Tag Archives: conference

## Minimum rank of graphs with loops

28 Jun

A few months ago I was one of two Distinguished Graduate Speakers at this awesome conference, Underrepresented Students in Topology and Algebra Research Symposium (USTARS).  The conference started in 2011, when a bunch of advanced grad students and postdocs (I don’t think anyone on the committee was a professor yet) decided that there needed to be a place for underrepresented students to see other people like them doing the math they did.  And now they’re all professors and still organizing this traveling conference!  So many people of color!  So many women!  So many homosexual people!  (Not so many trans people…) So many first-generation college students!  So great!  Too many slides!  (I pretty actively hate slide talks unless there are pictures that are impossible to make on a chalk board.)

Credit: Erik Insko for this picture of me giving a talk at USTARS 2016!

Anyway, I wanted to blog about some math I saw there, based on a talk by Chassidy Bozeman, a grad student at Iowa State and a fellow EDGE-r.  The talk is from a paper that resulted from her graduate research course, which Iowa offers to intermediate grad students and gets them started on research (hence the eight authors on that paper).  I thought it was fun and approachable!  And I have one crumpled page of two-month old notes, so we’ll see how this goes.

First, let’s remember what a graph is: a collection of vertices (dots) and edges (lines between dots).  A graph is simple if there’s at most one edge between any two vertices, and if no vertex has an edge to itself (aka a loop).  A loop graph allows loops, but still no multiple edges.

Left is a simple graph, right is a loop graph

You can associate an infinite family of symmetric matrices to a loop graph.  These square matrices will have the number of vertices of the graph as the number of columns and rows, and the entry $a_{ij}$ will be 0 if and only if there is no edge between the corresponding vertices and  in the graph.  The rest of the entries will just be real nonzero numbers.  This infinite family is useful for defining the minimum rank of a loop graph: it’s the minimum rank over all of this infinite family of matrices.  The rank of a matrix is a measure of how large it is.  For definition by example, $\left( \begin{array}{cc} 1 & 0 \\ 1 & 0 \end{array} \right)$ has rank 1, and $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$ has rank 2.

So the goal of Chassidy’s talk was to characterize all loop graphs whose minimum rank is the full size of the graph.  A couple more definitions before the theorem: a generalized cycle is a subgraph whose connected components are either a loop, a single edge, or a cycle.  It’s spanning if it touches every vertex in the graph.  Spanning generalized cycles don’t always exist!

Components of a generalized cycle: loop, edge, cycle

No spanning generalized cycle, womp womp

Theorem: A loop graph has minimum rank equal to its size if and only if there exists a unique spanning generalized cycle.

Quick nonexample: here’s a picture of a loop graph that has more than one spanning generalized cycle, so from the theorem we know it doesn’t have minimum rank four.

This is theorem 3.1 in their paper.  It’s also comically the end of my notes on this talk.  Here are a few facts they use to prove the theorem:

• If is an induced subgraph of G (that means the vertices of H are a subset of the vertices of G, and its edges are all the edges that appear in G within that subset), then it minimum rank is bounded above by the minimum rank of G.  Succinctly, $mr(H)\leq mr(G)$.
• If is a disjoint union of a bunch of connected components, the its minimum rank is the sum of the minimum ranks of those components.  $mr(G)=\sum mr(G_i)$.

Here’s a picture of my notes!  Read the actual paper if you want to know more!

## Combinatorics fun with complexes

5 Apr

Last weekend I spoke at the Graduate Student Combinatorics Conference in Clemson, in one of those parallel sessions, so there were two other speakers slotted for the same 25 minute slot.  But those two didn’t show up, so I ended up being a “plenary speaker”!  Everyone came to my talk, which is funny because I’m in geometric group theory, not combinatorics.  But I got a lot of compliments afterward even though I only got through 2/3 of the prepared material (oops that’s what happens when you don’t practice/finish the talk 2 hours before showtime).  Related: I don’t think this is humblebragging, I believe in real bragging- you’re awesome, shouldn’t you tell people about it?

Anyways, I enjoyed the first graduate talk I saw, by a grad student at KU named Bennet Goeckner [I still am not sure about etiquette for blogging but I think I’ll start using peoples’ names instead of just linking to their websites.  If they get mad at me for google hits then I’ll go back to just links.]  It was based on a paper he coauthored with three professors, so I thought I’d tell you a bit about it.

