Tag Archives: research

The employers’ argument for parental leave

22 Dec
As you may be aware, I’m having a baby in three months.  My spouse’s work offers one week of parental leave to non-primary caretakers of new babies, so I decided to put an hour in of internet research to see if I could make a little report that he could send on to HR or someone who might be interested.  And then since I did that, I figured I’d post it to here!  Happy holidays dear reader!  Maybe this will be helpful to someone.
The main sources are all freely available on the internet.
Overall takeaways:
• Jobs that offer paid parental leave are increasingly important for Millennials and young workers and increase employee retention, as seen in the CWF study as well as the California experiment.
• Paid parental leave has no negative economic impact on employers, as in California experiment.
• Longer parental leave increases employee satisfaction with work-life balance, which increases worker happiness and thus productivity.
• For employees who are partners with other employees, leave for the non-gestating employee increases work-life balance satisfaction for the gestating employee and retention for that employee.
Here’s the Department of Labor briefing on paternity leave: https://www.dol.gov/asp/policy-development/paternityBrief.pdf.  Selected quotes (sources mentioned are available at the end of this short briefing:)
• “In one study of working fathers in the U.S., those who took leaves of two weeks or more were much more likely to be actively involved in their child’s care nine months after birth – including feeding, changing diapers, and getting up in the night.6 Studies from other countries have confirmed that fathers who take more paternity leave have higher satisfaction with parenting and increased engagement in caring for their children.7”
•  “Fathers are increasingly concerned about work-life balance, and nearly half of men surveyed report that the demands of work interfere with family life.11”
• “In a 2014 study of highly educated professional fathers in the U.S., nine of out ten reported that it would be important when looking for a new job that the employer offered paid parental leave, and six out of ten considered it very or extremely important. These numbers were even higher for millennial workers.23”
Here’s a 2010 California study on Paid Family Leave, which was implemented in 2004 and offered employees six weeks of leave at 55% of salary (followed by New Jersey and Washington): http://cepr.net/documents/publications/paid-family-leave-1-2011.pdf.  This offers concrete evidence of the effects of paid family leave in general.
• Most employers report that PFL had either a “positive effect” or “no noticeable effect” on productivity (89 percent), profitability/performance (91 percent), turnover (96 percent), and employee morale (99 percent).
• Relevant pages: 7-10
• Look at this chart, based on over 1000 worker fathers who were mostly (over 90%) well-educated professionals:
• “Only 20% of the study participants felt that all of the time off should be taken consecutively beginning with the birth of their children. More than 75% preferred the option to take the paid time off when it was most needed after the birth, within a specified period of time such as six months. For example, over a six month period after the birth of their child, they could take two weeks at the beginning and then additional days off as needed up to the maximum amount allowed.”
• Pages 1-14 are about worker desires, pages 15-20 are about employers’ implementations, including spotlights on Ernst and Young, Deloitte, and American Express.  “• For policies that didn’t differentiate between primary and secondary caregivers, fathers were given an average of two weeks of paid leave • For policies that did have designated provisions for fathers who were primary or secondary caregivers, fathers as primary caregivers were given an average of about eight weeks, which was approximately three times as much as the leave offered to secondary caregivers”
• Takeaways: “Nearly three quarters of the fathers believed that the most appropriate amount of time for fathers to have off for paternity leave is between two and four weeks… 76% of fathers would prefer the option of not taking all their time off immediately following the birth of their children”
• Pages 27-28 are concrete recommendations for employers.
If no one wants to read this post , here’s a great and easy to read article from ThinkProgress that similarly cites a bunch of studies and includes anecdotal evidence titled “How Everyone Benefits When New Fathers Take Paid Leave.”  I highly recommend this article.

Connecting hyperbolic and half-translation surfaces, part II (math)

1 Mar

Last week we saw the standard definition for a hyperbolic surface.  You can tweak this standard definition to define all sorts of surfaces, and we tweaked it for a definition of half-translation surfaces.  Here are the two definitions:

•  A hyperbolic surface is a topological space such that every point has a neighborhood chart from the hyperbolic plane and such that the transition maps are isometries.
• A half-translation surface is a topological space such that all but finitely many points* have a neighborhood chart from the Euclidean plane and such that the transition maps are combinations of translations and flips.  These finitely many points are called singularities.

Precision note: according to Wikipedia, we need to add the adjective “Hausdorff” to our topological space.  We won’t worry about this or give a precise definition of it; you can just know that Hausdorff has something to do with separating points in our space.

