A picture log of my 2017 post-consulting life

December 18, 2017

So it’s been almost a year since I set sails away from the sophistication of the corporate world to find more primitive adventures. Although not all things went as planned, it’s been perhaps the most memorable year I’ve had: 14 countries in 5 continents, 2 broken bones, tried many new things and met many new friends. Main while, I guess the main plot of my life has somehow unfolded semi-unexpectedly: I became a Canadian permanent resident and married some French model theorist.

Updated map (places visited, by country)

Chapter 1. Snow, ice & mountains

January: Alps (France & Swisserland)

I started my journey in the French Alps, without having much clue about how the mountains work and where the best ski places are. I did the fool-proof thing of googling where Mont Blanc is and made my way there (logic being I probably won’t miss out by starting at the highest point, there’s gotta be cool stuff around it, right?) So I winded up passing through Moscow, Lyon and found myself in the ski/mountaineering mecca of Chamonix, where I tried paragliding, snowshoeing and took some first courses in snowboarding.

Then I headed to Swiserland and spent a week climbing frozen waterfalls. We ended up driving around Swiserland finding different falls everyday and moved from single pitch to multi pitch and towards higher grades. Conditions were generally -15ish degrees plus windchill. There’s nothing like hanging on vertical ice with painfully frozen fingers and toe trying to do pull-ups.

February: Alps to UK (Austria, France, UK, Ireland)

At the beginning of February I went from Austria back to France, doing more snowboarding on the way. Then I broke my wrist on the day which there’s a photographer following the class and we were asked to go down an easy slope to get photos taken. Of course I tried to look like I can ride better than I actually can, and fell hard. Luckily the doctors in Chamonix are extremely experienced with such injure, ‘We get this 20 times a day…everyday’ he told me while mechanically casting my arm.

After attempting to snowboard with the casted wrist (which worked relatively well), I decided to cut off the cast after 10 days and head to the Scottish winter mountaineering class as planned (which did not go well as my hand started to turn bright blue halfway up the mountain on the first day). So I spent the rest of the week wondering around the Scottish countryside, where I learned to embrace the slowness and developed a new understanding of where Turner got his amazing landscape colours from.

Then I did some museum-going and spent much of the day at coffee shops in major cities throughout England, Wales, Northern Ireland and Ireland.

March: Japan

After a short stop back in Shanghai I decided to head out for some more Snowboarding, this time in Hokkaido, Japan.

Towards the end I was able to go off-piste and into the woods

Did some touristy things while there as well (by now I have officially been to all Disney parks in the world!)

Chapter 2: Ocean

April: Indonesia

After putting down my ski goggles and digging out my diving mask, I went straight to the warm ocean for some surfing and diving.

Found the new world of free-diving and got completely hooked (I feel like I’ll never want to use a scuba tank again)

May: Western Australia

Going further south I arrived in the vast empty lands of Western Australia.

Arrived in Exmouth – A small town in which the main industry is whale-shark tourism (i.e. taking boats of tourists out and drop them down to swim with wild whale-sharks in the ocean). Here I had a wonderful few weeks volunteering as a crew member on a whale-shark boat, of course applying some newly-acquired free-diving skills =)

June: Indonesia

I went back to Bali and volunteered for the freediving shop in June, painted logos, fishes and instructor’s fins while furthering my freediving training. After a month I was able to hold breath for 4’30” and reached -40m (which is sufficient depth for getting instructor license). Also received my custom-made carbon fibre fins from Greece, in red like all my other gear of course =)

Chapter 3: Land &amp sky

July: Western USA

The plan for the summer was to get into the dessert, get skydiving license and start big-wall rock climbing in Yosemite. It did not exactly go as planned…

With a good friend I met snowboarding in Japan, we had a classical american road trip through Nevada and Arizona, camp and hiked in death valley and grand canyon.

August: Atlanta, USA

Then I went to Atlanta to get my AFF skydiving license, met lots of awesome people in the skydiving community, learned how to pack my own parachute and broke my ankle on the 8th jump.

September: Berkeley, California, USA

My original plan was to spend about a month in Yosemite to try breaking into the rock climbing community (by, perhaps, showing up at the infamous CAMP 4 and stay there until I find some cool guys with a van). The goal was to start big-wall climbing, hopefully attempt some classic multi-day routes by the end (El Cap being my lifetime rock climbing goal).