I didn’t know anything about combinatorics before going to this conference.  Like, I didn’t even know it was a field of study, despite posting about an open problem in it almost exactly two years ago.  So I was very happy that in this first talk he defined an abstract simplicial complex, which is a basic object of study in combinatorics (at least, it came up in a ton of later talks).  This is a subset D of the set of subsets of {1, 2, 3,…, n} that follows a rule: if a is in D, and b is a subset of a, then b is also in D.  Example: if the set {1,2,3} is in D, then that means all of these sets are in D too: {1},{2},{3},{1,2},{2,3},{1,3}.  Here’s why we call it simplicial: you can see all of this information if you draw a triangle (a.k.a. a 2-simplex)!

Big picture of triangle contains all the info on the right (+smile!)

Might as well be thorough: the set of subsets is called the power set, so the power set of {1,2,3} is what we listed above, plus the entire set and the empty set.  Note that in our example, we have 8 sets in the power set of a 3-element set.  This is not a coincidence!  In general, a power set will have $2^n$ elements, if the original set had elements.

Also, a n-simplex is the convex hull of (n+1) points in space: so 3 points in 2 space makes a triangle.  4 points in 3 space makes a tetrahedron, a.k.a. a 3-simplex.  5 points in 4 space makes a 4-simplex.

0,1,2, and 3-simplices

So whenever you have abstract objects that satisfy the simplicial condition, you can build an abstract simplicial complex out of them.

Here’s an example from the talk: X=<1 2 3, 2 3 4>.  Convince yourself that this is two triangles glued together along an edge labeled by 2 and 3.  We can build a lattice that encodes the subset information in a different way then the triangles picture.  I also love this example because the lattice looks like a heart, and I ❤ lattices!

Red lines indicate that the bottom set is contained as a subset in the top set

We say a simplicial complex is partitionable if you can cut it up into Boolean intervals that end in the top layer (but start at some layer).  The picture shows you the partitioning, and you can kind of tell by looking what a Boolean interval is (it describes the skeleton of a n-cube for some n).

This simplicial complex was partitionable… but my heart isn’t (it belongs to GGT)

It’s a little hard to show that things aren’t partitionable.  Here’s an example that probably showed up in the talk but I didn’t write it down: Y= <1 2 3, 3 4 5>.

Simplex and lattice, plus a happy person wearing a bowtie!

If we make one of the partitions that contains the bottom empty set and one of the top sets, we can’t make the rest into partitions that start at the top.

No way to partition remaining 5 sets

Their paper answers a conjecture from 1979, which asked if all Cohen-Macaulay simplicial complexes are partitionable (Cohen-Macaulay has something to do with homology, which we haven’t done here but my friend Jeremy has a mathy post about it).  They said haha no!  They took a counterexample in something else, called Ziegler’s Ball, chopped it up a little bit, glued a bunch of copies of it to itself, and built something surprisingly nice (with 16 vertices) that is not partitionable.  This has applications in commutative algebra, besides being a fun combinatorial thing.  The paper is relatively short and approachable if you’re a grad student looking for something fun to read, and they ask three questions for further research at the end!

26 Jan

This is, as far as I can tell, a great way to succeed in math academia.  But I’m only partway through the process and I’m not married to the idea of being in mathademia (I’m married to my spouse!).  Side story: some years back a professor was surprisingly denied tenure at a university where his wife’s family lived nearby.  He and his wife (and kids) then moved to a different country so he could be a tenured math professor there [it’s a good job].  I do not identify with this story.  I do identify with this: when I was a kid my mom would annually schlep us three kids to California from Minnesota for all two weeks of her allotted vacation time.

Outline of mathademia [I did not know all this til grad school]: you spend 4 years in undergrad somewhere that you want to go/live, then 5-6 years getting a Ph.D. somewhere you learn to enjoy going to/living (though no one cares how long you spend in grad school; I know one professor who took a six year break before going on), then 1-3 years doing a postdoc somewhere you often don’t want to live/go to, then possibly more of those postdocs until you get a tenure-track job somewhere that you better want to live.  After 6-7 years in your tenure track job, you either get tenure and will live there forever, or not and will have to go find somewhere else.  It’s all fairly civilized and organized.  Also, it gets more and more selective the further you go: I regularly hear about people applying to 80 jobs and getting 2-3 interviews and one offer.  Also, if you are romantically involved with someone else in academia, good luck with the two-body problem; almost every academic couple I know has had years of long-distance dating or marriage.