Half-translation spaces come with something nifty that occurs in Euclidean space.  You know how when you look at a piece of notebook paper, it has all these nice parallel lines on it for writing?  Or if you look at a big stack of paper, each sheet makes a line which is parallel to the hundreds of others?

This is from clip art panda.  How useful!

Mathematicians call that a foliation: each sheet of paper is called a leaf.  This is an intuitive definition; we’re not going to go into a technical definition for foliation.  Just know that Euclidean space comes with a foliation of all horizontal lines y=r, where is some real number.  Then since transition maps of half-translation spaces come from either straight translations or flips, the foliation carries over to the half-translation space (though orientation might have flipped, we don’t care about those in this application).

All the slides are generously shared by Aaron Fenyes

Notice in the picture in the lower left that there are a few points where the horizontal foliation doesn’t quite work.  Those are the singularities that show up in the definition of a half-translation surface (we need them if we want our surface to be anything besides an annulus).

At those singular points, we glue together patches of Euclidean space.  The orange color in this picture shows the path of the critical leaf as it winds all the way around the surface some number of times.

Last week we had those nify gifs to show us how to think about curvature as positive, negative, or zero.  Here’s the example of zero curvature, because the last arrow is the same as the first arrow:

We can actually get precise numbers instead of just signs for curvature.

Here the triangle encompasses -π/3 curvature.  Notice that it embeds straight down into the hyperbolic surface, so we see an actual triangle down in the lower left.  If we made this triangle bigger and bigger, eventually it’d wrap around the surface and we wouldn’t see a triangle, just a bunch of lines hinting at a triangle up in the hyperbolic plane.  That’s the next picture.

Curvature can range from -π to π.  Here’s an example of an extremely negatively curved triangle which has -π curvature:

Such a triangle in hyperbolic space has all three corners on the boundary/at infinity.  This is called an ideal triangle.  So all ideal triangles encompass -π curvature.  You can see also how in the surface, we have a collection of lines whose preimage is the ideal triangle.

We can also use the same process to find curvature in other places.  For instance, if we make a little hexagon around a singularity of a half-translation surface, we can go around it with the same parallel transport process to figure out how much curvature the singularity contains.  We’ll make use of that horizontal foliation we saw earlier.

This looks very similar to our ideal triangle: the arrow starts off pointing up, and ends up pointing exactly the opposite direction.  So this singularity has -π curvature too, just like the ideal triangles.

Now for the math part!  Here’s the question: given a hyperbolic surface, how can we construct an associated half-translation surface?

Answer: we’ll use those foliations that we had before, as well as something called a geodesic lamination: this is when you take a closed subset of your surface, and give it a foliation.  So it’s like a foliation, only there’ll be holes in your surface where you didn’t define how the pages stack.  The first example of a geodesic lamination is a plain ol’ geodesic curve in your surface: the curve itself is a closed subset, and the foliation has exactly one leaf, the curve itself.  After this example they get real funky.

You don’t even have to take me to funkytown; geodesic laminations are already there!

Given a book, we might want to know how many pages we’ve read once we stick our finger in somewhere.  Luckily there are page numbers, so we can subtract the page number we started at from the page number we’re standing at.  Similarly, given a foliation, we might want to have a measure on it, transverse to the leaves.  If we have one, it’s called a measured foliation.  These exist.

So let’s start with our hyperbolic surface, and choose a maximal measured geodesic lamination on it.  Maximal means that the holes are the smallest they could possibly be.  Turns out this means they’re the images of ideal triangles under the atlas.

Told you they were funky.

Also, there are only finitely many of these triangle-shaped holes down in the surface (we’re sweeping some math under the rug here).  Now we need to get from this surface to a half-translation surface.  We’ll keep that foliation given by the lamination, and we need to get rid of those complementary triangles somehow.  So the lamination’s foliation will become the horizontal foliation of the half-translation surface, and the ideal triangles will correspond to singular points.  We can’t just collapse the ideal triangles to singular points, because as we saw earlier, images of ideal triangles are really funky and wrap around the surface.  We need to find smaller triangles to turn into singular points.  Here’s the picture:

Upstairs, we made a new purple foliation (transverse to the lamination) of the complementary ideal triangles, by using arcs of circles perpendicular to the boundary circle (these circles are called horocycles).  So now we have teensier triangles in the middle of the ideal triangles, called orthic triangles.  To make a half-translation surface, we’ll quotient out the horocycles, which means that in each ideal triangle, we identify an entire purple arc with one point.

Quotienting out horocycles a.k.a. identifying the pink lines all as individual pink points.  That means each side of the orthic triangle is identified with a point, so the orthic triangle disappears.

