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.

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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.

WHY?

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:

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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!

Overview

`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.

Structure

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

Outcome

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. ^^


Weekly updates: Zoo sketches, acrylic, figure eight snake and Alexander horned deer

July 24, 2013

Hi everyone, so I’ll probably be posting my progress in learning how to draw and paint here every week, hope they are at least entertaining to look at~ =P

Some head study from photos, the rest I did live in the zoo~ (well…animals are moving, but I think I’m getting used to that ^^)

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I was not too happy with the rendering of animals, hence I started copying style from some cool people:

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And just a reminder that tigers probably going to extinct soon~

 

 

2 (4 of 5)

Then we had to re-design animals as characters =P

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Real va Stylized. (Figure eight knot snake!)

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Then I spent a whole lot of tome figuring out how to put the Alexander horned sphere as horns >.<

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This week I tried using acrylic (for the first time besides that one i high school^^), it was FUN!

 

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An oil again…I tried using less crazy colors to achieve a more classical feeing…doesn’t quite work yet >.<

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Reviewed Bambi for the first time since I was 5 years old! It was AMAZING…did thirty Thumpers~

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Back to the drawing board — Month #1 (and a bit more)

July 17, 2013

So this is the first post after my blog (and life) transition! Just wish to record some progress in my concept art studies after not drawing and painting for 6 years. Feel so good that sometimes I can’t believe this is going to be what I do full-time! (Why didn’t I start this 1.5 years ago?!)

In any case, the goal of this blog is to have bad paintings and drawings at the beginning and hopefully show improvements over time. My ultimate hope is for this to envolve into a log that shows dreams can be achieved no matter how distant it is from where you are.

Anyways, here are some selected pages of what I’ve been doing over the last month or so =P

First, some animal thumbnails~ (went to the zoo last week, but those are mostly from photo as I’m still not very comfortable drawing while standing in the crowd >.<)

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Gorillas are super fun to draw!
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Bears (and some polar bears)

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African collection

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Birds

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This is me trying to figure out how does animal legs bend…

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Copied animal poses before the zoo, to get a better idea of how to capture the gesture.

Now onto human figure invention~

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Those are given a background and invent a character to put into the scene.

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This was super fun to draw~ I am happy to see that now I can draw figures without finding a photo reference ^^

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Take a pose and change the character posing. (we actually had a model doing in and was asked to change him into ‘superhero’, ‘sexy women’ etc =P)

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I think I really like drawing fat guys…

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Hand expressions, roughly a half of them copied from master animators from Disney, the other half drawn according to my own (left) hand =P

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Designing a ‘space pirate’.

Okay, below are some absolutely uncategorized random stuff I just decided to throw in:
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A little layout. (obviously inspired by Ratatouille)

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Some plants (drawn on spot in Caltech)

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Balls, painted in Photoshop

Okay…onto…oil painting! I have to say that I have never done oil in my life…(and I only ever painted a couple times back in high school >.<) But it’s so exciting to start!!!

I started off by going to those life model sessions where there is a model posing and everyone drops in and paints:

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First-time still life.

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Some thumbnail landscapes from my travel photos.

Anyways, painting is something I am most looking forward to improve! Just started to systematically learning it last week, so stay turned!

A sketch of a piece I really wanted to paint once I get better at it:

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The beginning of a long voyage — from Princeton to Pixar

July 16, 2013

Dear blog readers:

I am both amazed and touched to see that the blog kept getting traffic long after I stopped using it, hopefully you have found something here interesting or useful. The intension of this post is to announce I decided to start blogging here again. But in a completely different context, to go along with my new life beyond mathematics. Along the way I’d like to give my choice a little explanation (mainly towards my dear mathematical readers) and perhaps give my two cents on life and dreams.

As mentioned in this post 1.5 years ago, I decided to restart my career with the end goal of working at Pixar or Disney as a concept artist.

Definition: Concept art is a form of illustration where the main goal is to convey a visual representation of a design, idea, and/or mood for use in films, video games, animation, or comic books before it is put into the final product.

Roughly speaking, for life action movies one reads the script and selects locations, props and actors; in animation, however, most of the times the world which the story take place doesn’t exist! This means that every single detail needs to be designed from imagination: from a chair, a lamp to whole islands, cities and characters. A concept artist is the person who reads the story and designs the world to stage it!