1. Go to college.  While there (4 year liberal arts school or a university that offers PhDs both seem fine), major in math and take as many math courses as you can.  If possible, take graduate courses in math as an undergraduate.  If none are available, ask to take a reading course with a professor or a graduate student [for instance, several schools have grad-undergrad student reading programs like UT, UMD, UCHicago, Rutgers, UConnBerkeley, MIT, Yale,  and more all the time].  For studying abroad, consider Budapest.  I did it and it was great!  I’m still in touch with friends from BSM and there are several in my field.
2. While in college, do research.  Ask a professor for advice on doing a senior thesis project.  During your junior and/or sophomore summer, DO AN REU and get a little money to go to math research camp for a few weeks and hopefully get a peek into the publishing world.
3. GO TO OFFICE HOURS.
4. Also, study for the Math GRE during that same summer if not earlier.  Even if you’re a math hotshot it’s a hard test and you should study for it.  It’s only offered three times a year.  Note: some grad programs don’t require this.  Take the regular GRE too but there’s probably no need to study for it.
5. Take the Putnam.  Do any extracurricular math activities you can.
6. If you followed steps 1 and 2 you should be able to get some strong letters of recommendation for applying to grad school.  So, apply to grad school.  There’s lots of advice out there about this.  Also, while you’re applying to grad school, APPLY TO THE NSF GRFP and possibly NDSEG as well.  Note: this step is a lot of work!

1. Maybe you took some time between undergrad and grad school.  In that case, you’d better brush up on your math!  Some people have success in looking at their old notes/books/homework.  You could also check out these books: All the Math You Missed, Mathematician’s Survival Guide.
2. If you’re a woman, consider doing this incredible summer brush up program that I did.  It’s great.
3. Take 15 minutes and do this exercise.  First, quickly write down a list of five things that you value (for me: food, family, learning, math, creativity/writing).  Then choose one of the things and take five full minutes (time yourself) to write about why you value that thing.  Do so for two other things too.  I did this my first semester of graduate school with my roommate based on an article I can’t find anymore, but roughly was this one.  The point is that affirming your values and sense of identity can help you cope with graduate school.
4. Figure out a physical exercise that you enjoy doing enough to actually do it.  If you’re already a gym rat or sports person, great.  If not, may I suggest Couch 2 5K.  Other things I’ve seen grad students do: rock climbing, cross country skiing, ballroom dancing, triathlons, marathons, swimming, yoga.  Physical activity really helps with the mental strain of first year of grad school.

1. Swim in your own lane.  First and second year it’s easy to compare to other students who are in your same classes/have better or worse preparation than you.  After that it gets a lot harder and a lot more tempting to compare with your cohort.  Try to avoid this.
2. Finish prelims/qualifying exams as quickly as possible, so you can focus on research = primary goal of graduate school.  This is how I studied for prelims: took the courses if applicable, downloaded all the previous exams (generally available on the department website), did one or two exams a week and checked answers with a study group once a week.  Study group = invaluable for problems that you aren’t sure how to solve.  Made a binder of all exams and all solutions (neatly written up) to reread at my leisure before exam.
3. Talk to other graduate students, especially older ones.  Many programs have a big sib/little sib program for first years.  Exploit this.  Grad students know so many things that aren’t on the internet (which professors are good to TA for, shortcuts between classroom buildings, who wants what for exams or reading courses, what seminars to attend, who to ask for help).
4. Publish if possible.  Find collaborators and publish results.  This is far easier said than done.
5. Read read read.  Trawl arxiv every day (takes a few minutes) just to prime some words into your head/see who’s who in your field.  Read.  Reading math is HARD.  You need to do a lot of it in mathademia so you should try to learn how you do it best.  I take extensive notes while reading, others don’t.
6. Give talks.  Every program has somewhere you can give a talk (1st/2nd year seminar, grad student colloquium, junior topic seminars) and you should give at least one before you start doing job talks.  It’s terrifying and then gets better the more you do.
7. Go to seminars!  Every professor who has given me advice has said this to me.  I am not great about it but I think I have sleep apnea/mild narcolepsy.  My advisor is always falling asleep in talks too, which makes me feel better, but then he wakes up and asks intelligent questions which makes me feel worse.
8. Take care of yourself.  Mental health days are legit.  Get physical exercise.  Eat well.  DO YOUR LAUNDRY, for all of our sakes.
9. Go to at least one conference so people know you.  Follow your advisor around and have her introduce you to people in your field.  Try to give some talks at conferences.
10. Applying to jobs is pretty much a full-time extremely stressful job.  So that fall semester of your last year of grad school, don’t expect to get a lot of math done.
11. Write for the future version of yourself who doesn’t understand past you’s cryptic notation.  Write write write.  This is the ultimate goal of graduate school, to write a thesis.