Obviously that plan was completely scarped once I broke the ankle…So what ended up happening was that I got ‘stuck’ in Berkeley for 6 weeks, during which I picked up some reading, mathematics, cooking as well as the above-mentioned French model theorist whom I ended up marrying a couple months later.

October: Canada to Turkey

My Canadian permanent resident application got approved! So I went quickly to Hong Kong and took care of the documents, went back to Shanghai to pack up, then officially ‘landed’ in Canada as new immigrant!

A week later I took my first ‘trip aboard’ as a Canadian PR =) Not wanting to push the ankle too hard yet, I decided to do some traditional backpacking without extreme sports, with a good friend I met earlier on the road, we set off for Turkey.

Did some standard touristic sightseeing (the greek and roman ruins far exceeded my expectations).

…and did some mellow exploration of nature

November: Morocco & Western Sahara, then back to Canada

After Turkey I arrived in another long over-due popular backpacking destination: Morocco. Starting off by city-hopping from Casa Blanca to Rabat, Fez, Marrakech, Essaouira.

Then into the desert:

Then I did something slightly less conventional: I took an overnight bus down south to the costal city Agadir, rested for a night there and took another overnight bus ride to a small town called Laayoune, in the rather controversial area of Western Sahara (former spanish colony, now officially belonging to Morocco). I went, to pay my tribute to Taiwanese author Echo Chen, whose books about the desert and living life as a wanderer and nomad had an undeniable influence on me from back in my teenage years.

The town is remote and rather unremarkable in general, with lots of UN personnels from the special mission which has been here since 1990. But I found various sites which appeared in the books, the author’s mud hut, the spanish church, the postal-office and even stayed at the legendary (and only) fancy hotel in town which remained exactly the same as 50 years ago.

– For those who are curious, Echo Chen was a Taiwanese lady who studied in Madrid, one day she came across a photograph of the Sahara desert in a magazine and felt a special connection, subsequently decided to move there. Those Chinese-language books are semi-fictional stories about her life in the desert. The books had a lasting popularity as inspiration for young people to ‘follow dreams’ and ‘wander around the world’.

From there I returned to Canada and got married at the end of November. It was quite fun, we found a street-corner with some trees and Christmas lights do perform the ceremony and finished right when it started to rain…

December: Canada, US, Bahamas, Belize

(Coming soon)


Farewell, McKinsey & Hello, World!

December 30, 2016

Yesterday I returned my phone, computer and access card to the Firm, officially puting an end to my 1.5 years of consulting. I have been planning for this day, packed up all my earthly belongings and locked them up in a 1 cubic meter box in the outskirts of the city, got a Hong Kong sim card to by-pass the Chinese internet blockage, calculated savings from the McKinsey period, placed them into s separate account & marked it ‘to be spent’. Right now I’m on a plane, heading northwest, to start my journey.


My farewell e-mail to everyone I know at the Firm states that I am to ‘take an indefinite amount of time catching up on all the cool stuff I have not been doing enough; to re-gain inspiration and embrace (once again) the endless possibilities of the world’. 

Some has concluded this is a ‘gap year’; but I’d rather think of my time in the corporate world as an adventure outside of my usual routine. Sometimes it all feels like a role-playing game: wake up, put on my ‘consultant costume’ and head out to complete a bunch of pre-set tasks for the day. This world infacinating for an accidental tourist, almost as if there is an invisible bubble dividing us from reality. In this alternative world one gets delivered between planes, fancy offices and fancy hotel rooms; spending their days on calls, e-mails and meetings; making decent amount of money. This can be comforting and satisfying to many, leading to losing track of one’s dreams and goals, to wind up just tagging along and climb the ladder of conformality and prestige. So naturally I found myself steering away and setting sails into uncharted waters, once again.

When trying to come up with a list of things (anything anywhere in the world) I think way back into childhood and ask myself what I used to dream of and haven’t done (or done enough of) yet. Right now I found the major shortage would be building proficiency in outdoor activities – as a kid i remembered that I was no less passionate about the outdoors (from normal camping to expeditions into deep ocean and high mountains) than I was in, say, math or art. One difference, though, is that I’ve realized my mediocre talent in that area from very early on. Hence, I thought, since it’s not likely that I’ll be a great mountaineer or sportsman, this better wait till I have done a few years of stuff I’m actually good at. But this ‘itch’ has always been there. Time to take actions!

4 major themes of activities I would like to improve my proficiency in (some I have no experience in, such as ice climbing).