In this way we get tripods from triangles.  The middles of these tripods are singular points of the half-translation surface.  The measure from the measured lamination gives a measure on the foliation of the half-translation surface.

But Euclidean space actually comes with horizontal and vertical distances defined (remember, half-translation surfaces locally look like Euclidean space).  So far we have a way to get one direction.  How do we get the transverse distance?  We use the fact that we chose a geodesic lamination of our hyperbolic surface.  Geodesics are curves of shortest length; in particular they have length.  So if I’m in my translation surface and moving along a leaf of the foliation, I can look back at where I was in the lamination of the hyperbolic surface and use that distance.  [There’s some rug math here too.]  So we’ve made neighborhoods in the half-translation surface look like Euclidean space.

So that’s that!  You can also go backwards from a half-translation surface to a hyperbolic surface by blowing up the singular points into ideal triangles.  [More math, especially when the singularities of the half-translation surface are messy or share critical leaves].  Aaron claims this is folklore, but a quick google search led me to this paper (in French) and this one by the same author who connects flat laminations (on half-translation surfaces) to the geodesic ones we see in hyperbolic surfaces in section 5.

*I lied about finitely many points.  You can have infinitely many singularities in a half-translation surface; they just have to be discrete (so you should be able to make a ball around each other disjoint from the others, even if the balls are different sizes).  Examples of discrete sets: integers, $2^x, x>0$. Examples of not-discrete sets: rational numbers, $2^x, x<0$.

Connecting hyperbolic and half-translation surfaces, part I (definitions)

24 Feb

I love talks that start with “I haven’t seen this written down explicitly anywhere, but…” because that means someone is about to explain some math folklore!  There are some statements floating around in mathland that specialists in those fields believe, so the rest of us believe them because the specialists said so, but no one knows a citation or a written proof for them.  That’s folklore.  Two weeks ago I gave a talk and someone asked a question (are RAAGS uniquely determined by their defining graphs?) and I said “probably, but I have no references for you.”  I found a reference a day later and emailed it to her, but the reference was way hard and had way more machinery than I was expecting.  The power of folklore!

Anyways, this series of posts will be based on a talk by a grad student at UT, Aaron Fenyes.  This was a great, great talk with lots of pretty slides, which Aaron has generously allowed me to put up here.  We’ll review curvature and surfaces, and then talk about how to go back and forth between two kinds of surfaces.

We’ve chatted about hyperbolic space v. Euclidean and spherical space before in terms of Euclid’s postulates, but let’s chat a bit about curvature. We say the Euclidean plane/real space has curvature 0, that hyperbolic space is negatively curved, and spherical space is positively curved.  There’s a nice way to see curvature: draw a triangle in your space (that old link also has some triangle conditions in it), and imagine standing at a point on that triangle and looking toward one corner of the triangle.  By “looking out” I mean your gaze should lie tangent to the triangle.  Remember:

Left: tangent; line hits circle at exactly one point.
Center: not tangent, line hits circle at two points
Right: not tangent, line doesn’t hit circle

Now walk toward the corner you’re facing, and then walk down the second side of the triangle still facing that direction (so you’re sidestepping), and walk around the next corner (so you’re now walking backwards) and keep going until you end up where you started.  This is called parallel transport.  If your triangle was in Euclidean space, then you’re facing the same way you were when you started.

Gif!  Note the ending blue vector is identical to the beginning purple vector.  So we have curvature 0.

If your triangle was slim, then you might find yourself facing the opposite way that you started!  Or if your triangle isn’t that curved, you’ll find yourself facing a direction counterclockwise from your original one.

Aaron Fenyes made the pictures; I gif’d it!  My first gif!

This is the summary of the gif: you can see how the ending light blue vector is pointed away from the original purple vector

From the original to the new vector: it’s a pi/3 counter-clockwise turn.

Similarly, if your triangle was fat, you’ll end up facing a direction clockwise from your original.

Here’s the picture of just the first and last arrows:

The green arrow is now clockwise from the red arrow, which means this has positive curvature.

So curvature is a way to measure how far clockwise you’ve turned after doing this parallel transport.

I love that description of curvature vs. the way I did it before, but they’re all good ways of seeing the same thing.  Next we need to review surfaces.  When we first met hyperbolic surfaces, we built them by gluing pairs of pants together, which themselves were stitched together from right angled hexagons which lived in hyperbolic space.  Redux:

Now if I take a little patch from my hyperbolic surface, I can trace back through one or two pairs of pants to find the original hexagon(s) in hyperbolic space where my patch came from.  So I have a map from hyperbolic space to my patch of hyperbolic surface, describing the metric and geometry around that patch.  This map is called a chart, and every point on a hyperbolic surface will have a chart associated with it, sending some part of hyperbolic space to a neighborhood of that point.

