A few days ago an historic 160-page paper with a very short title MIP*=RE was uploaded to the arXive by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. I am thankful to Dorit Aharonov and Alon Rosen for telling me about it. The paper simultaneously settles several major open problems in quantum computational complexity, mathematical foundations of quantum physics, and mathematics. Congratulations to Zhengfeng, Anand, Thomas, John, and Henry!
A tweet by Ryan
The new paper dramatically improved the 2019 result by Anand Natarajan, and John Wright asserting that “NEEXP in MIP*”.
In this post I will talk a little about two corners of this work and neglect many others: I will describe a few conjectures about infinite groups related to the new result, and I will give a very gentle beginning-of-an-introduction to interactive proofs. I will also give some useful links to papers, presentations and blog posts.
Boris Tsirelson is one of my mathematical heroes. The new paper gives a negative answer to an important problem posed by Tsirelson. (Here is a great interview with Boris.)
My Qstate: a master project (Vidick’s memories of the road since Julia Kempe offered him the problem 14 years ago; with a lot of scientific content). Older related posts on My Qstate I, II, III.
A Notices AMS article by Vidick: From Operator Algebras to Complexity Theory and Back.
Shtetl Optimized: MIP*=RE; GLL (Ken Regan) Halting Is Poly-Time Quantum Provable ; Other posts on blogs: WIT (Boaz Barak); CC (Lance Fortnow); WN; Posts on an earlier 2019 result MIP* contains NEEXP
Quanta Magazine: An article by Kevin Hartnett about an earlier result MIP*=NEEXP; An article about the new result.
Older posts here: about Vidick- 2012 paper (among various updates); a 2008 post mentioning sofic groups (among various updates);
Videotaped lectures: from our recent winter school Thomas Vidick on quantum protocols Video 1, Video 2, Video3.
(I am thankful to Alex Lubotzky for telling me about the algebra background.)
Links: Finitary approximations of groups and their applications, by Andreas Thom, Andreas’ ICM 2018 videotaped lecture. And a great video: Best of Andreas Thom. See also this paper Stability, cohomology vanishing, and non-approximable groups by Marcus De Chiffre, Lev Glebsky, Alex Lubotzky, and Andreas Thom.
The assertion of Connes’ embedding conjecture refuted in the MIP*=RE paper would imply several (outrageous :)) stronger conjectures that are still open. One is the conjecture of Connes that every group is “hyperlinear.” Another famous conjecture (an affirmative answer to a question posed by Gromov) is that every group is sofic. As sofic groups are hyperlinear we can now expect (ever more than before) that non-sofic and even non hyperlinear groups will be found. Here is a rough short explanation what these conjectures are about. (Kirchberg’s conjecture, is another group theoretic question of this nature.)
Every finite group is a permutation group and is a linear group. This is not the case for infinite groups and there are various interesting notions of “approximately-permutation-group” (this is “sofic”) and “approximately linear” (this is called “hyperlinear”).
Given a group Γ we want to find a sequence of functions
Such that asymptotically as n grows these functions are “almost homomorphisms” with respect to certain metrics DIST on or respectively. This means that for every two elements
,
tends to zero when n tends to infinity.
Now,
And there are various other metrics that were also considered. The assertion of the famous embedding conjecture by Connes on von-Neumann algebras (now refuted by the new result) implies that every group is hyperlinear.
A remaining wish list: Find a non sofic group; find a non-hyperlinear group; refute Kirchberg’s conjecture (if it was not already refuted).
Links: here are slides of a great talk by Yael Kalai: The evolution of proof in computer science; an a blog post on this topic by Yael Kalai, and a post here about Yael’s 2018 ICM paper and lecture.
A decision problem is in P if there is a polynomial time algorithm (in terms of the input size) to test if the answer is yes or no. A decision problem is in NP if there is a proof for a YES answer that can be verified in a polynomial time.