Note: not my images, for illustration & aspiration only

Ice & Snowimg_0762Landimg_0761Oceanimg_0760Skyimg_0759In addition of course there are less active hobbies I will continue to peesue, such as painting/sculpting and continue to expand my world map =)


It’s difficult to plan for an open-ended journey, so I have attacted like a large oil painting: one starts from a very rough sketch of the composition, then work up the ‘resolution’, one small section at a time, the parts closer to the painter gets more details filled in first while the objects further away remains a sketch with few brush strokes and lots of room for revision. Here is the current ‘sktech’ of 2017:

  • January – Feburary: Snow sports in the Alps, snowboarding, ice climbing and some mixed Alphine climbing
  • Feburary – March: UK, mainly outdoors in Scotland
  • Spring/Summer 2017: a subset of the following activities

– Australia & New Zeland for skydiving/hang gliding/hiking

– Himalayas to observe the mountaineering season and get some high-altitude experience (no Everest)

– Caribbeans/Hawaii/SE Asia for more advanced scuba diving and surfing

  • Summer/fall 2017: Southwest USA for pilot license, burning man and rock climbing (time to take steps towards my El Capitan dream)
  • Fall 2017: a subset of the following

– Europe for oil painting/drawing/sculpture workshops

– Convensional sight-seeing in a few parts of the world I have not been yet (Japan, Turkey, Morocco, Middle-East

– Patagonia for kayaking, climbing and hiking

  • Winter 2017: hopefully I may conclude the year in Antartica

Note that this is what I can come up with through day-dreaming behind an office desk, looking to improve it as I go and meet cool people; So if you have ideas/ recommendations/ couch to host random person in a cool place; shoot me an e-mail! =)

Finally, my New Year’s Eve greetings from Red Square, Moscow as on my way to the Alps~img_0763Let 2017 be a magical year for all!



A brief recollection of my 2016

December 10, 2016

When the crisp breeze replaced roast chestnut smell on the streets of Shanghai, I realized another year has slipped away…

So I’ve been in this management consulting gig for approximately 1.5 years now. Apologize that I have not been able to post here, there is a number of reasons for that…

Anyways, since this is likely the only full-year I’ll spend in ‘serious business’, I guess giving a brief summary could be interesting to those in & out of academia. (if not, this would at least be interesting to a future me =P) Note that I have to avoid discussing all client and project work for McKinsey.

Here we go~

January: Did an interesting side project of producing a short video for work


February: Got excited about LUNAR (an industrial design Firm we McKinsey has acquired), started to review industrial sketching and went to San Francisco to meet LUNAR


March: My birthday, and through work I somehow got involved in the furniture design scene in Shanghai, made some designer friends and went to various shows

April: Spent the whole month and some more in Korea on a project, but hardly ever got out of the hotel/office (well, except for some shopping at the end)


May: Got on a random project in a 4th-tier city in the middle of China, decided to reward myself to a weekend in Paris (or shall I say Paris Disneyland? >.<)

June: Short business trip in Copenhagen =)


Then took a detour to Sweden and bumped into a very, very old friend (from middle school!)

July: On what’s probably my favorite project in McKinsey, confidential stuff aside, went on a team event with some hardcore partying in Macau (infact I got out from the party in a wheelchair, oh well, good crazy times =)

August: Company retreat in Tibet (we shipped over 500 consultants from 4 offices to Lhasa, largest group in Tibet ever; many passed out due to altitude upon arrival XD) Mostly just spent days staying in a fancy hotel with other consultants, I don’t quite get the point of being in Tibet for that, except for getting a room with huge bathroom, and I got bored so experimented with traditional Tibet costume

September: Project in Beijing, stayed in the trendy W hotel with a round bathtub & view from the office is the CCTV building; but in fact had a rather miserable few weeks

Then went to Singapore for a training

October: Went surfing in Indonesia (after all, I still find super basic hostels with an ocean view better than any glamours place)


Took a week off to go to Australia (completely an impulsive decision, woke up and bought a same-day flight =P) Generally tried various flying (skydiving, hang-gliding, gliding…) during the day and caught shows in the evenings

November: Went to Yangshuo again to enhance my rock climbing skills, finally comfortable with leading some lower grades, rewarded myself with that red quick-draw set I’ve been wanting for years =)

December: I finally visited the Shanghai Disneyland! (ok…I procrastinated for 5 months because I was too afraid it’s not going to live up to expectations…turns out was a good park overall =) Spotted most of my work from 2014! (anyone remember that statue from this 2014 post when I was working in Disney?)

