Prime subset sums

Efrat Bank‘s interesting number theory seminar here before break was about sums of arithmetic functions on short intervals in function fields.  As I was saying when I blogged about Hast and Matei’s paper, a short interval in F_q[t] means:  the set of monic degree-n polynomials P such that

deg(P-P_0) < h

for some monic degree-n P_0 and some small h.  Bank sets this up even more generally, defining an interval in the space V of global sections of a line bundle on an arbitrary curve over F_q.  In Bank’s case, by contrast with the number field case, an interval is an affine linear subspace of some ambient vector space of forms.  This leads one to wonder:  what’s special about these specific affine spaces?  What about general spaces?

And then one wonders:  well, what classical question over Z does this correspond to?  So here it is:  except I’m not sure this is a classical question, though it sort of seems like it must be.

Question:  Let c > 1 be a constant.  Let A be a set of integers with |A| = n and max(A) < c^n.  Let S be the (multi)set of sums of subsets of A, so |S| = 2^n.  What can we say about the number of primes in S?  (Update:  as Terry points out in comments, I need some kind of coprimality assumption; at the very least we should ask that there’s no prime factor common to everything in A.)

I’d like to say that S is kind of a “generalized interval” — if A is the first n powers of 2, it is literally an interval.  One can also ask about other arithmetic functions:  how big can the average of Mobius be over S, for instance?  Note that the condition on max(S) is important:   if you let S get as big as you want, you can make S have no primes or you can make S be half prime (thanks to Ben Green for pointing this out to me.)  The condition on max(S) can be thought of as analogous to requiring that an interval containing N has size at least some fixed power of N, a good idea if you want to average arithmetic functions.

Anyway:  is anything known about this?  I can’t figure out how to search for it.

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Israel

The main part of our trip was to Jerusalem, where I met my new nephew and organized a workshop about new developments in the polynomial method.

  • The vote on UN resolution 2334 was held while I was there but nobody I talked to seemed really focused on it.  One Israeli businessman told me “the Arabs won’t destroy Israel, Netanyahu won’t destroy Israel, the only thing that can destroy Israel is the residue of Bolshevism.”  Then he told me things about taxes that curled my hair.  Apparently if you do work for person X, and bill them for 100,000 shekels, you owe taxes on the 100,000 shekels, whether or not person X pays you!   They go bankrupt or just stiff you, you’re screwed.  If you go to the tax agency to say “how can I pay taxes on income I didn’t get” they say “the problem is between you and person X, go sue them if you need that money.”
  • Tomer Schlank told me my accent was really good!  I don’t speak Hebrew, by the way.  But I’m actually very good at imitating accents, which is a problem, because when I carefully think about what I’m going to say (in Spanish, French, German, Hebrew, whatever) and then say it, I can sometimes fool the person I’m talking to into thinking I’m going to understand their response.  Fortunately, I’m also very good at the blank look.
  • My kids really wanted to go to the science museum and I was reluctant — there are science museums all over the world! — but I relented and actually it was kind of great.  Culturally interesting, first of all, because the place was completely packed with Orthodox families; it was Hanukkah, a rare time when kids are off school but it’s not chag, which makes it massive go time for kid-oriented activities in Jerusalem.  I was happy for my kids to experience that feeling of immersion in a crowd that was on the one hand Jewish but on the other hand quite culturally alien, in a way that secular cosmopolitan schwarma Israel really isn’t.  As for the museum itself:   “Games in Light and Shadow” was a really charming exhibit, sort of a cross between interactive science and a walk-through art space a la Meow Wolf.
  • Sadly, we didn’t make it back to Cafe Itamar this time.  But we did return to Morduch.  I know it’s a tourist destination but in this case the tourists have it right, the Iraqi Jewish food there is incredible.  Get the kubbe soup, get the hummus basar.  This would be the best food I ate in Israel were it not for my Mizrachi machetunim in Afula.  So it might be the best food you can eat in Israel.
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France

Back from France!  Just there for 2 1/2 days on the way back from Israel.