To illustrate the idea let’s see a couple of examples:

Examples:
As we know cars (2) was a movie that takes place in an imaginary universe with residents being cars instead of people and animals; Here’s some of Pixar’s brilliant designs:

This was one of the paintings hanging in the palace, note the UK coat of arms logo on the floor~

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All landmark architectures around the world are re-designed to have car elements in them.

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And here’s what a busy London street looks like (I just love the way buses and taxis look so… British! Note that London double deck buses actually have that ‘sigle lens glass’ on them =P)

London_Bus_Scene_SimpleNew

Anyways, hopefully I have given some ideas on what animation concept design is about~ Now onto some properties of a concept designer:

Propositions: A concept artist is typically:
– Super creative
– Has childlike imagination
– Thinks and communicates visually
– Draws and paints well

Hmm…I am totally convinced that this is what I am made for more than anything else! In fact, this is what I wanted way before I got into mathematics, but like many people, childhood dreams sometimes get buried and forgotten in the attic. Luckily I found it back that winter in Disney World and will never, ever, lose it again.

Once I figured out what I wanted the rest is very simple: just do everything possible to get there! In this case I am perhaps at the antiportal point to where I wanted to be: there is just no obvious route from mathematics to animation design, which means I get to create my own path! So I started with making some route maps and took some steps to look into how they work–

Route #1:
Hang around the Princeton computer science department and specialize in computer graphics
–> get a PhD in computer science
–> get into the software research group at Pixar
–> try to get to know people in the art and design group
–> go from there

Current status: So I started by taking graduate computer graphics and algorithms course at Princeton in spring 2012, apparently I’m not bad at them (especially algorithms) and that we Princeton CS department actually has good connections to Pixar. However, after chasing down Tony DeRose in Stony Brook when he was giving a public lecture, I found that unfortunately I do need to first have a PhD in CS and that the software group does not really talk to the design group that much…Hence I estimated this route is not the most efficient.

Route #2:
Get my PhD in mathematics
–> leave academia and work in finance for a few years
–> save enough money
–> go to Art Center, major in entertainment design.
–> graduate and get into Pixar

Current status: The major problem with this plan is of course it involved doing relatively uninteresting and unrelated stuff for a few years. Plus although I’m sure I would have love to attend art school, getting another bachelor’s degree is time consuming. Hence after some research on how I might go about getting a job in finance, I decided to move onto the third plan and perhaps come back to this if nothing seem more efficient.

Route #3: (and this is what I am doing right now!)
Move to California
–> take courses from independent studios (such as concept design academy)
–> become technically at least as good as people who went through art school
–> build a portfolio
–> apply to Pixar directly

Current status: Deep down I have always known this is actually the best and fastest way to go, but I didn’t go for it till this summer because in this case the last six years I spent in mathematics is officially absolutely useless. But now I figured that trying to utilize them would only result in making the process taking longer. Looking into this, the choice is actually very simple: I should have no pity in completely starting over and not look back. ‘Being good at something should only work towards one’s advantage’ sounds like a tautology, but in reality it’s striking to see how often abilities and past accomplishments become burdens that prevents people from chasing dreams and, eventually, prevents them from getting to where they wanted to be.

So here I am in Pasadena since June, I’m thrilled to say that I have never felt more alive since I finished undergrad! Not only that I got to draw and paint all the time, seeing improvements on a daily basis but also I have finally found the group I belong to by being around truly creative people! It will take some time, but I know this is what I want to do and I will get there!

In any case, if Mike Wazowski ended up as a scarer, what am I concerned about? Inspired by the ending of Monsters University, current plan: I’ll work on getting as close as I can to Picar till 2015, if it hasn’t worked out by then I’m going back to Shanghai and start by sweeping the floor at the Shanghai Disneyland.

From now on I will record my progress as an concept artist in training here. Hence if you are here to read mathematics, please unsubscribe…and wish me luck! ^^

With hope, onward
-C


On Tao’s talk and the 3-dimensional Hilbert-Smith conjecture

May 6, 2012

Last Wednesday Terry Tao briefly dropped by our little town and gave a colloquium. Surprisingly this is only the second time I hear him talking (the first one goes back to undergrad years in Toronto, he talked about arithmetic progressions of primes, unfortunately it came before I learned anything [such as those posts] about Szemeredi’s theorem). Thanks to the existence of blogs, feels like I knew him much better than that!