Here are two examples: The question if graph has a perfect matching is in P. The question if graph has an Hamiltonian cycle is in NP. If the answer is yes a prover can give a proof that requires the verifier a polynomial number of steps to verify.
IP is a complexity class based on a notion of interactive proof where, based on a protocol for questions and answers, the prover can convince the verifier (with arbitrary high probability) that the answer is yes. Following a sequence of startling developments Adi Shamir proved that IP is quite a large complexity space P-space. When we consider several non-interacting provers (two provers suffice) the computational power denoted by MIP is even larger: László Babai, Lance Fortnow, and Cartsen Lund proved that MIP=NEXP! NEXP is the class of decision problems where if the answer is yes a prover can give a proof that requires the verifier (at most) an exponential number of steps to verify.
We replace the model of classical computation with quantum computation. Each of the two provers, Prover1 and Prover2, have access to separate sets of m qubits but they can prepare in advance a complicated quantum state on those 2m qubits. When we run the verification protocol each prover has access only to its m qubits and, like in the classical case, the two provers cannot communicate. These types of verification protocols represent the complexity class MIP*. In 2012 and Tsuyoshi Ito and Thomas Vidick proved that MIP* contains NEXP. In this 2012 post I reported an unusual seminar we ran on the problem.
Interactive quantum lecture: We had an unususal quantum seminar presentation by Michael Ben-Or on the work A multi-prover interactive proof for NEXP sound against entangled provers by Tsuyoshi Ito and Thomas Vidick. Michael ran Vidick’s videotaped recent talk on the matter and from time to time he and Dorit acted as a pair of prover and the other audience as verifier. (Michael was one of the authors of the very first paper introducing multi-prover interactive proofs.)
Let me mention also a related 2014 paper by Yael Kalai, Ran Raz, and Ron Rothblum: How to delegate computations: the power of no-signaling proofs. They considered two provers that are limited by the “non-signaling principle” and showed that the power of interactive proofs contains NEXP. (Here is a videotaped lecture by Ran Raz.)
In April 2019, Anand Natarajan and John Wright uploaded a paper with a proof that MIP* contain NEEXP. (NEEXP is the class of decision problems where if the answer is yes a prover can give a proof that requires the verifier (at most) doubly exponential number of steps to verify.)
Here is a nice quote from the Harnett’s quanta paper regarding the Natarajan-Wright breakthrough:
Some problems are too hard to solve in any reasonable amount of time. But their solutions are easy to check. Given that, computer scientists want to know: How complicated can a problem be while still having a solution that can be verified?
Turns out, the answer is: Almost unimaginably complicated.
In a paper released in April, two computer scientists dramatically increased the number of problems that fall into the hard-to-solve-but-easy-to-verify category. They describe a method that makes it possible to check answers to problems of almost incomprehensible complexity. “It seems insane,” said Thomas Vidick, a computer scientist at the California Institute of Technology who wasn’t involved in the new work.
Now with the new result, I wonder if this bold philosophical interpretation is sensible: There is a shared quantum state that will allow two non-interacting provers (with unlimited computational power) to convince a mathematician if a given mathematical statement has a proof, and also to convince a historian or a futurist about any question regarding the past or future evolution of the universe.
The negative answer to Tsirelson problem asserts roughly that there are types of correlations that can be produced by an infinite quantum systems, but that can’t even be approximated by a finite system. Connes’ 1976 embedding conjecture (now refuted) from the theory of von Neumann algebras asserts that “Every type von Neumann factor embeds in an ultrapower of a hyperfinite factor.”
The abstract of the new paper mentions a few other works that are important for the new proof.
Anand Natarajan and Thomas Vidick. Low-degree testing for quantum states, and a quantum entangled games PCP for QMA. (FOCS, 2018.)
Anand Natarajan and John Wright. NEEXP ⊆ MIP∗ (FOCS 2019) (We mentioned it above.)
Joe Fitzsimons, Zhengfeng Ji, Thomas Vidick, and Henry Yuen. Quantum proof systems
for iterated exponential time, and beyond.