  1.  Cheeses eaten:  Bethmale, Camembert, unidentified Basque sheep’s milk, Chabechou, Crottin de Chavignol, 28-month aged Comté, Vieux Cantal, Brie de Nangis.  The Brie and the Chabechou (both from La Fermette) were the highlights.
  2.  On the other hand, Berthillon ice cream not as amazing as I remembered — possibly because American ice cream has gotten a lot better since 2004, the last time I was in Paris?
  3. The Louvre is by far the most difficult major world museum to navigate.  Why, for instance, the system where the rooms have numbers but the room numbers aren’t on the map?
  4. I wonder museums with long entry lines have considered opening a separate, presumably shorter line for people willing to pay double (3x, 5x?) the usual entry fee.
  5. In the most Thomas Friedman moment of my life, an Algerian cab driver heard my accent and immediately began telling me how much he loved Trump.  If you want to know the Algerian cab driver conventional wisdom on this topic, it’s that Trump is a businessman and will “calmer” once he becomes President.  I am doubtful.  Tune in later to see who was right, the Algerian cab driver or his skeptical passenger.
  6. AB really loved the Rodin sculptures in the Musee d’Orsay.  I think because her height is such that she’s quite close to their feet.  Rodin sculpted a hell of a foot.
  7. I learned from Leila Schneps that the masters of the Académie Française have declared that “oignon” is to be spelled “ognon” from now on!  I can’t adjust.

Robin laid a gun

OK here’s a weird piece of kid culture AB brought home:

Jingle bells, Batman smells

Robin laid a gun

Shot a tree and made it pee in 1981

This dates back at least to 2007 apparently.

It scans and rhymes very nicely but makes so sense at all.  What can it mean?

It seems like we are witnessing a kind of cultural hybrid; the “Jingle bells / Batman smells” of my childhood has here combined with a “Jingle bells / shotgun shells” tradition I was unaware of until now, which is actually older than the Batman version.  A lot of the “shotgun shells” versions found online involve Santa meeting his death in a hail of bullets, but “shot a tree and made it pee” is not uncommon.  I wonder how many utterly nonsensical kids rhymes we know are actually hybrids of different songs, each of which at some point sort of made sense?

 

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Women in math: accountability

I’ve talked about women in math a lot on this blog and maybe you think of me as someone who is aware of and resistant to sexism in our profession.  But what if we look at some actual numbers?

My Ph.D. students:  2 out of 15 are women.

Coauthors, last 5 years: 2 out of 23 are women.

Letters posted on MathJobs, last 2 years:  3 out of 24 are women.

That is sobering.  I’m hesitant about posting this, but I think it’s a good idea for senior people to look at their own numbers and get some sense of how much they’re actually doing to support early-career women in the profession.

Update:  I removed the numbers for tenure/promotion letters.  A correspondent pointed out that these, unlike the other items, are supposed to be confidential, and given the small numbers are at least partially de-anonymizable.

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Hast and Matei, “Moments of arithmetic functions in short intervals”

Two of my students, Daniel Hast and Vlad Matei, have an awesome new paper, and here I am to tell you about it!

A couple of years ago at AIM I saw Jon Keating talk about this charming paper by him and Ze’ev Rudnick.  Here’s the idea.  Let f be an arithmetic function: in that particular paper, it’s the von Mangoldt function, but you can ask the same question (and they do) for Möbius and many others.

Now we know the von Mangoldt function is 1 on average.  To be more precise: in a suitably long interval ([X,X+X^{1/2 + \epsilon}] is long enough under Riemann) the average of von Mangoldt is always close to 1.  But the average over a short interval can vary.  You can think of the sum of von Mangoldt over  [x,x+H], with H = x^d,  as a function f(x) which has mean 1 but which for d < 1/2 need not be concentrated at 1.  Can we understand how much it varies?  For a start, can we compute its variance as x ranges from 1 to X?This is the subject of a conjecture of Goldston and Montgomery.  Keating and Rudnick don’t prove that conjecture in its original form; rather, they study the problem transposed into the context of the polynomial ring F_q[t].  Here, the analogue of archimedean absolute value is the absolute value

|f| = q^{\deg f}

so an interval of size q^h is the set of f such that deg(f-f_0) < q^h for some polynomial f_0.

So you can take the monic polynomials of degree n, split that up into q^{n-h} intervals of size q^h, and sum f over each interval, and take the variance of all these sums.  Call this V_f(n,h).  What Keating and Rudnick show is that

\lim_{q \rightarrow \infty} q^{-(h+1)} V(n,h) = n - h - 2.

This is not quite the analogue of the Goldston-Montgomery conjecture; that would be the limit as n,h grow with q fixed.  That, for now, seems out of reach.  Keating and Rudnick’s argument goes through the Katz equidistribution theorems (plus some rather hairy integration over groups) and the nature of those equidistribution theorems — like the Weil bounds from which they ultimately derive — is to give you control as q gets large with everything else fixed (or at least growing very slo-o-o-o-o-wly.)  Generally speaking, a large-q result like this reflects knowledge of the top cohomology group, while getting a fixed-q result requires some control of all the cohomology groups, or at least all the cohomology groups in a large range.