This time he talked about Hilbert’s 5th problem, Gromov’s polynomial growth theorem for discrete groups and their (Breuillard-Green-Tao) recently proved more general analogy of Gromov’s theorem for approximate groups. Since there’s no point for me to write 2nd-handed blog post while people can just read his own posts on this, I’ll just record a few points I personally found interesting (as a complete outsider) and moving on to state the more general Hilbert-Smith conjecture, very recently solved for 3-manifolds by John Pardon (who now graduated from Princeton and became a 1-st year grad student at Stanford, also appeared in this earlier post when he gave solution to Gromov’s knot distortion problem).

Warning: As many of you know I never take notes during talks, hence this is almost purely based on my vague recollection of a talk half a week ago, inaccuracy and mistakes are more than possible.

All topological groups in this post are locally compact.

Let’s get to math~ As we all know, a Lie group is a smooth manifold with a group structure where the multiplication and inversion are smooth self-diffeomorphisms. i.e. the object has:

1. a topological structure
2. a smooth structure
3. a group structure

It’s not too hard to observe that given a Lie group, if we ‘forget’ the smooth structure and just see it as a topological group which is a (topological) manifold, then we can uniquely re-construct the smooth structure from the group structure. From my understanding, this is mainly because given any element in the topological group we can find a unique homomorphism of the group \mathbb{R} into the manifold, sending 0 to identity and 1 to the element. resulting a class of curved through the identity, a.k.a the tangent space. Since the smooth structure is determined by the tangent space of the identity, all we need to know is how to ‘multiply’ two such parametrized curves.

The way to do that is to ‘zig-zag':

Pick a small \varepsilon, take the image of \varepsilon under the two homomorphisms, alternatingly multiplying them to obtain a sequence of points in the topological group. As \varepsilon \rightarrow 0 the sequence becomes denser and converges to a curve.

The above shows that given a Lie group to start with, the smooth structure is uniquely determined by the topological group structure. Knowing this leads to the natural question:

Hilbert’s fifth problem: Is it true that any topological group which are (topological) manifolds admits a smooth structure compatible with group operations?

Side note: I had a little post-colloquium discussion with our fellow grad student Sam Lewallen, he asked:

Question: Is it possible for the same topological manifold to have two different Lie group structures where the induced smooth structures are different?

Note that neither the above nor Hilbert’s fifth problem shows such thing is impossible, since they both start with the phase ‘given a topological group’. My *guess* is this should be possible (so please let me know if you know the answer!) The first attempt might be trying to generate an exotic \mathbb{R}^4 from Lie group. Since the 3-dimensional Heisenberg group induces the standard (and unique) smooth structure on \mathbb{R}^3, I guess the 4-dimensional Heisenberg group won’t be exotic.

Anyways, so the Hilbert 5th problem was famously solved in the 50s by Montgomery-Zippin and Gleason, using set-theoretical methods (i.e. ultrafilters).

Gromov comes in later on and made the brilliant connection between (infinite) discrete groups and Lie groups. i.e. one see a discrete group as a metric space with word metric, ‘zoom out’ the space and produce a sequence of metric spaces, take the limit (Gromov-Hausdorff limit) and obtain a ‘continuous’ space. (which is ‘almost’ a Lie group in the sense that it’s an inverse limit of Lie groups.)

Hence he was able to adapt the machinery of Montgomery-Zippin to prove things about discrete groups:

Theorem: (Gromov) Any group with polynomial growth is virtually nilpotent.

Side note: I learned about this through the very detailed and well-presented course by Dave Gabai. (I thought I must have blogged about this, turns out I haven’t…)

The beauty of the theorem is (in my opinion) that we are given any discrete group, and all that’s known is how large the balls are (in fact, not even that, we know how large the large balls grow), yet the conclusion is all about the algebraic structure of the group. To learn more about Gromov’s work, see his paper. Although unrelated to the rest of this post, I shall also mention Bruce Kleiner’s paper where he proved Gromov’s theorem without using Hilbert’s 5th problem, instead he used space of harmonic maps on graphs.

Now we finally comes to a point of briefly mentioning the work of Tao et.al.! So they adopted Gromov’s methods of limiting and ‘ultra-filtering’ to apply to stuff that’s not even a whole discrete group: Since Gromov’s technique was to take the limit of a sequence of metric spaces which are zoomed out versions of balls in a group, but the Gromov-Hausdorff limit actually doesn’t care about the fact that those spaces are zoomed out from the same group, they may as well be just a family of subsets of groups with ‘bounded geometry’ of a certain kind.