The abstract also mentions two papers about the connections with Tsirelson problem and Connes embedding conjecture
Tobias Fritz. Tsirelson’s problem and Kirchberg’s conjecture. Reviews in Mathematical
Physics, 24(05):1250012, 2012.
Marius Junge, Miguel Navascues, Carlos Palazuelos, David Perez-Garcia, Volkher B Scholz, and Reinhard F Werner. Connes’ embedding problem and Tsirelson’s problem. Journal of Mathematical Physics, 52(1):012102, 2011.
Let me also mention
Narutaka Ozawa. About the Connes embedding conjecture. Japanese Journal of Mathematics, 8(1):147–183, 2013.
Authors: Bartłomiej Dudek, Paweł Gawrychowski, Tatiana Starikovskaya
Download: PDF
Abstract: In the problem of $\texttt{Generalised Pattern Matching}\ (\texttt{GPM})$
[STOC'94, Muthukrishnan and Palem], we are given a text $T$ of length $n$ over
an alphabet $\Sigma_T$, a pattern $P$ of length $m$ over an alphabet
$\Sigma_P$, and a matching relationship $\subseteq \Sigma_T \times \Sigma_P$,
and must return all substrings of $T$ that match $P$ (reporting) or the number
of mismatches between each substring of $T$ of length $m$ and $P$ (counting).
In this work, we improve over all previously known algorithms for this problem
for various parameters describing the input instance:
* $\mathcal{D}\,$ being the maximum number of characters that match a fixed character,
* $\mathcal{S}\,$ being the number of pairs of matching characters,
* $\mathcal{I}\,$ being the total number of disjoint intervals of characters that match the $m$ characters of the pattern $P$.
At the heart of our new deterministic upper bounds for $\mathcal{D}\,$ and $\mathcal{S}\,$ lies a faster construction of superimposed codes, which solves an open problem posed in [FOCS'97, Indyk] and can be of independent interest.
To conclude, we demonstrate first lower bounds for $\texttt{GPM}$. We start by showing that any deterministic or Monte Carlo algorithm for $\texttt{GPM}$ must use $\Omega(\mathcal{S})$ time, and then proceed to show higher lower bounds for combinatorial algorithms. These bounds show that our algorithms are almost optimal, unless a radically new approach is developed.
Authors: Marc Hellmuth, Carsten R. Seemann, Peter F. Stadler
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Abstract: Binary relations derived from labeled rooted trees play an import role in
mathematical biology as formal models of evolutionary relationships. The
(symmetrized) Fitch relation formalizes xenology as the pairs of genes
separated by at least one horizontal transfer event. As a natural
generalization, we consider symmetrized Fitch maps, that is, symmetric maps
$\varepsilon$ that assign a subset of colors to each pair of vertices in $X$
and that can be explained by a tree $T$ with edges that are labeled with
subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$
if and only if $m$ appears in a label along the unique path between $x$ and $y$
in $T$. We first give an alternative characterization of the monochromatic case
and then give a characterization of symmetrized Fitch maps in terms of
compatibility of a certain set of quartets. We show that recognition of
symmetrized Fitch maps is NP-complete but FPT in general. In the restricted
case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since
such maps coincide with class of monochromatic Fitch maps whose
graph-representations form precisely the class of complete multi-partite
graphs.
Authors: Kohei Yamada, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda
Download: PDF
Abstract: The longest common subsequence (LCS) problem is a central problem in
stringology that finds the longest common subsequence of given two strings $A$
and $B$. More recently, a set of four constrained LCS problems (called
generalized constrained LCS problem) were proposed by Chen and Chao [J. Comb.