Now for Hast and Matei’s paper.  Their observation is that the variance of the von Mangoldt function can actually be studied algebro-geometrically without swinging the Katz hammer.  Namely:  there’s a variety X_{2,n,h} which parametrizes pairs (f_1,f_2) of monic degree-n polynomials whose difference has degree less than h, together with an ordering of the roots of each polynomial.  X_{2,n,h} carries an action of S_n x S_n by permuting the roots.  Write Y_{2,n,h} for the quotient by this action; that’s just the space of pairs of polynomials in the same h-interval.  Now the variance Keating and Rudnick ask about is more or less

\sum_{(f_1, f_2) \in Y_{2,n,h}(\mathbf{F}_q)} \Lambda(f_1) \Lambda(f_2)

where $\Lambda$ is the von Mangoldt function.  But note that $\Lambda(f_i)$ is completely determined by the factorization of $f_i$; this being the case, we can use Grothendieck-Lefschetz to express the sum above in terms of the Frobenius traces on the groups

H^i(X_{2,n,h},\mathbf{Q}_\ell) \otimes_{\mathbf{Q}_\ell[S_n \times S_n]} V_\Lambda

where $V_\Lambda$ is a representation of $S_n \times S_n$ keeping track of the function $\Lambda$.  (This move is pretty standard and is the kind of thing that happens all over the place in my paper with Church and Farb about point-counting and representation stability, in section 2.2 particularly)

When the smoke clears, the behavior of the variance V(n,h) as q gets large is controlled by the top “interesting” cohomology group of X_{2,n,h}.  Now X_{2,n,h} is a complete intersection, so you might think its interesting cohomology is all in the middle.  But no — it’s singular, so you have to be more careful.  Hast and Matei carry out a careful analysis of the singular locus of X_{2,n,h}, and use this to show that the cohomology groups that vanish in a large range.  Outside that range, Weil bounds give an upper bound on the trace of Frobenius.  In the end they get

V(n,h) = O(q^{h+1}).

In other words, they get the order of growth from Keating-Rudnick but not the constant term, and they get it without invoking all the machinery of Katz.  What’s more, their argument has nothing to do with von Mangoldt; it applies to essentially any function of f that only depends on the degrees and multiplicities of the irreducible factors.

What would be really great is to understand that top cohomology group H as an S_n x S_n – representation.  That’s what you’d need in order to get that n-h-2 from Keating-Rudnick; you could just compute it as the inner product of H with V_\Lambda.  You want the variance of a different arithmetic function, you pair H with a different representation.  H has all the answers.  But neither they nor I could see how to compute H.

Then came Brad Rodgers.  Two months ago, he posted a preprint which gets the constant term for the variance of any arithmetic function in short intervals.  His argument, like Keating-Rudnick, goes through Katz equidistribution.  This is the same information we would have gotten from knowing H.  And it turns out that Hast and Matei can actually provably recover H from Rodgers’ result; the point is that the power of q Rodgers get can only arise from H, because all the other cohomology groups of high enough weight are the ones Hast and Matei already showed are zero.

So in the end they find

H = \oplus_\lambda V_\lambda \boxtimes V_\lambda

where \lambda ranges over all partitions of n whose top row has length at most n-h-2.

I don’t think I’ve ever seen this kind of representation come up before — is it familiar to anyone?

Anyway:  what I like so much about this new development is that it runs contrary to the main current in this subject, in which you prove theorems in topology or algebraic geometry and use them to solve counting problems in arithmetic statistics over function fields.  Here, the arrow goes the other way; from Rodgers’s counting theorem, they get a computation of a cohomology group which I can’t see any way to get at by algebraic geometry.  That’s cool!  The other example I know of the arrow going this direction is this beautiful paper of Browning and Vishe, in which they use the circle method over function fields to prove the irreducibility of spaces of rational curves on low-degree hypersurfaces.  I should blog about that paper too!  But this is already getting long….

 

 

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Reader poll: how many times have you worn a necktie?

This Fermi question is probably easy for some people:  I’m guessing there are some women for whom the answer is zero (am I right?), and some men whose necktie-wearing is dominated by “every day at work, 5 days a week, 50 days a year.”  For me it’s much harder.  I guess I’d say — three or four times a year, as an adult?  More when I was younger and went to more weddings.  I’m gonna estimate I’ve put on a tie 150 times in my life.

Followup question:  how many ties do you own?  I think I probably have about 8, but 5 are in the “will never wear again” category, and one of the 3 “sometimes wear” is the American flag tie I only wear on July 4.