Last page of sketchbook…

January 25, 2016

Recently I got very excited about industrial design, partially due to McKinsey’s acquisition of this absolutely awesome design studio named LUNAR in 2015. Of course I would need to understand our design language better, as a first step, I found some time this weekend and did some quick studies of LUNAR’s work.

Then I realized, it’s the last page of my sketchbook… time to start a new one, to draw more and draw better…But I certainly feel this old friend deserved some sort of farewell, flipping the pages is like looking back into life; Marks, lines, forms contains subtle expressions of the mood when creating them.

So I decided to put the whole book here at once, in reverse order, although some drawings have appeared in earlier posts, some are bad/unfinished/empty, but this is the book as is, representing bits and pieces of life (mostly life spent in various coffee houses in Shanghai).

Last page, words on the right was an experiment I did trying to see if markers can be used for caligraphy…obviously a master copy =P

IMG_0237Got all excited about the Force Awakens a few weeks ago

Bought the LEGO Classic and Mindstorm EV3 (still in the planning stage of my Turing machine project with EV3) I guess one can see traces of my trying to pick up industrial sketching starting here


And I did a study of the types of basic bricks…

 One day I was waiting in Starbucks and saw a new set of coffee things

Some very small auto doodles…love where they are going! Feels like 60s again.

  In Italy obsessed with those red ink drawings from 1500s…and some museum sketching

 Some more museum sketchings…

  And some plaza sketchings

 That day I forgot to bring paper to life drawing session in Florence…

Between Disney and Art Nouveau (I learned they are opening a fancy Remy restaurant in the Disney Wonders cruise)

 On Beauty and Beast

 On Alice in Wonderland

IMG_0226  On Pirates of the Caribbean

And some more pirates from the original (and best) Disneyland when the ride opened in the 70s

IMG_0227An unfinished train

 Master copies: Left-Chris Sanders; Right-Picasso

 More Chris Sanders and ransoms in a certain cafe

Somehow got interested in steampunk for a bit…the character is a copy from the wonderful book ‘walking my octopus’

Read newest ivory poaching reports…

  Some Shanghaiese architectures

  More Shanghai, a little field trip to the acient city of Fenghuang

More from the road

 Temples and hip coffee shops

In the park…

 Some imaginary plants

 More hip coffee shopsTanks form life

Antique cars

Cubes walking over to the horizon (first test page to get used to the material)


Opening spread of the book: hats


Well…that’s the book…cover to cover. Hope you enjoyed it =P

Year of 2015 – life in the strange world of consulting

December 31, 2015

I am surprised and touched after logging into here after more than a year tans found continuously high traffic volume…so much so that I’m thinking of picking this back up…no promises, though. =P

In any case, I have to say that it has been one of the most exciting periods of my life since I last posted (which, I guess, contributed to my not posting here).

Trajectory: t \in (\mbox{Dec 2014},\mbox{July 2015})


During this time, I:

  • left the prop painting job in Disneyland;
  • signed an offer with consulting firm McKinsey & Co.;
  • got my advanced scuba diving license and dived for a month in Thailand and the Philippines;
  • went to study representational painting in Florence;
  • climbed in China’s top rock climbing spot Yangshuo;
  • went to a ‘mini-MBA’ organized by McKinsey in Amsterdam;
  • backpacked from Holland to Poland;
  • got back to Princeton and defended my PhD thesis (yes, I finally did that);
  • volunteered in a wildlife conservation area in Kenya;
  • came back to Shanghai and settled down to start consulting;


The above took us from November 2014 to July 15th, 2015. Since July 15th I have been role-playing as a consultant in the strange corporate world. According to Fortune magazine, I am officially working for  “the most well-known, most secretive, most high-priced, most prestigious, most envied, most trusted, most disliked management consulting firm on earth,” it’s understandable that no client or content related to work can be mentioned here, so I’ll just talk about cool stuff I do outside of work, or life in general = )

Consultant costume:


Let’s pick up where we left off: I was painting props in Disneyland, moving from a sub-contractor to a general contractor, then hopefully to Imagineering (i.e. moving along the path described in ). All looking pretty promising in October 2014, until I hit a brick wall called internal politics >.< (won’t go into details) The result is that I decided to leave and find another way.