Post doc

1. Write write write.  Publish publish publish.  Collaborate.
2. Make sure people care about your work, somehow (attend conferences, give talks???)
3. ????

Tenure track job

1. Write write write.  Publish publish publish.
2. From my perspective, be superhumanly amazing and incredible.

Tenure

1. Profit from your hard work!  Keep working hard because if you made it this far, you really love math and your work.  I’ve heard many professors say “why would I retire?” but I also know that teaching is draining.

ALTERNATE TRACK, STARTING IN GRAD SCHOOL

1. Become a really good TA, and try to instruct your own courses.  Become conversant in things like flipped classrooms, IBL, clickers, and various pedagogy.  Consider doing math circle, DRP mentorship, tutoring.
2. Write an incredible teaching statement, and personalize your cover letter to each teaching school you’re applying to.  Convince them that you love teaching.  Teach the letter reader something they didn’t know/think about before.  See notes I took from a talk by the president of the MAA:
3. Be an awesome teacher, and continue to do research on the side (depending on your position).

Note that I had a lot more advice for undergraduates than I did for after where I am now (I didn’t follow all that undergraduate advice either).  I also have no metrics really of “success” besides getting tenure, which is not for everyone.  I feel very successful life-wise with my family and blog and triathlons, but as to professional success I am pretty emo.  This blog post is about professional success==tenure.  Or you could be a badass and become a freelance mathematician or a mathematical writer  or anything else you want to be!  You’re getting a Ph.D. in math; the world is your oyster!

Also!  Writing this blog has been very fun and rewarding and one of the best parts is when undergraduates or beginning grad students or other people write to me to ask for advice or just say hey.  I love hearing from you!  I’m on email (yenergy), twitter (yenergy), and instagram (yenergyyy) so hit me up!

## Not a sociologist or ethnographer, but I am a curious person (about gender and race)

2 Jul

Inspiration for this post: this tweet.

So I’ve written before about being a woman in math, and this will not be my last post on the subject either.  First, some background.  One really, really awesome thing about my field (geometric group theory) is its webpage.  Some time ago, a great professor at UCSB made this website which includes a list of all active geometric group theorists in the world (self-reported), a list of all departments in the world with said people, lists of publishers and interesting links/software, and most importantly for me, a list of all conferences in the area.

Long aside: said professor once gave me some great advice which I have since forgotten/warped in my memory to mean: do what you want to do.  This is probably not what he said, but he did use this amazing website as an example: at the time, people said that making the site was a waste of his time, and now its a treasured resource for researchers around the world.  Everyone in GGT knows this site (because they or their advisor is on it!)  So that’s part of the reason I have this blog, and started that women in math conference- it’s maybe a “waste” of my time, but it’s something I want to do and now people are starting to know me for it.  At both the Cornell and the MSRI programs I went to these past two months, a graduate student has come up to me and told me she reads my blog, so yay!  I love you, readers!  Also, side note in this aside: the video lectures from the summer graduate school in geometric group theory are already posted (in the schedule part of this link), so if you like videos and GGT I’d recommend them.  Lots of first and second year graduate students in the audience, so they’re relatively approachable.