Definition: An K-approximate group S is a (finite) subset of a group G where S\cdot S = \{ s_1 s_2 \ | \ s_1, s_2 \in S \} can be covered by K translates of S. i.e. there exists p_1, \cdots, p_K \in G where S \cdot S \subseteq \cup_{i=1}^k p_i \cdot S.

We shall be particularly interested in sequence of larger and larger sets (in cardinality) that are K-approximate groups with fixed K.

Examples:
Intervals [-N, N] \subseteq \mathbb{Z} are 2-approximate groups.

Balls of arbitrarily large radius in \mathbb{Z}^n are C \times 2^n approximate groups.

Balls of arbitrarily large radius in the 3-dimensional Heisenberg group are C \times 2^4 approximate groups. (For more about metric space properties of the Heisenberg group, see this post)

Just as in Gromov’s theorem, they started with any approximate group (a special case being sequence of balls in a group of polynomial growth), and concluded that they are in fact always essentially balls in Nilpotent groups. More precisely:

Theorem: (Breuillard-Green-Tao) Any K-approximate group S in G is covered by C(K) many translates of subgroup G_0 < G where G_0 has a finite (depending only on K) index nilpotent normal subgroup N.

With this theorem they were able to re-prove (see p71 of their paper) Cheeger-Colding’s result that

Theorem: Any closed n dimensional manifold with diameter 1 and Ricci curvature bounded below by a small negative number depending on n must have virtually nilpotent fundamental group.

Where Gromov’s theorem yields the same conclusion only for non-negative Ricci curvature.

Random thoughts:

1. Can Kleiner’s property T and harmonic maps machinery also be used to prove things about approximate groups?

2. The covering definition as we gave above in fact does not require approximate group S to be finite. Is there a Lie group version of the approximate groups? (i.e. we may take compact subsets of a Lie group where the self-product can be covered by K many translates of the set.) I wonder what conclusions can we expect for a family of non-discrete approximate groups.

As promised, I shall say a few words about the Hilbert-Smith conjecture and drop a note on the recent proof of it’s 3-dimensional case by Pardon.

From the solution of Hilbert’s fifth problem we know that any topological group that is a n-manifold is automatically equipped with a smooth structure compatible with group operations. What if we don’t know it’s a manifold? Well, of course then they don’t have to be a Lie group, for example the p-adic integer group \mathbb{Z}_p is homeomorphic to a Cantor set hence is not a Lie group. Hence it makes more sense to ask:

Hilbert-Smith conjecture: Any topological group acting faithfully on a connected n-manifold is a Lie group.

Recall an action is faithful if the homomorphism \varphi: G \rightarrow homeo(M) is injective.

As mentioned in Tao’s post, in fact \mathbb{Z}_p is the only possible bad case! i.e. it is sufficient to prove

Conjecture: \mathbb{Z}_p cannot act faithfully on a finite dimensional connected manifold.

The exciting new result of Pardon is that by adapting 3-manifold techniques (finding incompressible surfaces and induce homomorphism to mapping class groups) he was able to show:

Theorem: (Pardon ’12) There is no faithful action of \mathbb{Z}_p on any connected 3-manifolds.

And hence induce the Hilbert-Smith conjecture for dimension 3.

Discovering this result a few days ago has been quite exciting, I would hope to find time reading and blogging about that in more detail soon.


A train track on twice punctured torus

April 22, 2012

This is a non-technical post about how I started off trying to prove a lemma and ended up painting this:

One of my favorite books of all time is Thurston‘s ‘Geometry and Topology of 3-manifolds‘ (and I just can’t resist to add here, Thurston, who happen to be my academic grandfather, is in my taste simply the coolest mathematician on earth!) Anyways, for those of you who aren’t topologists, the book is online and I have also blogged about bits and parts of it in some old posts such as this one.

I still vividly remember the time I got my hands on that book for the first time (in fact I had the rare privilege of reading it from an original physical copy of this never-actually-published book, it was a copy on Amie‘s bookshelf, which she ‘robbed’ from Benson Farb, who got it from being a student of Thurston’s here at Princeton years ago). Anyways, the book was darn exciting and inspiring; not only in its wonderful rich mathematical content but also in its humorous, unserious attitude — the book is, in my opinion, not an general-audience expository book, but yet it reads as if one is playing around just to find out how things work, much like what kids do.