By some completely random sequence of events, I learned about this thing called consulting, in which I heard one can move around the world and across industries to solve problems. I decided that it sounds pretty cool and went for an interview, then two more follow-up interviews, then I signed the offer.

McKinsey turns out to be, umm, not much like what I imagined (for example, the ‘traveling around the world’ part have not happened yet; in fact I have been completely stuck in Shanghai since July >.<). But one thing I am very grateful for is that they let me pick when to join, given a window of one year since the contract. I picked July 2015 to have 7 months to complete many things I’ve always wanted to do, planned and budgeted with the aim of using up the whatever small amount of saving I had. You’ve already seen the results above ^_^

A few things I learned from the 7 months pre-consulting:

  • It’s important to make time and systematically complete things one wants to do, so that they don’t accumulate
  • Spending all of one’s money on cool stuff feels great, I’ll aim for doing this periodically
  • Going out and do things often unlocks next levels for pursuing the interest (a lot like video games)

A few things I learned from the 5 months with McKinsey:

  • Unlike in mathematics where I stare at a blank paper for weeks before putting down a stroke, people here start ‘doing’ things immediately; This takes a lot of pressure away and wish I have done things more like this in academia
  • The corporate world is not very creative, but strangely spending time on manual tasks satisfying and increases my urge to be creative during personal time
  • When things are not ideal (e.g. being stuck in China) one should: 1) try as hard as one can to move within the orginazation in the right direction 2) do things outside of work to completement what’s non-ideal

Plans going forward (laundry list, mostly for myself to check back when I get bored and can’t think of things to do) – it goes roughly from shorter term to longer and more involved items which I have less clue about how to proceed:

  • Do interesting side-projects on weekends, such as:
    • Design and build a working Turing machine with my recently obtained LEGO Mindstorm EV3 (I would need some help on this from my mathematical blog readers: Anyone have a good set of turing machine notes? I remember spending lots of time writing the states back in my undergrad days, need to refresh those memories but lost the notes)
    • Re-start improving my digital painting with my new iPad pro and apple pencil (so far I believe it’s going to totally substitute Wacom!)
    • Think about topology – sometimes I feel the need to read/think math in order to verify that I’m not becoming more stupid in this job =P 
  • Climb Kilimajaro in February 2016 – this was originally part of my African trip this June, but due to lost passport in Kenya it didn’t happen. Just booked the flight
  • Oil painting – follow up with connections built in Florence, possible next painting workshop in Rome, Venice, Paris or Sweden
  • Sculpture – would be great to start that at some point, ideally marble, in Florence or London
  • Flying – Go to Arizona and actually complete my pilot license (did’t get very far at all in Princeton due to weather and the fact that Airport is 30 min biking away from school >.<), can be done in a month with good weather and intensive flying
  • Diving – Need to do more boat diving in interesting spots, see whales, sharks and shipwreck
  • Rock climbing – Back to Yangshuo, follow up with connections built there with the rock climbing community, improve ratings, eventually move to large walls and climb El-Captain
  • Wildlife conservation – 1) Follow up with LUMO in Kenya, re-visit, try to fund my cattle program to prevent locals to kill Lions  2) Good NGOs to try connecting with: WildAid in China for stopping Ivory and shark fin demand and Save the Elephant in Kenya  3) explore how might I get McKinsey involved
  • Industrial design – Potentially get a masters in that, either ArtCenter or Europe
  • Disney (Imagineering or PIXAR) – Remains a huge void and question mark I need to figure out… Have to get there at some point in life… Ideally through building a portfolio and reputation in  visual development

Hopefully that would keep me occupied for some time =P

I would aim to write here whenever there is any thing intellectually interesting going on. Let me know if you have any suggestions for bits of things I can think of/write about!

Best regards,

“Dr.” Conan Wu

Recent progress on the Imagineering quest

August 17, 2014

As we all know, I set my life goal of doing this Disney/Pixar stuff 2.5 years ago during that epic vacation in Walt Disney World. So after learning that they are building a park in my hometown Shanghai, I had this idea of coming back and start from there which got realized half a year ago after I had an injure trying to renovate the studio space in LA. (well…that’s another story >.< I one just needs to know that I'm fully recovered and am jumping around again, as you can tell from below)

After arrival I didn't really know how to proceed, so I spent some time catching up on academic drawings:

2014-03-25 21.31.53

2014-04-02 18.21.28

2014-03-30 21.32.57

and paintings:

2014-04-06 22.41.45

2014-04-12 21.57.35

Then I spent most of the Spring wandering the roads of Shanghai:


2014-04-15 01.32.02

2014-04-15 01.29.17

Hanging out in coffee shops:


and did some domestic traveling:



Finally in May, I was recovered and ready to head to the park (ok…plus start road biking…):

2014-05-12 18.24.33

I started off by finding a random job painting props (such as rocks, wood, and sculptures that are made of fiberglass) with a local sub-sub-subcontractor under Disney. This was quite an adventure, I got to live in the construction worker’s dorm inside the park and worked 6 days a week, it was FUN:

2014-05-22 07.39.27

(Note: everything in the painting shop are top secrets and cannot be taken photos of =P)

So I went to paint items in the Shanghai version of the classical Pirates of the Caribbean ride.

After a few weeks they somehow discovered I’m pretty good at this and decided to make me an art supervisor (oh, and I received a salary multiplier of 2.5). After another few weeks I got an offer from a not-too-sub contractor I dealt with on the site (with another salary multiplier of 2.5, now let’s see, what will happen if one keeps this exponential growth… =P Ok just joking…But I decided to turn them down and went for a job interview with Disney (finally!)

That’s pretty much where I am right now. Hopefully I’ll soon be calling myself an Imagineer, if not I’ll just keep trying until that happens =P. Wish me luck!

A posthumous paper: Random Methods in 3-manifolds

January 29, 2014

Hi all, life has taken some dramatic turns since I last posted: I did not get to teach topology in Art Center, so I took a different approach in job-finding and ended up making pastries in a local bakery overnight (11pm-7am) for two months until some (very complicated) personal affairs arise, due to an irreversible influence from certain individual, I decided that I should forget about applied/digital art and just paint classically instead; So I think I’ll start by become a painter who also works in random jobs (such as dishwashing). Oh, and I’m getting married sometime this year~

Ok, enough random things about me…I’m here to give a little teaser of a posthumous paper of mine in mathematics before it goes on the ArXiv, which I finally received a complete draft from my wonderful co-authors Alex Lubotzky and Joseph Maher. I hope this summary from my point of view could serve as my tribute to this interesting piece of work.


Let’s start with an ‘unrelated’ piece of history: Once upon a time, many standard or number-theoretically significant graphs (such as Ramanujan graphs, as I might have mentioned when talking about expanders before) were not known to exist, then there comes Paul Erdos, after whom they were known to exist and is literally ‘everywhere’, but we still didn’t manage to ‘catch’ any particular one of them, at least not for another twenty something years. So we know that in mathematics it’s sometimes easier to prove ‘most’ objects satisfy some properties than to pick one out, for establishing existence.

While in Israel, Alex presented to me this fascinating idea he had about proving existential results in topology using random methods:

(crush-course for those who don’t know topology)
1. All closed 3-manifolds can be written as two many-holed solid donuts glued together along their surface.


It’s easy to believe all smooth manifolds can be chopped into tiny tetrahedrons.
Take the triangulation -> take it’s 1-skeleton -> take a small neighborhood of the 1-skeleton This is a neighborhood of a graph, hence a handlebody. Now what’s the complement of that 1-skeleton neighborhood?
…also a neighborhood of a graph~! …hence another handlebody…(note that the two donuts must have same number of holes since the gluing is clearly a homeomorphism)

This is called a Heegaard splitting of the 3-manifold.

OK, we now know all 3-manifolds arise from such gluings when we use some (probably large) genus donuts. We can fix a genus and ask what are all possible gluings occurring in that genus.

Now two homotopic homeomorphisms clearly give the same 3-manifold, hence we only need to consider the homotopy classes of surface homeomorphisms, which forms the infamous mapping class group of the surface.

To summarize, we have in hand a discrete group in hand whose elements parametrize (with repetitions) all 3-manifolds given by gluing donuts of that genus.

What can we do on infinite discrete groups? Well, actually many things, but in particular we may put a probability measure on its generators and random walk!

Now we can ask all sorts of things regarding what happens after walking for a long time, such as:

After taking N steps,

How likely are we landing on a gluing map that gives a hyperbolic 3-manifold? (property 1)

How likely is the resulting gluing a Heegaard splitting with minimal genus? (property 2)

Topologists might have the intuition that ‘most’ 3-manifold should be hyperbolic and guess that ‘most’ Heegaard splittings are minimal genus; if so, I’m glad to tell you that…your intuition is correct!