Back to topic: I went through the list of conferences that had occurred so far this year and “ran some numbers,” by which I mean I divided.  I did this because I noticed that at the past few conferences I’ve attended, there seem to be disproportionately many female speakers (in a good way).  For instance, at this summer school I counted 12/60 female students (though later someone said there are 14 of us so don’t rely on my counting) and 1/4 female speakers.  But the numbers at that level are so low that the data is essentially meaningless: 25% vs. 20% isn’t that meaningful when the other choices are 0, 50, 75, or 100% female speakers.  But if you collect enough data, it probably becomes meaningful.  See my table below.*

If I were a sociologist or ethnographer, I would do this for all the conferences and interview a random sample of attendees and organizers in order to come to some data-backed conclusions about the phenomena here.  I’m not, so I’ll just make some guesses.  It looks like American conferences artificially inject more gender diversity into their invited speakers lists, while foreign ones don’t (YGGT in Spa a notable exception).  I’d also guess that conferences that target graduate students have more women speakers than conferences that don’t.

Three things that support my “artificial diversity” theory: to attend an MSRI summer school, graduate students are nominated by their schools.  Schools can nominate two students, and a third if she is a woman or an underrepresented minority.  The NSF, which is a huge source of funding for American conferences, is really into “Broadening Participation”, which means including participants who are women, African-American, Native American, Hispanic, or disabled.  And, as seen in table above, the percentage of female domestic speakers is twice that of foreign speakers.

I think this is great!  It’s much easier to do something if you see someone who looks like you/has gone through similar struggles doing so.

A response to myself from a few years ago, when I felt feelings about the burden of representing all women at a table full of men: I felt bad recently for wanting to ask a Hispanic female graduate student what she thought about increasing numbers of Hispanic women in math, because I thought I was placing this exact burden on her.  I was expecting her to speak for all Hispanic women.  But another graduate student solved this conundrum for me- her experience is invaluable in trying to understand the plight of her demographic, but we shouldn’t be too hasty to generalize from it.  And more importantly, someone needs to ask these questions.  My discomfort is relatively stupid and small compared to the issue at hand- we should try to solve these problems together and respectfully, but there’s bound to be missteps along the way, and that’s OK.

I don’t have solutions, and I’ve barely stated the problem or why we should care about it, but at least I’m trying to ask questions.

## Prees, prees, pretty prees

13 May

Last weekend I had a wonderful time at the Cornell Topology Festival- I went because my internet and now real life friend tweeted about it!  Good things can come out of the internet!

Another very talented friend made an icosohedron out of balloons (following a template by Vi Hart, so you can do it too!) and now I have a picture of me holding it:

Anyways, apologies for delay in posting.  I’ll try to double up this week to make up for it.  There were a bunch of great talks at the conference, and here’s a post about one of them that really intrigued me, “Universal Groups of Prees” by Bob Gilman.  I thought “prees” was a typo and meant to say “trees”, but nope, the word is “pree.”

Hopefully you remember or know what a group is.  A pree is something on the way to being a group, but not quite: it’s a set with partial multiplication, identity, inverses, and the associative law when defined.  So in a group, whenever you have two elements and b, closure ensures that the product ab exists and is an element of the group.  In a pree, it’s not necessarily the case that ab exists.  But if it does, and bc exists and a(bc) exists, then (ab)c must also exist, and equal a(bc).  [If it’s hard to think of something non-associative, check out this wikipedia article on the cross product and use your right hand and a friend’s right hand].  We can show this visually:

Start from the top vertex.  If you follow the arrow right and the arrow up, you’ll end up at the bottom left vertex.  This tells us that we should label the edge from the bottom left vertex to the top by ab.

This part of the triangle just shows that a*b=ab, that is, the product of and are defined in our pree.  Next we’ll add a triangle for b*c:

Notice that the arrow for b is the same direction as the previous picture; I just took the arrows out of the big triangle for cleanness

Now we add the blue triangle in to the big triangle.  There are two different ways to read the last face, and that fact means that those two expressions better be equal.  This is associativity.

Using the big triangle labels, we get a(bc). Using the small triangle, we have (ab)c. These are the same edge, so they must be equal

If you’ve done any group theory you might have the same reaction I did: “THIS IS SO WEIRD TELL ME MORE I WANT TO KNOW!”  This combinatorial pictorial thing is reminiscent of vector multiplication, but with group elements and it intrigues me no end.  For short, they say this is an axiom of associativity:

The official axiom probably lacks smiley eyes, but that’s clearly an oversight

So you can think of prees as partial multiplication tables, and they determine a graph.