To give a taste of what I’m talking about, one of the tiny details which totally caught my heart is this page (I can’t help smiling each time when flipping through the book and seeing the page, and oh it still haunts me >.<):

This was from the chapter about Kleinian groups, when the term ‘train-track’ was first defined, he drew this image of a train(!) on moving on the train tracks, even have smoke steaming out of the engine:

To me such things are simply hilarious (in the most delightful way).

Many years passed and I actually got a bit more into this lamination and train track business. When Dave asked me to ‘draw your favorite maximal train track and test your tube lemma for non-uniquely ergodic laminations’ last week, I ended up drawing:

Here it is, a picture of my favorite maximal train track, on the twice punctured torus~! (Click for larger image)

Indeed, the train is coming with steam~

Since we are at it, let me say a few words about what train tracks are and what they are good for:

A train track (on a surface) is, just as one might expect, a bunch of branches (line segments) with ‘switches’, i.e. whenever multiple branches meet, they must all be tangent at the intersecting point, with at least one branch in each of the two directions. By slightly moving the switches along the track it’s easy to see that generic train track has only switches with one branch on one side and two branches on the other.

On a hyperbolic surface S_{g,p}, a train track is maximal if its completementry region is a disjoint union of triangles and once punctured monogons. i.e. if we try to add more branches to a maximal track, the new branch will be redundant in the sense that it’s merely a translate of some existing branch.

As briefly mentioned in this post, train tracks give natural coordinate system for laminations just like counting how many times a closed geodesic intersect a pair of pants decomposition. To be slightly more precise, any lamination can be pushed into some maximal train track (although not unique), once it’s in the track, any laminations that’s Hausdorff close to it can be pushed into the same track. Hence given a maximal train track, the set of all measured laminations carried by the train track form an open set in the lamination space, (with some work) we can see that as measured lamination they are uniquely determined by the transversal measure at each branch of the track. Hence giving a coordinate system on \mathcal{ML})(S).

Different maximal tracks are of course them pasted together along non-maximal tracks which parametrize a subspace of \mathcal{ML}(S) of lower dimension.

To know more about train tracks and laminations, I highly recommend going through the second part of Chapter 8 of Thurston’s book. I also mentioned them for giving coordinate system on the measured lamination space in the last post.

In any case I shall stop getting into the topology now, otherwise it may seem like the post is here to give exposition to the subject while it’s actually here to remind myself of never losing the Thurston type childlike wonder and imagination (which I found strikingly larking in contemporary practice of mathematics).


Filling and unfilling measured laminations

April 10, 2012

(images are gradually being inserted ~)

I’m temporarily back into mathematics to (try) finish up some stuff about laminations. While I’m on this, I figured maybe sorting out some very basic (and cool) things in a little post here would be a good idea. Browsing through the blog I also realized that as a student of Dave’s I have been writing surprisingly few posts related to what we do. (Don’t worry, like all other posts in this blog, I’ll only put in stuff anyone can read and hopefully won’t be bored reading :-P)

Here we go. As mentioned in this previous post, my wonderful advisor has proved that the ending lamination space is connected and locally connected (see Gabai’08).

Definition: Let S_{g,p} be a hyperbolic surface of genus g and p punctures. A (geodesic) lamination L \subseteq S is a closed set that can be written as a disjoint union of geodesics. i.e. L = \sqcup_{\alpha \in I} \gamma_\alpha where each \gamma_\alpha is a (not necessary closed) geodesic, \gamma is called a leaf of L.

Let’s try to think of some examples:

i) One simple closed geodesic

ii) A set of disjoint simple closed geodesics

iii) A non-closed geodesic spirals onto two closed ones

iV) Closure of a single simple geodesic where transversal cross-sections are Cantor-sets

An ending lamination is a lamination where
a) the completement S \backslash L is a disjoint union of discs and once punctured discs (filling)
b) all leaves are dense in L. (minimal)

Exercise: example i) satisfies b) and example iv) as shown satisfies both a) and b) hence is the only ending lamination.

It’s often more natural to look at measured laminations, for example as we have seen in the older post, measured laminations are natural generalizations of multi-curves and the space \mathcal{ML}(S) is homeomorphic to \mathbb{R}^{6g-6+2p} (Thurston) with very natural coordinate charts (given by train-tracks).