At this point I would like to sidetrack a little bit and point out that, many of those traditional combinatorics/number theory/graph theory random method arguments goes like this: take a smartly chosen class of objects, put a carefully constructed probability distribution on it, and Boom~ ‘most’ (asymptotic probability one) many objects are our desired objects! so they exist!

Now of course we already know that hyperbolic 3-manifolds exist in every Heegaard genus…but we figured that this random implying existence method can be pushed much further than merely most imply exist. After all, it is a group which we are walking on~

First of all, property 1 and 2 are not only generic in the sense of having asymptotic probability 1, but actually the set that does not satisfy property 1 and 2 decreases exponentially, i.e. the exceptional set for both properties have size O(e^{-cN}) for some c>0 after $N$ steps.

The above leads one to think of the possibility of estimating decay rates of various 3-manifold properties under this random walk and thus drew conclusions such as “if property A decays exponentially, property B decays polynomially but not faster, then even if ‘most’ objects satisfy neither A now B, we can still conclude that there exist objects that’s B but not A.

Now this is all very nice but useless unless we can find and prove some manifold properties with interesting, non-exponential decay rates. For that we may take advantage of the group structure: homomorphisms between groups project random walks, hence invariants that take value in a (hopefully simpler) group would have level sets in the mapping class group having decay rates given by return probabilities of the projected Markov processes on the simpler group, which can be polynomial.

In that spirit, we apply our random method to find hyperbolic genus g homology 3-spheres with particular Casson invariants. (I will not get into Casson invariants here, let’s just keep in mind that it’s a classical integer invariant of homology 3-spheres, it is generally pretty hard to construct non-trivial examples with particular Casson invariants) Namely we prove:

Theorem: For any integers g, n with g \leq 2, there exists hyperbolic homology 3-spheres with Heegaard genus g and Casson invariant n.

The subgroup of the mapping class group consisting of all elements that give raise to homology 3-spheres is called the Torelli group. So Casson invariant assigns integers to Torelli group elements. With some work one can show that this is somewhat close to a homomorphism to \mathbb{Z}. More precisely, it’s a homomorphism on what’s called the Johnson kernel, which is a normal subgroup of the Torelli.

Unfortunately little is known about the Johnson kernel, in particular we don’t know if it’s finitely generated. But for our purpose we can pick out three elements from the group and consider the subgroup H generated by them. (two Pseudo Anosov elements with distinct stable and unstable laminations, plus a third element that guarantees Casson homomorphism is surjective.)

Now the two Pseudo Anosov elements makes the argument of exponential decay carry through (i.e. property 1 and 2 still holds outside of an exponentially small set in H); The Casson invariant is a homomorphism hence projects the random walk in H to a Markov process on \mathbb{Z}. Asymptotically such process hits returns to 0 with probability \sim 1/N^2; making all integers achieved with a quadratic asymptotic decay rate. i.e. all level sets of the Casson homomorphism has decays only quadratically in H.

From the above we can conclude there are manifolds with any Casson invariant which falls outside both the exception set of hyperbolic and Heegaard genus g.

Some slightly more recent results =P:

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Progress update: Painting, drawing etc. 11/25/2013

November 25, 2013

A life painting~ (this time I got a longer pose)

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Newest unfinished master copy:
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Rembrandt head, finished (compare to the last version in this post)
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Some plants from the LA Arboretum:

2 (7 of 7)

2 (6 of 7)

2 (4 of 7)

2 (3 of 7)

Turning sphere inside out~ (well…still gotta have some nerdy stuff, right?)

2 (5 of 7)

Composition: Kids found an abandoned boat and stolen stuff from home to decorate it as a pirate ship =P

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Mathematics, with Imagination – a course proposal

November 6, 2013

Since this summer I have been secretly working on pitching a course, in fact, something that has been growing in my mind throughout my time in mathematics. This is a course that presents mathematics, especially geometry and topology, as artistic inspiration rather than a practical tool. I finally decided to write about it in this post. I would be more than happy to hear your response and suggestions on both the course itself and its topic selections!

The current state of this is that I have finally found the perfect home for the course: The Art Center in Pasadena. After a few month of poking around I made it into the administration and attracted quite some interest from various people. Two days ago I was asked to speak about curvature for half an hour in the faculty meeting. So I might start teaching there this spring and will find out soon~ Wish me luck!