Optim, 2011]. In this paper, we consider the substring-excluding constrained
LCS (STR-EC-LCS) problem. A string $Z$ is said to be an STR-EC-LCS of two given
strings $A$ and $B$ excluding $P$ if, $Z$ is one of the longest common
subsequences of $A$ and $B$ that does not contain $P$ as a substring. Wang et
al. proposed a dynamic programming solution which computes an STR-EC-LCS in
$O(mnr)$ time and space where $m = |A|, n = |B|, r = |P|$ [Inf. Process. Lett.,
2013]. In this paper, we show a new solution for the STR-EC-LCS problem. Our
algorithm computes an STR-EC-LCS in $O(n|\Sigma| + (L+1)(m-L+1)r)$ time where
$|\Sigma| \leq \min\{m, n\}$ denotes the set of distinct characters occurring
in both $A$ and $B$, and $L$ is the length of the STR-EC-LCS. This algorithm is
faster than the $O(mnr)$-time algorithm for short/long STR-EC-LCS (namely, $L
\in O(1)$ or $m-L \in O(1)$), and is at least as efficient as the $O(mnr)$-time
algorithm for all cases.
Authors: Lars Jaffke, Mateus de Oliveira Oliveira, Hans Raj Tiwary
Download: PDF
Abstract: It can be shown that each permutation group $G \sqsubseteq S_n$ can be
embedded, in a well defined sense, in a connected graph with $O(n+|G|)$
vertices. Some groups, however, require much fewer vertices. For instance,
$S_n$ itself can be embedded in the $n$-clique $K_n$, a connected graph with n
vertices. In this work, we show that the minimum size of a context-free grammar
generating a finite permutation group $G \sqsubseteq S_n$ can be upper bounded
by three structural parameters of connected graphs embedding $G$: the number of
vertices, the treewidth, and the maximum degree. More precisely, we show that
any permutation group $G \sqsubseteq S_n$ that can be embedded into a connected
graph with $m$ vertices, treewidth k, and maximum degree $\Delta$, can also be
generated by a context-free grammar of size $2^{O(k\Delta\log\Delta)}\cdot
m^{O(k)}$. By combining our upper bound with a connection between the extension
complexity of a permutation group and the grammar complexity of a formal
language, we also get that these permutation groups can be represented by
polytopes of extension complexity $2^{O(k \Delta\log \Delta)}\cdot m^{O(k)}$.
The above upper bounds can be used to provide trade-offs between the index of
permutation groups, and the number of vertices, treewidth and maximum degree of
connected graphs embedding these groups. In particular, by combining our main
result with a celebrated $2^{\Omega(n)}$ lower bound on the grammar complexity
of the symmetric group $S_n$ we have that connected graphs of treewidth
$o(n/\log n)$ and maximum degree $o(n/\log n)$ embedding subgroups of $S_n$ of
index $2^{cn}$ for some small constant $c$ must have $n^{\omega(1)}$ vertices.
This lower bound can be improved to exponential on graphs of treewidth
$n^{\varepsilon}$ for $\varepsilon<1$ and maximum degree $o(n/\log n)$.
Authors: Joon-Seok Kim, Carola Wenk
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Abstract: Simplification is one of the fundamental operations used in geoinformation
science (GIS) to reduce size or representation complexity of geometric objects.
Although different simplification methods can be applied depending on one's
purpose, a simplification that many applications employ is designed to preserve
their spatial properties after simplification. This article addresses one of
the 2D simplification methods, especially working well on human-made structures
such as 2D footprints of buildings and indoor spaces. The method simplifies
polygons in an iterative manner. The simplification is segment-wise and takes
account of intrusion, extrusion, offset, and corner portions of 2D structures
preserving its dominant frame.
Authors: João F. Doriguello, Ashley Montanaro
Download: PDF
Abstract: In this work we revisit the Boolean Hidden Matching communication problem,
which was the first communication problem in the one-way model to demonstrate
an exponential classical-quantum communication separation. In this problem,
Alice's bits are matched into pairs according to a partition that Bob holds.
