.  So I decided to wimp out and run the 50k, which has twice as many kilometers (math!) but smoother trails and less climbing per distance (but still 6,000′ of gain).

Elevation profile for the Elk-King’s 50k.

Yesterday’s run was not my prettiest.  My mileage had been relatively low since Volcanic because of general fatigue, an achilles tendinosis flare up, and a doozy of a head cold.  I was running tired, but I still ran strong, and it was enough for the win.

Keeping it together in the Elk-King’s 50k. Photo: Paul Nelson Photography.

I took the lead in the first mile, believing that the flat-ish terrain in the first eight miles favored my road running background.  By the first aid station (mile 4), I had a one-minute lead.  By the second (10.6 miles), I was up by five minutes.  By mile 20 I had 13 minutes on second place, and I would build that to 17 minutes by the finish line.

The first 20 miles are punctuated by 1,300′ and 400′ climbs, which are then repeated in reverse on the way back to the Tillamook Forest Center.  My energy and pace felt pretty solid until the 20 mile turnaround, at which point muscular fatigue started to set in.  Slopes I had handled at 10-minute pace on the way out became 13-minute pace on the return.  My adductor threatened to cramp, but was quelled by a GU and some long thirsty drinks from my handheld.  I was living for the final downhill only to be cruelly met with a fierce side stitch as I crested the final 1,300-footer.  I ran the next three miles bent over at the side, mimicking Young Frankenstein’s Igor, except that I would have admonished Gene Wilder’s Frederick “No, run this way.”  But the stitch subsided and I was able to enjoy the last couple of miles along the Wilson River, finishing on the bridge to the Forest Center amidst the raucous cheers of eight or nine people.

Crossing the finish line at the Tillamook Forest Center. I had envisioned a more enthusiastic celebration….

Per usual with events put on by Go Beyond, there was an excellent party after the race, including a growing number of familiar faces.  It was great to share some time off the trails with this excellent and welcoming community, and ultimately celebrate as I picked up some awesome hardware (created by Portland’s own Matt Helms).

Picking up my trophy from Todd after a great season of trail running.

The Trail Series win comes with a sweet jacket and free entry into next year’s suite of races.  I hope to see everyone there, running hard.

My hips move forward, weight shifting past my feet as gravity pushes me towards David’s heels.  My left foot responds sluggishly.  We’ve been running for seven hours and my toes fail to extricate themselves from the rock-littered path which descends through Paintbrush Canyon.  My shoulders and head continue their push towards the end of the Teton Crest Trail while my lower extremities remain rooted until suddenly I am airborne, sailing horizontal down the slope.  By accident or luck or a perverse sort of skill, I execute a three point landing, distributing the fall’s force across my right hand, hip, and ankle.  Later, David reports that I “made a lot of sounds,” but I don’t remember them.  I pause, then stand up and find myself almost unscathed — some minuscule scrapes, nothing more.  We continue.

Table Mountain from the South Fork of Cascade Canyon.

The Teton Crest Trail stretches forty miles from the Phillips Canyon trailhead on Highway 22 to String Lake in the northeast corner of Grand Teton National Park.   It gains 9,200 vertical feet (2,800 meters).  Its high point is Paintbrush Divide (10,720′ / 3,267m), followed closely by Hurricane Pass (10,338′ / 3,151m).  Amongst other features, it traverses the Death Canyon Shelf and plummets through the South Fork of Cascade Canyon.

The cliffs above Death Canyon Shelf.

Most Crest visitors — impeded by heavy packs filled with tent, sleeping bag, food, sometimes even a change of clothes — hike the route in 3-4 days.  To quote Melville, “Oh, ye foolish! throw all these thunder-heads overboard, and then you will float light and right.”  Our running vests stuffed with energy gels, two liter water bladders, iodine pills, a map, light jackets, and bear spray, David and I took off at 6:30am on August 20, 2015.  I carried a SPOT GPS messenger which transmitted our location to a few family members and could alert SAR if we found ourselves in a pickle.  My phone was tucked into a pocket, serving as lightweight camera.

Grand, Middle, and South Teton, and Schoolroom Glacier from Hurricane Pass.

We also carried hopes, but despite the big objective (40 miles! in the Tetons! self-supported!) our goals were modest, and for good reason.  I am running the Volcanic 50 (a rugged, 32-mile circumnavigation of Mt St Helens) on September 5, and David will take on the Wasatch 100 on September 11-12.  Additionally, David had run (in fact, won) the Bridger Ridge Run the previous Saturday, and was recovering slowly.  We agreed that anything between eight and twelve hours would be fine, and that we should stay completely within ourselves during the run.  It was totally OK to take our eyes off our footing, stop, and look around whenever we felt like it.