`The best mathematics uses the whole mind, embraces human sensibility, and is not at all limited to the small portion of our brains that calculates and manipulates symbols. Through pursuing beauty we find truth, and where we find truth we discover incredible beauty.’ — William Thurston

Much like art, mathematics is all about idealizing and simplifying the real world. I have always believed that, when exposed to the right set of topics, artists in all disciplines can get inspiration from mathematics. The objective of this course is to present a set of visually interesting topics from a wide range of advanced mathematics in a fashion that would be appealing to artistically creative minds.


The first half of the semester will consist of lectures on one topic per week, the topics are typically at advanced undergraduate to graduate level, but presented in a way that’s tailored especially for artists (i.e. lots of imagination required, absolutely no numbers and formulas). Some rigorous proofs will be presented followed by discussions. Every week there will be some interesting homework problems related to the topic in order to solidify student’s understanding, as well as some more creative homework that helps generating ideas for art inspired by the topic. Starting from week 7 we will develop a final project in which students can pick one of the thumbnail ideas from the pervious weeks and develop it into a project. I will present some inspirational projects (such as hyperbolic geometry inspired fashion design, 3D printings, fine art sculptures, digital fractal art, screen prints and film/animation projects).

In the eighth week we will talk about individual projects, make sure they are scientifically sound and meaningful as well as resolving practical difficulties. The remaining part of the semester will consist of lecture and discussion sessions on some more abstract topics which would serve as exposition rather than demanding precise understanding, no problem sets will be given on those. We will touch base on the progress of projects at the end of each class. Many potential in-class activities could be included, for example one could spend a class having students collaborate on building a human sized four-dimensional polytope out of the geometric construction tool `Zome’, or play teamwork games on knots and links. The last class will be a presentation and review of projects.

Weekly plan

Week 1: Surfaces from an ant’s perspective

Week 2: From peeling orange to metric structures

Week 3: Knots and links

Week 4: Fractals, natural and man-made

Week 5: Geometry of paper folding

Week 6: Pathological spaces

Week 7: Inspirations for final project

Week 8: Project planning

Week 9: Cubes and polytopes, in all dimensions

Week 10: Collaborative Zome tool construction

Week 11: Shapes of Universes

Week 12: Real estate in hyperbolic space

Week 13: Infinity and beyond

Week 14: Project presentation and discussion


This course will develop student’s skills in imagining abstract spaces and visual problem-solving as well as giving them a brief tour into the fascinating world of contemporary mathematics. The final project would ideally serve as a demonstration of student’s ability to integrate sophisticated scientific ideas into a piece of beautiful artwork which we will submit to the annual Joint Mathematics Meeting art exhibit, and the International Bridges Conference which links mathematics with Music, Art, Architecture and Culture. A successful project would make an excellent portfolio piece.

Progress in painting: 08/172013-09/17/2013

September 17, 2013

Hi people~ apologies for not updating, I was busy trying to end things with mathematics and Princeton + figuring out how to remain in the country legally + moving to the arts district in downtown LA + went to San Francisco and had a wonderful visit to Pixar (Many thanks to Matthias Goerner who successfully moved from Topology to Pixar, for getting me into their top security campus and showing me around!) + helping some awesome people in marketing their private art classes.

Anyways, I guess all that is taken care of now and I have started a new semester last week! I decided to spend this fall to systematically study traditional drawing and painting before continuing the more hip, animation related stuff, I think so far it’s going pretty well ^^

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(A Rembrandt master copy I’m currently working on, check out ‘Johannes Wtenbogaert’ for original)

This post will be some updates on paintings progress I made over the last month…

So basically I started off dealing only with light and dark:

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After a couple wood blocks we moved on to head casts:

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I moved into my small room with a wonderful view of downtown LA, did a quick black and white study looking out from my window:

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Then instead of completely monotone we added a couple colors (burnt ember, ultramarine blue and white) to establish cool and warm:

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Used the same colors to do a master copy: (this was quite interesting, I was given a black and white photocopy of the painting and was told the color of things and the time of the day, the task is to estimate the color temperature used in the original painting)

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Move on the full color, a thumbnail study of a piece by John Singer Sargent:

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Went to a model session and played with some saturated color:

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Tried a classical technique that lays down the paint and wipe out the highlights:

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OK~ so this bought us to roughly where I am in painting right now~ I’m back into focusing on image-making again and hopefully I’ll be back here soon to talk about drawing and some other classes and projects I’ve been working on. ^^