A hike in the Himalayas (& directing undergraduate research)

A hike in the Himalayas

In the previous post I suggested six attributes of a good undergraduate research problem. I claimed that a good research problem for undergraduates

1. should be accessible,
2. has "legs",
3. leads to higher math,
4. dreams of mathematical heights,
5. uses "magic",
6. offers alternatives.

In this post I review those characteristics using a common metaphor.

I have never been to the Himalayas.  My wife and I used to backpack in the Colorado Rockies and so I've dreamed about seeing the Himalayas someday.  (Is this a "bucket list" goal?  Possibly.)  Suppose instead of just visiting the Himalayas, one were to truly explore the Himalayas -- not just hike on well-worn trails, but really go into the backcountry and find new trails to build, new peaks to climb, new rivers to raft!

What would this involve?  It would be good to have a guide, a person with local experience, who can suggests areas to explore and who has some understanding of the paths into the wilderness.

Now let's turn the story around.  Suppose that you are the local guide.  What does it take to give your visitor, your client, a good wilderness experience?

I will offer six characteristics of a good guide to the Himalayas.  (If I were to ever get an opportunity to truly explore the Himalayas, here is what I would expect my guide to be able to do.)  These six characteristics deliberately parallel the six characteristics of a good problem from the previous post.

Frankly, I know a lot more about finding good math problems than I do about hiking in the Himalayas.  Still, this is my blog and the Himalayas are my dream so....  (If one prefers a slightly different metaphor, imagine hunting for new snowboarding routes in the Grand Tetons.  Now there is some exciting "research"!)

A good guide meets the trekker client where they are and helps them acclimate

As you (the good guide) meet your trekker customer, you want to assess their abilities and help them acclimate to the mountains.  Begin with some small hikes as your guests adjust to altitude and to the hiking experience.  A good guide knows of nice hikes that are safe and part of a larger experience.  Of course, these hikes are not themselves explorations -- they are just serious walks in the mountains.  Many people have traveled these trails before.  But these introductory hikes set the stage for future exploration.

Be prepared for a serious expedition

As your hiking customer gains experience, he/she will want to do some serious exploration.  Be prepared to lead that trip.  Know the equipment necessary; have access to good resources; be ready to take healthy, energetic explorers deep into the wilderness.

Sure, there are other guides who can take people on enjoyable walks.  But your customers want to explore new territory.  You should have ideas on where to go and how to begin the trip and you should be ready to take them as far as they are capable of going.

Take your clients to interesting places

As you, the expedition guide, adjust to your customers, you want to show them some interesting sights.  It is not enough to hike ten miles somewhere, set up a tent, walk around for a few days and then hike back.  You want them to enjoy their experience and return again to the mountains.  So you should be able to say, "Beyond that ridge is a range of high peaks.  Let's climb that ridge and then see what might be reachable on the other side."

If this is a genuine expedition, you may not know what is on the other side, but you have some ideas.  There are some big mountains out there and so there are lots of interesting smaller ones!

Plan your trip so that your customers ooh and ahh at the scenes they encounter.

Even if you can't climb the top peaks, can we get good views of them?

I don't plan on climbing Mt. Everest.  But if I were hiking in the Himalayas, I would like one day to come around a corner and hear my guide say, "See, there, on the left, that faraway mountain with the plume?  That's Everest."

I've hiked in Colorado and I enjoyed the high peaks there.  So why go to the Himalayas?  Because of Everest.  I don't need to climb Everest myself, but that highest of all peaks draws thousands of visitors to Nepal every year, just so they can see it!

A good guide uses all available resources

Sherpa guides on Everest sometimes climb the mountain without access to fixed ropes or manmade bridges.  But if I have a guide in the Himalayas, I want him (her) to be able to guide me to bridges across ravines, bolted ladders up steep faces, any type of equipment that will help me get further into the wilderness.  As a visitor and guest, I need these manmade (magical?) aids and I hope my guide will not be shy about offering them.  (Indeed, I will surely need an occasional gasp from an oxygen tank, long before we reach even 20,000 feet in elevation!)

A good guide suggests offers alternatives

So I'm not climbing Everest.  But I'd like to climb some ridges and hills, maybe a smaller, gentler peak.  A good guide will have an idea where those are and how to get there.  As the hike goes on, the good guide will recognize the customers' abilities and limits and will be prepared to suggest alternate goals for the expedition.

Find a photo opportunity, a place where all of us can stand together, squeezed into a single frame and later say, "See, I was there!  [I didn't climb Everest but] I climbed this peak!"

"Climbing" the (a,b,c) order-of-products problem

Let's apply the last two posts (on good research problems and good hiking experiences) to the (a,b,c) order-of-products problem (posted Aug 26 & 29.)

The order-of-products problem begins with a simple question from geometrical symmetry or modular arithmetic.  These are ideas often taught in high school and certainly accessible to high school students.  So this problem has some nice entrypoints; there are good ways for undergraduate students to access this problem and acclimate to the mathematics needed.

The order-of-products problem leads fairly quickly to some elementary group theory and then seems to eventually require an understanding of group homomorphisms and subgroup structure, including Sylow Theorems.  It connects with Coxeter groups and groups of Lie type.  So it seems to lead to some interesting higher level problems.  In a search of the literature, I have not yet found any place where this problem is attacked in this generality, although there are places where versions of the problem have been solved.  So this problem seems to lead to some "high peaks" of mathematics and so should take researchers some distance.

I'm not aware of any big conjectures associated with the order-of-products problem.  I do not see an "Everest" on the horizon.   But this may be due to the fact that this problem is not directly in my area of research.  If I am to direct this problem further, I am uncomfortable with the fact that I don't personally know the tools used in creating finite groups.  Maybe I am not the best "guide" for this problem?  A better guide would probably know more about reflection groups, finite Coxeter groups and similar topics in group theory with a "geometric" flavor.

But I do see some smaller peaks; just the existence of triangle groups in hyperbolic geometry seems like an interesting "mountain range" to explore.  So, until better guides come along, I can take students for a good long walk in the mountains!

As students adjust to the order-of-products problem, they can use computer software (like GAP or Sage, which are free) and I am quite willing to give students "magical" results about simple groups and Sylow theorems.

All in all, the order-of-product problem seems like a good, but not great, problem for students.  This problem might be a good B level undergraduate research problem; if I am the director of this exploration, my lack of experience in the finite group theory probably puts the potential of this problem as a B- or C+.

Summary

Imagine breaking new trails across a secluded mountain valley, rafting an unexplored section of a river, or taking your snowboard down a virgin mountain slope.  That is exciting!  Every explorer, every wilderness guide, began as a novice enjoying smaller, safer trips.  Let's get our undergraduate students into short (mathematical) wilderness trips and see if they will catch on to the excitement of the mathematical quest.

Next time: Graph theory as one source for good undergraduate math problems.

[Mathematical prerequisites: This article requires no mathematics beyond high school.]

Follow Ken W. Smith on Twitter @kenwsmith54