Wildflowers in Phillips Canyon.

Our early start afforded some excellent wildlife viewing as we trotted towards Phillips Pass.  Within the first four miles, we saw five bull moose.  The first was the largest, huge antlers shedding velvet as he trotted across our path fifty feet in front of us.  His jowls jostled as he joined his friend downhill from us, far enough away that the digital zoom on my slowly drawn phone camera transformed him into a pixelated, blurry mess.

A bull moose near the trailhead.

As we ran, forest fires raged in California and Washington.  Though hundreds of miles away, winds had carried the smoke east and south, and the mountains and valley alike were shrouded in a thick haze.  The effect was not entirely unwelcome.  Though less sharp and imposing, the Grand and other peaks gained a certain mystique, rising like shrouded giants out of the Wyoming plain.

A smoke-hazed view of the mountains from Hurricane Pass.

The smoke may or may not have impacted our breathing during the run.  Regardless, the altitude kicked my butt.  I flew from Portland, Oregon (50′) to Bozeman, Montana (5,000′) the afternoon August 17, and despite a day acclimatizing there and two days near the Tetons (7,000′), I felt completely unarmed when confronted with serious climbing above 9,000′.  Slopes that I would eat for breakfast were they nestled in the Columbia River Gorge became arduous power hiking endeavors.  It is strange to feel your heart pound in your ears like a bass drum at 170 beats per minute when all you are doing is hands-on-knees trudging up a 10% grade.

Mount Woodring from Paintbrush Divide.

The aforementioned fall may not have impacted me physically, but it sapped my confidence and left me tired and wobbly for the rest of our run into String Lake.  While David’s feet floated over and skipped off of rocks, I felt like a stroke victim struggling to regain motor control.  “David, can we slow it down a bit?”  We were passing a volunteer trail crew and I wanted nothing more than to impress them with our trail running acumen, our feline grace.  But I felt like I was held together with Wrigley’s spearmint gum and my head was a hum of negativity.  When would this end?  Would it end?  Crescendos of fatigue almost overwhelmed me.  How did they do this?  How do you do this for sixty more miles, 25,000 more vertical feet?  Who did I think I was, trampling this ground, an arrogant interloper?

I’ve not felt lower in a run, but unlike the smoke in the air, this haze lifted as we crossed the footbridge on the north side of String Lake.  Backpackers gave way to day hikers and stand-up-paddleboarders.  My fatigue gave way to relief.  The trail gave way to pavement, and we decided to make a construction cone on the edge of the parking lot our finish line.  Eight hours and thirty-nine minutes of Teton lows and mostly highs.

Teton Crested.

Acknowledgments: Special thanks to David’s Subaru, which stops just fine even when the brake pads are totally worn through.  Additional thanks to Ana for gracious hospitality and letting me steal David for three days, and to Sarah for holding down the fort (and the other fort).

Dedication: This run is dedicated to Aesop, Sequoia, and Fiasco — may our fingers never wag as much as their tails.

Bonus video of the route:

(Sorry, Suunto, I won’t be taking 120 hours of rest.)

### Five Weeks, Four Races

Moving to Oregon, I was excited about a number of things:  rain, mushrooms, conifers, coffee — pretty much anything that reminded me of Twin Peaks.  Running was another anticipated highlight.  With year-round runnable weather and amazing trail systems in Forest Park and the Columbia River Gorge, I was sure that the home state of Prefontaine, Hayward Field, and Nike would not disappoint.

And what better way to acclimate to the new running scene than to race way too much, way too soon, off of too little training, while nursing a nagging injury?  This is a time-tested strategy of countless runners, and who was I to spoil tradition by considering the results of said tests?  So I sat down at my laptop, fired up Race Center Northwest, and crafted the following very reasonable racing schedule: Reed College 5k (9/20), Multnomah Falls Trail Run (9/28), Portland Half Marathon (10/5), Elk-King’s Mountain Traverse (10/18).

### Results

My crazy race schedule worked out far better than it had any right to.  I had to scale back some training after a flare up of my achilles injury, but managed to stay healthy and enjoy the process of both recovering and racing.  Being a little injured and a little undertrained gives you the freedom to play in your races, and I really enjoyed that.  Here’s what happened:

 Race Distance Elevation Place Time Reed College 5k 5 km 188′ 4th 17:51 Multnomah Falls Trail Run 5.9 miles 2,600′ 1st 40:10 Portland Half Marathon 13.1 miles 200′ 4th* 1:16:52 Elk-King’s Mountain Traverse 15.5 miles 5,800′ 6th 2:39:03

*Officially 3rd.  I’ll explain below.