But I tagged this post “group theory”, and prees do relate to groups.  In fact, it’s a theorem that any finitely presented group (see here for reminder of definition) is a universal group of a finite pree.  This group is defined as having generators equal to the elements of the pree, and relators are the products of the pree.

Here’s the first example.  If your pree (which is a set with partial multiplication) consists of two groups K and L which share a subgroup A, then the universal group of that pree is $K \ast_A L$, the free product of K and L amalgamated over A.  If you don’t know what that means, don’t worry about it.  We’ll do amalgamated products some other time and I’ll add a link here for that.

You can also do this with letting your pree be a graph of groups, and get the correct corresponding group.

Something whacky! This isn’t an open problem, but an undecidable problem: whether a finite pree embeds in its universal group (this means that there’s a function sending the elements of the pree into the group which respects the pree multiplication and doesn’t send two different pree elements to the same group element).  So even if you might be able to tell, given a specific pree, whether it embeds in its universal group, there’s no algorithm that works for all finite prees.

Here’s one of the main theorems of the talk: if, every time you have a collection of elements that can be put together to form a triangulated rectangle or pentagon, as in the picture below, one of the orange lines exists, then the resulting universal group is biautomatic.

At least one diagonal of the rectangle and at least one chord of the pentagon exist. Colors don’t mean anything

Remember, prees only have partial multiplication.  So in the rectangle case, if we have ab and bc along the diagonal, the orange line means that ac also exists.

Like you, dear reader, I also don’t know what biautomatic means, and Gilman didn’t explain it during his talk.  But he did draw lots of pictures of this sort-of group-like thing.

Here is a survey article on prees.

On deck if I get around to it: more blog posts from this conference- talks by Denis Osin and Mladen Bestvina.  Also, I really need to bake something new.  I’ve made that super easy lime pie a bunch by now; I even made it at this conference with ingredients from a mini-mart.  The limes had no juice so this was mostly a sweetened condensed milk pie, which was still delicious but too sweet.

## Kissing numbers, current research in hyperbolic surfaces

30 Mar

I just got back from the fantastic Graduate Student Topology & Geometry Conference, where I gave a talk and also brought my baby.  I tried to google “bringing baby to academic conference” as I’ve seen one baby at a conference before (with his dad), and I knew this kid would be the only baby at ours.  But it was cold enough/uncomfortable enough that I just had him stay in the hotel with my mom, and I ran back during breaks to feed him.  Also, it was my first time being “heckled” by both of these two brother professors famous for “attacking” speakers- they happen to know just about everything and are also suckers for precision, which I am not (and should be).  But I got a lot of good feedback on my talk, and I’m generally a very capable speaker (though I was not as prepared as I would’ve liked, thanks to somebody who likes to interrupt me every five minutes…)  Anyways, this is not about me, this is about my friend who gave one of the best talks of the conference and more importantly, her research.  This post is based on notes I took during her talk + skimming her paper (joint with her advisor) on which it is based.

Remember that we had our introduction to hyperbolic space.  This research is focused on hyperbolic surfaces, which are shapes that locally look like hyperbolic space- this means that if you look at one point on the surface and just a little area around it, you think you’re in hyperbolic space.  A good analogy is our world- we live on a sphere, but locally it looks like flat space.  If you didn’t know better, you’d think the earth is flat, based on your local data.  So how can we build a hyperbolic surface?

While hexagons in flat space always have angles that sum to 720 degrees, that’s not true in hyperbolic space.  In fact, you can make right angled hexagons, which means that every single corner has 90 degrees.  If you pick three lengths a,b,c>0 and assign these lengths to three sides of the hexagon like the picture, you’ll fully determine the hexagon- hyperbolic space is wacky!

Now glue two copies of a hexagon together along those matching a,b,c sides.  You’ll have a funny shape with three holes in it, and those holes will have circumference 2a, 2b, 2c.  This is called a pair of pants in topology.