These pairs are compressed using a Parity function and it is promised that the
final bit-string is equal either to another bit-string Bob holds, or its
complement. The problem is to decide which case is the correct one. Here we
generalize the Boolean Hidden Matching problem by replacing the parity function
with an arbitrary function $f$. Efficient communication protocols are presented
depending on the sign-degree of $f$. If its sign-degree is less than or equal
to 1, we show an efficient classical protocol. If its sign-degree is less than
or equal to $2$, we show an efficient quantum protocol. We then completely
characterize the classical hardness of all symmetric functions $f$ of
sign-degree greater than or equal to $2$, except for one family of specific
cases. We also prove, via Fourier analysis, a classical lower bound for any
function $f$ whose pure high degree is greater than or equal to $2$. Similarly,
we prove, also via Fourier analysis, a quantum lower bound for any function $f$
whose pure high degree is greater than or equal to $3$. These results give a
large family of new exponential classical-quantum communication separations.
Scott’s preface: Imagine that every time you turned your blog over to a certain topic, you got denounced on Twitter and Reddit as a privileged douchebro, entitled STEMlord, counterrevolutionary bourgeoisie, etc. etc. The sane response would simply be to quit blogging about that topic. But there’s also an insane (or masochistic?) response: the response that says, “but if everyone like me stopped talking, we’d cede the field by default to the loudest, angriest voices on all sides—thereby giving those voices exactly what they wanted. To hell with that!”
A few weeks ago, while I was being attacked for sharing Steven Pinker’s guest post about NIPS vs. NeurIPS, I received a beautiful message of support from a PhD student in physical chemistry and quantum computing named Karen Morenz. Besides her strong words of encouragement, Karen wanted to share with me an essay she had written on Medium about why too many women leave STEM.
Karen’s essay, I found, marshaled data, logic, and her own experience in support of an insight that strikes me as true and important and underappreciated—one that dovetails with what I’ve heard from many other women in STEM fields, including my wife Dana. So I asked Karen for permission to reprint her essay on this blog, and she graciously agreed.
Briefly: anyone with a brain and a soul wants there to be many more women in STEM. Karen outlines a realistic way to achieve this shared goal. Crucially, Karen’s way is not about shaming male STEM nerds for their deep-seated misogyny, their arrogant mansplaining, or their gross, creepy, predatory sexual desires. Yes, you can go the shaming route (God knows it’s being tried). If you do, you’ll probably snare many guys who really do deserve to be shamed as creeps or misogynists, along with many more who don’t. Yet for all your efforts, Karen predicts, you’ll no more solve the original problem of too few women in STEM, than arresting the kulaks solved the problem of lifting the masses out of poverty.
For you still won’t have made a dent in the real issue: namely that, the way we’ve set things up, pursuing an academic STEM career demands fanatical devotion, to the exclusion of nearly everything else in life, between the ages of roughly 18 and 35. And as long as that’s true, Karen says, the majority of talented women are going to look at academic STEM, in light of all the other great options available to them, and say “no thanks.” Solving this problem might look like more money for maternity leave and childcare. It might also look like re-imagining the academic career trajectory itself, to make it easier to rejoin it after five or ten years away. Way back in 2006, I tried to make this point in a blog post called Nerdify the world, and the women will follow. I’m grateful to Karen for making it more cogently than I did.
Without further ado, here’s Karen’s essay. –SA
by Karen Morenz
Everyone knows that you’re not supposed to start your argument with ‘everyone knows,’ but in this case, I think we ought to make an exception:
Everyone knows that STEM (Science, Technology, Engineering and Mathematics) has a problem retaining women (see, for example Jean, Payne, and Thompson 2015). We pour money into attracting girls and women to STEM fields. We pour money into recruiting women, training women, and addressing sexism, both overt and subconscious. In 2011, the United States spent nearly $3 billion tax dollars on STEM education, of which roughly one third was spent supporting and encouraging underrepresented groups to enter STEM (including women). And yet, women are still leaving at alarming rates.
Alarming? Isn’t that a little, I don’t know, alarmist? Well, let’s look at some stats.