Above, each time link will take you to GPS data from my race, and each place link points to official results for the event.

### Reed College 5k

I can’t think of any metrics by which this qualifies as a “good” race.  I went out hard (5:23 for the first, uphill mile) and had neither the physical fitness nor mental fortitude to keep it up. Regardless, it was a fun outing and I was happy to see Sarah take 2nd in her age group.  I also got to race against some students and the college president.  Overall a low-key rust buster, or so I thought.

### Multnomah Falls Trail Run

The day after the 5k I went out for a recovery jog only to discover a new and acute pain in my left foot.  Having never experienced such a pain before, I was certain I had a stress fracture in my fifth metatarsal.  Thankfully, I followed the sage advice of a colleague and got an emergency appointment with Dr. AJ Scherer at Back in Motion.  He quickly ruled out a stress fracture and identified a cascading chain of problems stemming from mobility issues with my left ankle.  This explained the long-term achilles/calf pain I’d been having, and the foot pain was the newest manifestation.  After an adjustment, some Graston, electro-stimulation, and ice, I was able to run at the end of our session.  I don’t believe in miracle cures, but this felt like one.  (In fact, it remains a continuing issue requiring persistent care.  Thankfully, though, it’s not keeping me from running like a stress fracture would.)

So it was with Dr. Scherer’s blessing that I laced up my running shoes for the Multnomah Falls Trail Run only six days after thinking my racing season was over.  The MFTR starts and ends at the Wahkeena Falls Trailhead, making a loop up the Multnomah Falls switchbacks and back down via Wahkeena.  The scenery is gorgeous: the 620 foot-tall Multnomah Falls gives way to secluded rocky trails bumbling through rocks and moss.  Then old growth conifers shelter ferny undergrowth and some smooth packed dirt.  The final descent tumbles through rocky, rooty, face-smacking terrain which I might remember better if I hadn’t been so focused on not falling.

The heavy-use switchbacks on the initial ascent are paved, making them far more runnable than the grade would predict, but this was still some serious climbing.  I used my reasonable climbing skills to put a good gap on the field here, and couldn’t see my nearest competitor by the time I reached the first trail junction in the course.  When I reached the turn downhill I was excited to enjoy a brisk trot to the finish tape and my first race win in the Pacific Northwest.

Not so fast:  Given my semi-pathetic descending skills on uneven terrain, it should come as no surprise that my brisk is other people’s moderate, and within a quarter mile of heading downhill I heard heavy breathing and loud footsteps behind me.  A quick glance over my shoulder revealed that Superman was trying to overtake me.  (No, really.  A guy wearing blue tights with a pair of red underwear over them and a red infant bib taped to his back was gaining quickly.  [Much to my discredit, I had decided not to partake in the superhero costume component of the race.])

And then something wonderful happened:  I finally had a legitimate excuse to throw caution to the wind and push hard on a technical descent.  My legs were independent pistons, launching me from rock to root to rock, over streams, even through streams.  I was pushing hard, bending gravity to my will.  I was maintaining a gap.  We must have been hitting mid-5-minute miles as we screamed down some gnarly trails.

I wasn’t aware at the time, but drama was unfolding behind me as well: Superman (Tom Ferrell) was nipped at the line by the hard-charging Levi Younger.  I managed to hang on for a first place finish, which felt great.

### Portland Half Marathon

The Portland Half Marathon filled up months before I even knew it existed, but Reed College (where I teach math) managed to secure some last-minute spots for faculty, students, and staff who wanted to run.  Of course I wanted to run!

This was a race I had designated as a “tempo” effort, a potentially meaningless category that would hopefully keep me from going out too hard.  The course was great (for a road race) and featured some nice sunrise views of Mt. Hood.  I kept the leaders in sight initially, sorely tempted to drop my race plan and try to keep up.  Sanity prevailed and I had a good time cranking out an evenly paced race.  There were some great cheerleaders along the course, including

• some older women wearing reindeer antlers pantomiming slow-motion running,
• a marimba band on an overpass, and
• a large cast of elaborately dressed pirates.

The Reed turnout at the race was impressive, too.  President John Kroger, Vice President Mike Brody, physics professors Johnny Powell and Alison Crocker, one of my calculus students, math professor emeritus Rao Potluri, and many, many more.  We were all outfitted in custom Reed singlets with a running griffin on the front and a quote from the Odyssey on the back:

What greater glory attends a man, while he’s alive, than what he wins with his racing feet and striving hands?  Come and compete then, throw your cares to the wind!

(Fine, fine:  It takes a while to read, but what else are you going to do in an hour+ race?)