Obviously not all measured laminations are supported on ending laminations (e.g. example i) and ii) with atomic measure on the closed curves.) It is well known that if a lamination fully supports an invariant measure, then as long as the base lamination satisfies a), it automatically satisfies b) and hence is an ending lamination. This essentially follows from the fact that having a fully supported invariant measure and being not minimal implies the lamination is not connected and hence won’t be filling.

Exercise:Example iii) does not fully support invariant measures.

Scaling of the same measure won’t effect the base lamination, hence we may eliminate a dimension by quotient that out and consider the space of projective measured laminations \mathcal{PML}(S) \approx \mathbb{S}^{6g-7+2p}. Hence we may decompose measured laminations into filling and unfilling ones. i.e.

\mathcal{PML}(S) = \mathcal{FPML}(S) \sqcup \mathcal{UPML}(S)

where \mathcal{FPML}(S) projects to the ending laminations via the forgetting measure map \pi.

This decomposition of the standard sphere \mathbb{S}^{6g-7+2p} is mysterious and very curious in my opinion. To get a sense of this, let’s take a look at the following facts:

Fact 1: \mathcal{UPML} is a union of countably many disjoint hyper-discs (i.e. discs of co-dimension 1).

Well, if a measured lamination is unfilling, it must contain some simple closed geodesic as a leaf (or miss some simple closed geodesic). For each such geodesic C, there are two possible cases:

Case 1: C is non-separating. The set of measured laminations that missed C is precisely the set of projective measured laminations supported on S_{g-1, p+2}, hence homeomorphic to \mathbb{S}^{6g-13+2p+4} = \mathbb{S}^{(6g-7+2p)-2} we may take any such measured lamination, disjoint union with C, we may assign any ratio of wrights to C and the lamination. This corresponds to taking the cone of \mathbb{S}^{(6g-7+2p)-2} with vertex being the atomic measure on C. Yields a disc of dimension (6g-7+2p)-1.

Case 2: C is separating. Similarly, the set of measured laminations missing C is supported on two connected surfaces with total genus g and total punctures p+2.

To describe the set of projective measured laminations missing C, we first determine the ratio of measure between two connected components and then compute the set of laminations supported in each component. i.e. it’s homeomorphic to [0,1] \times \mathbb{S}^{d_1} \times \mathbb{S}^{d_2}/\sim where d_1+d_2 = 6g-2*7+2(p+2) = 6g-10+2p and (0, x_1, y) \sim (0, x_2, y) and (1, x, y_1) \sim (1, x, y_2).

Exercise: check this is a sphere. hint: if d_1 =d_2 = 1, we have:

Again we cone w.r.t. the atomic measure corresponding to C, get a hyper disc.

At this point you may think ‘AH! \mathcal{UPML} is only a countable union of hyper-discs! How complicated can it be?!’ Turns out it could be, and (unfortunately?) is, quite messy:

Fact 2: \mathcal{UPML} is dense in \mathcal{PML}.

This is easy to see since any filling lamination is minimal, hence all leaves are dense, we may simply take a long segment of some leaf where the beginning and end point are close together on some transversal, close up the segment by adding a small arc on the transversal, we get a simple closed geodesic that’s arbitrarily close to the filling lamination in \mathcal{PML}. Hence the set of simple closed curves with atomic measure are dense, obviously implying \mathcal{UPML} dense.

So how exactly does this decomposition look like? I found it very mysterious indeed. One way to look at this decomposition is: we know two \mathcal{UPML} discs can intersect if and only if their corresponding curved are disjoint. Hence in some sense the configuration captures the structure of the curve complex. Since we know the curve complex is connected, we may start from any disc, take all discs which intersect it, then take all discs intersecting one of the discs already in the set, etc.

We shall also note that all discs intersecting a given disc must pass through the point corresponding to the curve at the center. Hence the result will be some kind of fractal-ish intersecting discs:

(image)

Yet somehow it manages to ‘fill’ the whole sphere!

Hopefully I have convinced you via the above that countably many discs in a sphere can be complicated, not only in pathological examples but they appear in ‘real’ life! Anyways, with Dave’s wonderful guidance I’ve been looking into proving some stuff about this (in particular, topology of \mathcal{FPML}). Hopefully the mysteries would become a little clearer over time~!


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