A recent report by the National Science Foundation (2011) found that women received 20.3% of the bachelor’s degrees and 18.6% of the PhD degrees in physics in 2008. In chemistry, women earned 49.95% of the bachelor’s degrees but only 36.1% of the doctoral degrees. By comparison, in biology women received 59.8% of the bachelor’s degrees and 50.6% of the doctoral degrees. A recent article in Chemical and Engineering News showed a chart based on a survey of life sciences workers by Liftstream and MassBio demonstrating how women are vastly underrepresented in science leadership despite earning degrees at similar rates, which I’ve copied below. The story is the same in academia, as you can see on the second chart — from comparable or even larger number of women at the student level, we move towards a significantly larger proportion of men at the more and more advanced stages of an academic career.
Although 74% of women in STEM report “loving their work,” half (56%, in fact) leave over the course of their career — largely at the “mid-level” point, when the loss of their talent is most costly as they have just completed training and begun to contribute maximally to the work force.
A study by Dr. Flaherty found that women who obtain faculty position in astronomy spent on average 1 year less than their male counterparts between completing their PhD and obtaining their position — but he concluded that this is because women leave the field at a rate 3 to 4 times greater than men, and in particular, if they do not obtain a faculty position quickly, will simply move to another career. So, women and men are hired at about the same rate during the early years of their post docs, but women stop applying to academic positions and drop out of the field as time goes on, pulling down the average time to hiring for women.
There are many more studies to this effect. At this point, the assertion that women leave STEM at an alarming rate after obtaining PhDs is nothing short of an established fact. In fact, it’s actually a problem across all academic disciplines, as you can see in this matching chart showing the same phenomenon in humanities, social sciences, and education. The phenomenon has been affectionately dubbed the “leaky pipeline.”
But hang on a second, maybe there just aren’t enough women qualified for the top levels of STEM? Maybe it’ll all get better in a few years if we just wait around doing nothing?
Nope, sorry. This study says that 41% of highly qualified STEM people are female. And also, it’s clear from the previous charts and stats that a significantly larger number of women are getting PhDs than going on the be professors, in comparison to their male counterparts. Dr. Laurie Glimcher, when she started her professorship at Harvard University in the early 1980s, remembers seeing very few women in leadership positions. “I thought, ‘Oh, this is really going to change dramatically,’ ” she says. But 30 years later, “it’s not where I expected it to be.” Her experiences are similar to those of other leading female faculty.
So what gives? Why are all the STEM women leaving?
It is widely believed that sexism is the leading problem. A quick google search of “sexism in STEM” will turn up a veritable cornucopia of articles to that effect. And indeed, around 60% of women report experiencing some form of sexism in the last year (Robnett 2016). So, that’s clearly not good.
And yet, if you ask leading women researchers like Nobel Laureate in Physics 2018, Professor Donna Strickland, or Canada Research Chair in Advanced Functional Materials (Chemistry), Professor Eugenia Kumacheva, they say that sexism was not a barrier in their careers. Moreover, extensive research has shown that sexism has overall decreased since Professors Strickland and Kumacheva (for example) were starting their careers. Even more interestingly, Dr. Rachael Robnett showed that more mathematical fields such as Physics have a greater problem with sexism than less mathematical fields, such as Chemistry, a finding which rings true with the subjective experience of many women I know in Chemistry and Physics. However, as we saw above, women leave the field of Chemistry in greater proportions following their BSc than they leave Physics. On top of that, although 22% of women report experiencing sexual harassment at work, the proportion is the same among STEM and non-STEM careers, and yet women leave STEM careers at a much higher rate than non-STEM careers.
So,it seems that sexism can not fully explain why women with STEM PhDs are leaving STEM. At the point when women have earned a PhD, for the most part they have already survived the worst of the sexism. They’ve already proven themselves to be generally thick-skinned and, as anyone with a PhD can attest, very stubborn in the face of overwhelming difficulties. Sexism is frustrating, and it can limit advancement, but it doesn’t fully explain why we have so many women obtaining PhDs in STEM, and then leaving. In fact, at least in the U of T chemistry department, faculty hires are directly proportional to the applicant pool —although the exact number of applicants are not made public, from public information we can see that approximately one in four interview invitees are women, and approximately one in four hires are women. Our hiring committees have received bias training, and it seems that it has been largely successful. That’s not to say that we’re done, but it’s time to start looking elsewhere to explain why there are so few women sticking around.