You can glue together a bunch of pants to form a hyperbolic surface, by gluing them together along holes with the same length.  Any hyperbolic surface, conversely, can be cut up into pairs of pants (this pants decomposition is not unique, as you can see below).

You could also set one of those lengths equal to 0, so you’d get a right angled pentagon as one of the hexagon’s sides would collapse.  You can still do the pants thing here by gluing together copies of the pentagon, but instead of having a hole with circumference 2a like we had before, you’ll have a cusp that goes off to infinity- it’s like an infinite cone with finite volume.

Now we’ve built every hyperbolic surface (there are some more details, like how you glue together pants, but let’s just stick with this broad schematic for now). As long as the expression 2-2*(number of holes)-(number of cusps)<0, your surface is hyperbolic.  So, for instance, a sphere isn’t hyperbolic, because it has no holes and no cusps, so you get 2 which is not smaller than 0.  And a torus isn’t hyperbolic, because it only has one hole, so you get 2-1=1.  But all the surfaces in the pictures in this post are hyperbolic- try the formula out yourself!

One thing you can ask about a hyperbolic surface is: how long is its shortest essential curve?  By “essential,” we mean that it isn’t homotopic (this is a link to a previous post defining homotopy) to a cusp or a point.  This shortest curve is called the systole of the surface.  Systolic geometry is a whole area of study, as a side note.  But we’re interested in the question: how many systoles can a surface have?  This is called the kissing number of the surface.

A few notes: a “generic” surface has Kiss(S)=1, that is, there’s only one shortest curve if you happen to pick one “random” surface (scare quotes because no precise definitions).  And it’s relatively “easy” to make a surface with Kiss(S)=3*(number of holes)-3+(number of cusps).  Check for yourself that this number is exactly the number of curves in a pants decomposition of a surface.  Using some hyperbolic geometry you can prove that there won’t be any shorter curves if you make all of the pants curves very “short.”

So what Fanoni and Parlier do in their paper is come up with an upper bound on the kissing number of surfaces with cusps.  I won’t go into that, but I will try to explain part of a lemma they use on the way.

If your surface doesn’t have any cusps, then systoles can pairwise intersect at most once.  But if you do have cusps, then Fanoni & Parlier prove that your systoles can intersect at most twice (and they build examples of surfaces with cusps that have systoles that pairwise intersect twice).

First they show that two systoles which intersect at least twice can only intersect in the way pictured to the left below, and not as in the right:

This picture from the Fanoni-Parlier paper I did not make this!

This matters because it implies that two systoles which intersect at least twice must intersect an even number of times.  In particular, if two systoles intersect more than twice, then they intersect at least four times.

So assume for contradiction that two systoles and intersect more than twice.  So they intersect at least four times.  That means that there’s some intersection point somewhere such that the b-arcs coming out of it make up no more than half of the systole length (see picture below)

If the green arc is more than half the length of the circle, then the blue one is less than half the length of the circle.

So if you look at these short b-arcs, plus the path, and wiggle things around, you’ll see a four-holed sphere (two holes above” the curve, and two holes “below,” one of each inside a arc).

Left: a schematic of how a and b intersect. Black dots represent holes or cusps.
Center: the short b arcs plus the full a path
Right: the short b arcs plus the a path, after moving four dots to be holes of a 4-holed sphere

This four-holed sphere has a curve on it, determined by part of and the arcs, which is shorter than the original systoles.  This contradicts the definition of systole, so our premise must be wrong- two systoles can intersect at most twice.

This was proposition 3.2 in their paper- tomorrow I’m going to share propositions 3.1-3.3 with my advisor’s small seminar.  Hopefully I don’t get heckled too badly this time!

OOPS I ALMOST FORGOT: life update.  We bought a house and are moving to Austin, TX.  I’m still planning on finishing my Ph.D., just virtually.  [Up to a finite-index subgroup, obviously.  Bad math joke].  I’ll probably be flying up to Chicago every so often to meet with my advisor/eventually defend my thesis.  But yes, we’re driving in our minivan to Texas on Thursday.  So… we’ll see when we get the internet set up in the new house.  I’ll try not to make too long a break until my next post.