Reed results were good.  Alison took second in the women’s division of the half marathon, and I came in fourth behind Jack Flowers, a physics senior.  Jack was racing with someone else’s bib and was thus excluded from the results, hence my “official” 3rd place finish.  He ran a very different race, going out with the leaders and slowing a bit at the end, but it was still more than strong enough to hold off my bid for faculty supremacy in the final mile.

Overall, I was super pleased with this race.  Since it was a “tempo” run, I don’t have to worry about it being six minutes off my PR (eek!).

### Elk-King’s Mountain Traverse

The EKMT was my goal race for the fall.  I fell in love with it as soon as I saw the elevation profile:

Image: Frederic Bard

Over fifteen miles of rugged single track goodness with 5,800 feet of elevation gain (and loss) amongst the rain, mushrooms, and conifers of Tillamook State Forest.  (They even had Oblique coffee at the start line, making this a Pacific Northwest quadfecta.  Oh, and running:  quintfecta.)

All of the maroon segments of the elevation profile have a grade greater than 15%, and more than a few parts required flat out quadrupedal scrambling.  The Mazamas had been enlisted to set up some ropes at a couple of tricky sections (e.g. a steep, slick rock descent at the top of Elk) and I was happy to enlist the their aid.

Ascending Elk Mountain. Image: Paul Nelson Photography

I went out with the leaders but realized they had better ascending chops within 3/4 of a mile.  Given the gap they had on me by the finish line, they must have been better descenders as well.  I ended up in a truel with the eventual fifth and seventh place finishers, Nate Jaqua and Levi Younger.  (Yep, the same Levi who got second in the MFTR.)  On the Elk-King’s portion of the course we established a pattern in which I would build up a lead on uphill segments, then watch them zip by me on the technical downhills.  The one time I tried to really push on a descent and keep up, I came out of a corner a bit wide, planted firmly on the edge of the trail, and managed to kick a good foot of trail off the mountain.  One of the luxuries of running is that your momentum allows for some Wile E. Coyote moments in which you don’t plummet through empty space.

At the second aid station I was in no-man’s land, having lost contact with my competitors on the 2,300′ quad-pounding descent.  Nothing to do, of course, but keep my eyes on the trail and hope I would catch runners on the uphill and fend them off on the final downhill.  At this point, 50k competitors headed in the opposite direction started to appear.  I started asking how much gap there was to my competitors, and discovered I was closing in.  I caught Levi with about five miles to go; he was power hiking an incline I was still able to run.

One 50k’er estimated a 30-second gap to Nate shortly before mile 11, but I never caught or saw him.  In fact, shortly after the second aid station (mile 12), I was stopped short by a cramp in one of my adductors.  I walked it out for a couple minutes, constantly looking over my shoulder for a hard-charging Levi.  Thankfully, the adductor stopped spasming and I could continue on my way, finally settling into some relatively flat terrain permitting a small injection of pace.

And then, about a mile before I expected it, I could see the final bridge leading to the finish line.  No one in front of me, no one behind me, the sound of a cowbell clinking in the hand of an understimulated spectator.  I tried to look strong as I crossed the bridge and finish line, then promptly crumpled, arms and head supported by a plastic picnic table.  At some point I roused myself enough to down a couple liters of water (out of my snazzy finisher’s EKMT pint glass — so much better than a t-shirt) and congratulate Levi as he passed the finish line.  Despite the ensuing quad, hamstring, adductor, calf, and abdominal(!) cramps, I had no problem eventually enjoying the post-race barbecue, Super Dog IPA, and camaraderie.

I had come into the EKMT vaguely thinking that times between 2.5 and 3.5 hours would be reasonable, so my 2:39 felt just fine.  The finish could have been very fun and interesting if I hadn’t lost minutes to my leg cramp, so I’ll have to show up to my next trail race with a few more hills in my legs and maybe an extra banana in my race kit.

### And for my next trick

I’m not sure what’s next on my race docket.  Clearly some recovery running is in order, and I need to make sure my achilles fully heals.  David and I are still chewing on big plans for the spring, and I’d like to figure out if I can get into the Speedgoat 50k.  I guess the big question is whether I’ll find a few races to prod my running through the winter, or just focus on quality mileage and time on the trails.  Suggestions welcome!

I recently returned from a fun, scary, exhausting, and overall excellent trip to the Palisades.  Good times with Josh, Patrick, and Jon, and a new longest route for myself, as Josh and I climbed the short version of Moon Goddess Arete.  (We rapped into the escape gully one pitch above the notch and did the “fourth class” pitches in the gully to the top.)

I’ll leave a few pictures here and send you to Josh’s blog for a more complete trip report.