So why don’t more women apply?
Well, one truly brilliant researcher had the groundbreaking idea of asking women why they left the field. When you ask women why they left, the number one reason they cite is balancing work/life responsibilities — which as far as I can tell is a euphemism for family concerns.
The research is in on this. Women who stay in academia expect to marry later, and delay or completely forego having children, and if they do have children, plan to have fewer than their non-STEM counterparts (Sassler et al 2016, Owens 2012). Men in STEM have no such difference compared to their non-STEM counterparts; they marry and have children about the same ages and rates as their non-STEM counterparts (Sassler et al 2016). Women leave STEM in droves in their early to mid thirties (Funk and Parker 2018) — the time when women’s fertility begins to decrease, and risks of childbirth complications begin to skyrocket for both mother and child. Men don’t see an effect on their fertility until their mid forties. Of the 56% of women who leave STEM, 50% wind up self-employed or using their training in a not for profit or government, 30% leave to a non-STEM more ‘family friendly’ career, and 20% leave to be stay-at-home moms (Ashcraft and Blithe 2002). Meanwhile, institutions with better childcare and maternity leave policies have twice(!) the number of female faculty in STEM (Troeger 2018). In analogy to the affectionately named “leaky pipeline,” the challenge of balancing motherhood and career has been titled the “maternal wall.”
To understand the so-called maternal wall better, let’s take a quick look at the sketch of a typical academic career.
For the sake of this exercise, let’s all pretend to be me. I’m a talented 25 year old PhD candidate studying Physical Chemistry — I use laser spectroscopy to try to understand atypical energy transfer processes in innovative materials that I hope will one day be used to make vastly more efficient solar panels. I got my BSc in Chemistry and Mathematics at the age of 22, and have published 4 scientific papers in two different fields already (Astrophysics and Environmental Chemistry). I’ve got a big scholarship, and a lot of people supporting me to give me the best shot at an academic career — a career I dearly want. But, I also want a family — maybe two or three kids. Here’s what I can expect if I pursue an academic career:
With any luck, 2–3 years from now I’ll graduate with a PhD, at the age of 27. Academics are expected to travel a lot, and to move a lot, especially in their 20s and early 30s — all of the key childbearing years. I’m planning to go on exchange next year, and then the year after that I’ll need to work hard to wrap up research, write a thesis, and travel to several conferences to showcase my work. After I finish my PhD, I’ll need to undertake one or two post doctoral fellowships, lasting one or two years each, probably in completely different places. During that time, I’ll start to apply for professorships. In order to do this, I’ll travel around to conferences to advertise my work and to meet important leaders in my field, and then, if I am invited for interviews, I’ll travel around to different universities for two or three days at a time to undertake these interviews. This usually occurs in a person’s early 30s — our helpful astronomy guy, Dr. Flaherty, found the average time to hiring was 5 years, so let’s say I’m 32 at this point. If offered a position, I’ll spend the next year or two renovating and building a lab, buying equipment, recruiting talented graduate students, and designing and teaching courses. People work really, really hard during this time and have essentially no leisure time. Now I’m 34. Within usually 5 years I’ll need to apply for tenure. This means that by the time I’m 36, I’ll need to be making significant contributions in my field, and then in the final year before applying for tenure, I will once more need to travel to many conferences to promote my work, in order to secure tenure — if I fail to do so, my position at the university would probably be terminated. Although many universities offer a “tenure extension” in cases where an assistant professor has had a child, this does not solve all of the problems. Taking a year off during that critical 5 or 6 year period often means that the research “goes bad” — students flounder, projects that were promising get “scooped” by competitors at other institutions, and sometimes, in biology and chemistry especially, experiments literally go bad. You wind up needing to rebuild much more than just a year’s worth of effort.