Here Aaron picked a blue patch and an orange patch in the surface, and the picture shows their charts from the hyperbolic plane to the patches.

This picture might make you leery: what happens when images of charts overlap, like they do here?  The preimages in the hyperbolic plane are disjoint, but they map to the same yellow area in the surface.  We want to say there’s some reasonable relationship between the yellow preimage patches in the hyperbolic plane.  That relationship is the only one we know, isometry:

There’s an isometry of the hyperbolic plane sending the orange patch to the new orange patch, so that the yellow parts overlap exactly.

If we look only at the yellow patch, we can find another way to describe the map in the picture: first, do the blue chart sending the blue patch to the surface.  Then, do the inverse of the orange chart, which sends the orange surface patch to its preimage.  Restricted to the yellow overlap patch, this is the definition of a transition map.

So here’s another way to think of hyperbolic surfaces, instead of gluing hexagons together like before.  A hyperbolic surface is a topological space such that every point has a neighborhood chart from the hyperbolic plane and such that the transition maps are isometries.

If you change where the chart is coming from, we can change the adjective before surface.  For instance, a flat surface is when the charts come from the Euclidean plane.  Now we’re going to define half translation surfaces, where the charts come from the Euclidean plane, but we have some more conditions on the transition maps.  The isometries of the Euclidean plane all come from a combination of translations and rotations.  Instead of allowing all isometries, we’ll only allow some of them:

In this picture you can see the orange and blue patches on the surface which come from the Euclidean plane.  Now we’re allowing translations and pi (180 degree) rotations only for our transition maps.  That’s why they’re called half-translation surfaces: charts from the Euclidean plane, and transition maps are translations plus half-rotations (flips).  As an aside, a translation surface is when we allow translations only, and no flips.

In the next post in this series, I’ll go through Aaron’s explanation of how we can go from hyperbolic surfaces to half-translation surfaces and back, and we’ll get to revisit our old friend the curve complex.  It’ll be fun!

Quick post: research updates of friends

18 Aug

I noticed a few papers up on arXiv last week that correspond to some old posts, so I thought I’d make a quick note that these people are still doing math research and maybe you are curious about it!

We last saw Federica Fanoni and Hugo Parlier when they explored kissing numbers, and they gave an upper bound on the number of systoles (shortest closed curves) that a surface with cusps can have.  This time they give a lower bound on the number of curves that fill such a surface.  Remember, filling means that if you cut up all the curves, you end up with a pile of disks (and disks with holes in them).  So you can check out that paper here.

Last time we saw Bill Menasco, he was working with Joan Birman and Dan Margalit to show that efficient geodesics exist in the curve complex.  This new paper up on arxiv was actually cited in that previous paper- it explains the software that a bunch of now-grad students put together with Menasco when they were undergrads in Buffalo, NY (UB and Buffalo State) during this incredible sounding undergrad research opportunity– looks like the grant is over, but how amazing was that- years of undergrads working for an entire year on real research with a seminar and a semester of preparation, and then getting to TA a differential equations class at the end of your undergraduate career.  Wow.  I’m so impressed.  I got sidetracked: the software they made calculates distances in the curve complex and the paper explains the math behind it and includes lots of pretty pictures.

My friend Jeremy did a guest post about baklava and torus knots a long time ago, and of course he’s got his own wildly popular blog.  He also has a bunch of publications up on arXiv, including one from this summer.  They’re all listed in computer science but have a bunch of (not-pure) math in them.

The paper I worked on over that summer at Tufts with Moon Duchin, her student Andrew Sánchez (note to self: I need a good looking website I should text Andrew), my old friend Matt Cordes, and graduate student superstar Turbo Ho is up on arXiv and has been submitted: it’s on random nilpotent quotients.

Moon and Andrew and others from that summer have another paper which has been accepted to a journal, it’s also about random groups and is here.  It was super cool, I saw a talk at MSRI during my graduate summer school there and John Mackay (also a coauthor on that paper) was in the audience and this result came up organically during the talk.  Pretty great!

There’s another secret project from that summer which isn’t out yet, but I just checked two of the three co-authors webpages and they had three and four papers out in 2015 (!!!)  That’s so many papers!  So I don’t know when secret project will be out but I’ll post about it when it is.

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.