## Some thoughts on knots: current research

8 Mar

Over the weekend I went to the Third(!!!) annual Midwest Women in Mathematics Symposium (remember when I founded it?  Now it’s all fancy with funding and many attendees and event staff!)  As it turned out, not very much of the math in my parallel session was exactly up my alley, and also I was feeling lazy so I didn’t take many notes.  But here’s a small recap/introduction to knot theory from my memory.

Aside: I like using knot theory as an example when people ask me what math is for (this happened a lot as an undergraduate and less and less as the years go by).  I’m not even sure if this is true, but I tell people that mathematicians were studying knot theory for decades, and then biologists realized that they could use it to study how proteins fold and interact with other molecules.  APPLIED!  IN YOUR FACE, MATH DOUBTERS!  Unclear where I picked up this bit of folklore, but it’s my number one defense when people say that modern math research is useless.

So what is knot theory?  It’s certainly not not-theory, despite my claim as above that it can be applied.  Knot theory studies objects called knots.  A knot is some way that a circle is embedded in space- imagine taking a shoelace, knotting it up however you want to, and gluing the ends together.  (By space knot theorists mean $S^3$, but we can just think of it as $\mathbb{R}^3$, or the space we live in).  To talk about knots, knot theorists draw knots as diagrams using over and under crossings.  Two diagrams can represent the same knot, like in the picture below.

Even though they look different, these knots are the same.  They’re called the unknot.

If I didn’t mess up, the blue knot is the same as the orange knot- just follow the crossings and you’ll see that nothing is actually knotted; it’s just a pile of string lying on top of itself.  Below are some pictures of other knots.

It’s hard to tell if two diagrams represent the same knot.  Mathematicians can use a diagram to assign polynomials to a knot, and do it in such a way that if two diagrams represent the same knot, then they give the same polynomial.  Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial (which generalizes the previous two).  These still aren’t that great though, since two different knots can give the same polynomial (so while you can tell if your diagram ISN’T the unknot by seeing if the polynomial isn’t 1, you can’t tell if it IS the unknot if the polynomial gives you 1).

The first knot theory talk I saw connected knots to surfaces, so I was a fan.  It was given by Effie Kalfagianni, a professor at Michigan State.  One thing you can do with a knot is use it as the boundary of a surface.

From wikipedia: I was having a really hard time making my own pictures.

There are different ways to make a surface from a particular knot- draw a different diagram and you’ll get a different surface.  One thing you can study is the genus of a knot: this is defined as the minimum genus (# holes) of a surface bounded by that knot.  So for any diagram you draw, you can’t make a surface with a smaller number of holes.  The genus of the unknot is 0.  The genus of a knot using orientable surfaces is known, and there’s an efficient algorithm to find it.  BUT the problem is open for non-orientable surfaces (these are surfaces that don’t have two sides).

“Sometimes I feel like I can’t trust you… it’s like you’re two sided.”

So Kalfagianni’s research, joint with her student Christine Lee, puts a bound on the non-orientable genus of alternating knots, which are knots with diagrams that alternate between over and under crossings (alternating: the purple and red knots.  Not alternating: the unknot, either blue knot (there are two over crossings in a row)).  They use one of the factors in the Jones polynomial to do so.

So that was talk number one!  The second talk I saw was by Maggy Tomova, an assistant professor at University of Iowa.  I actually didn’t write any notes down for her talk, but I remember a cool concept from it.  A knot diagram is in bridge position if you can draw a line across the middle so that there are only local maxima above it and local minima below.

GET IT? It’s a visual pun!  The green is in bridge position.  The red is not.

One immediate note is that in general, bridge position is not unique: given a knot in bridge position, you might be able to find another diagram in bridge position that represents the same knot.  There are some properties that ensure that a bridge position is unique (this is a theorem that I don’t remember).  Tomova is working on some theorems that have to do with knots in bridge position, and I’m sorry that I can’t tell you more information.  She did her Ph.D. at UCSB though, with the same advisor as some delightful other people who are her co-authors on this project (the delightful only applies to the first link; I don’t actually know her other co-author but I really like Yoshi and the fact that he goes by Yoshi).  Also, one of her previous co-authors taught me abstract algebra when I was an undergraduate and he was a postdoc!  That link is to a piece he wrote on going to the “Dark Side,” a.k.a. leaving academia for Google.

So I am not a knot theorist, but there’s your post with thoughts on knots!