Answer in comments!

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Why did Terrill Thomas die of thirst?

Nobody decided to kill Terrill Thomas.  He kept flooding his cell at the Milwaukee County Jail and making a mess so they just turned off the water to his cell.  Then they left it off until he was dead.  It took six days.  Fellow inmates say he was calling out for water.  Corrections officers say they checked in on Thomas every half hour, and “he had made some type of noise or movement” every time.  Until the last time, when he didn’t make any type of noise or movement because he’d died of thirst.

How did this happen?  It didn’t happen because David Clarke — the sheriff of Milwaukee County, and a top candidate to lead the Department of Homeland Security in the Trump administration — wanted to kill a prisoner in the most agonizing way imaginable.  What kind of psycho would want that?  I don’t think Sheriff Clarke wanted to kill a newborn baby either.  The baby was the fourth person to die in the jail since April.

These people died because nobody really seems to care what happens in the Milwaukee County Jail.  The medical services there are run by Armor Correctional Health Services, a company which oversees healthcare for 40,000 inmates in 8 states.  What are you saying about your priorities if you call your health care company “Armor”?

Armor’s glassdoor page doesn’t make it sound like a great place to work.  One employee writes:   “Stop being bean counters and start listening to your employees. We are asked to do too much: too many patients, too many intakes, not nearly enough staff to be in compliance with your own rules!”  Armor was sued by the state of New York this year over 12 inmates who died in the Nassau County Jail, including Daniel Pantera, who died of hypothermia in solitary confinement; they settled the suit last month for $350,000 and are barred for three years from bidding for contracts in the state.  Armor does get a very nice endorsement, though, from Palm Beach County Sheriff Ric Bradshaw, who says right on the front of their website, “Armor stands out as an exemplary model of what partnership in correctional health should look like.”  Bradshaw’s department was held liable this year for $22.4m in damages to Dontrell Stephens, and this summer settled for $550,000 against a former U.S. Marshall who said he was roughed up by deputies after stopping to help the victims of a traffic accident.

Here in Wisconsin, Armor’s performance is overseen by Ronald Shansky, a court-appointed monitor and the first president of the Society of Correctional Physicians.  Some of what Shansky has to say, based on his visit to Milwaukee County correctional facilities last month:

screen-shot-2016-12-01-at-1-dec-4-12-pm

As for the deaths:

screen-shot-2016-12-01-at-1-dec-4-15-pm

Shansky also, I should say, has a lot of praise for some staff members at the jail, characterizing them as devoted to their jobs and patients and doing the best they can under strained circumstances.  And I believe that’s true.  Again:  the doctors of Armor didn’t want Terrill Thomas to spend six days dying of thirst.  Neither did the CEO of Armor.  Neither did David Clarke.  But it happened.  And everyone participated in creating the circumstances under which it happened, and under which it’s likely to happen again:  public services outsourced to companies without the staff or resources to do the job right.

It starts with jails.  But it goes on to schools, to parking, to Medicare, to policing, to the maintenance of our bridges and roads.  You’ll hear people say those services should be run like businesses.  We can see in Milwaukee County what that looks like.  Does it look good?

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Call for nominations for the Chern Medal

This is a guest post by Caroline Series.

The Chern Medal is a relatively new prize, awarded once every four years jointly by the IMU and the Chern Medal Foundation (CMF) to an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics. Funded by the CMF, the Medalist receives a cash prize of US$ 250,000. In addition, each Medalist may nominate one or more organizations to receive funding totalling US$ 250,000, for the support of research, education, or other outreach programs in the field of mathematics.

Professor Chern devoted his life to mathematics, both in active research and education, and in nurturing the field whenever the opportunity arose. He obtained fundamental results in all the major aspects of modern geometry and founded the area of global differential geometry. Chern exhibited keen aesthetic tastes in his selection of problems, and the breadth of his work deepened the connections of geometry with different areas of mathematics. He was also generous during his lifetime in his personal support of the field.

Nominations should be sent to the Prize Committee Chair: Caroline Series, email: chair(at)chern18.mathunion.org by 31st December 2016. Further details and nomination guidelines for this and the other IMU prizes can be found here.  Note that previous winners of other IMU prizes, such as the Fields Medal, are not eligible for consideration.

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Double standards in baby names

People love to make fun of George Foreman because he named all his sons George Foreman.  But former Secretary of State Lawrence Eagleburger named all his sons Lawrence Eagleburger and nobody raises a peep!  There’s no justice.

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