At no point during this time do I appear stable enough, career-wise, to take even six months off to be pregnant and care for a newborn. Hypothetical future-me is travelling around, or even moving, conducting and promoting my own independent research and training students. As you’re likely aware, very pregnant people and newborns don’t travel well. And academia has a very individualistic and meritocratic culture. Starting at the graduate level, huge emphasis is based on independent research, and independent contributions, rather than valuing team efforts. This feature of academia is both a blessing and a curse. The individualistic culture means that people have the independence and the freedom to pursue whatever research interests them — in fact this is the main draw for me personally. But it also means that there is often no one to fall back on when you need extra support, and because of biological constraints, this winds up impacting women more than men.
At this point, I need to make sure that you’re aware of some basics of female reproductive biology. According to Wikipedia, the unquestionable source of all reliable knowledge, at age 25, my risk of conceiving a baby with chromosomal abnormalities (including Down’s Syndrome) is 1 in about 1400. By 35, that risk more than quadruples to 1 in 340. At 30, I have a 75% chance of a successful birth in one year, but by 35 it has dropped to 66%, and by 40 it’s down to 44%. Meanwhile, 87 to 94% of women report at least 1 health problem immediately after birth, and 1.5% of mothers have a severe health problem, while 31% have long-term persistent health problems as a result of pregnancy (defined as lasting more than six months after delivery). Furthermore, mothers over the age of 35 are at higher risk for pregnancy complications like preterm delivery, hypertension, superimposed preeclampsia, severe preeclampsia (Cavazos-Rehg et al 2016). Because of factors like these, pregnancies in women over 35 are known as “geriatric pregnancies” due to the drastically increased risk of complications. This tight timeline for births is often called the “biological clock” — if women want a family, they basically need to start before 35. Now, that’s not to say it’s impossible to have a child later on, and in fact some studies show that it has positive impacts on the child’s mental health. But it is riskier.
So, women with a PhD in STEM know that they have the capability to make interesting contributions to STEM, and to make plenty of money doing it. They usually marry someone who also has or expects to make a high salary as well. But this isn’t the only consideration. Such highly educated women are usually aware of the biological clock and the risks associated with pregnancy, and are confident in their understanding of statistical risks.
The Irish say, “The common challenge facing young women is achieving a satisfactory work-life balance, especially when children are small. From a career perspective, this period of parenthood (which after all is relatively short compared to an entire working life) tends to coincide exactly with the critical point at which an individual’s career may or may not take off. […] All the evidence shows that it is at this point that women either drop out of the workforce altogether, switch to part-time working or move to more family-friendly jobs, which may be less demanding and which do not always utilise their full skillset.”
And in the Netherlands, “The research project in Tilburg also showed that women academics have more often no children or fewer children than women outside academia.” Meanwhile in Italy “On a personal level, the data show that for a significant number of women there is a trade-off between family and work: a large share of female economists in Italy do not live with a partner and do not have children”
Most jobs available to women with STEM PhDs offer greater stability and a larger salary earlier in the career. Moreover, most non-academic careers have less emphasis on independent research, meaning that employees usually work within the scope of a larger team, and so if a person has to take some time off, there are others who can help cover their workload. By and large, women leave to go to a career where they will be stable, well funded, and well supported, even if it doesn’t fulfill their passion for STEM — or they leave to be stay-at-home moms or self-employed.
I would presume that if we made academia a more feasible place for a woman with a family to work, we could keep almost all of those 20% of leavers who leave to just stay at home, almost all of the 30% who leave to self-employment, and all of those 30% who leave to more family friendly careers (after all, if academia were made to be as family friendly as other careers, there would be no incentive to leave). Of course, there is nothing wrong with being a stay at home parent — it’s an admirable choice and contributes greatly to our society. One estimate valued the equivalent salary benefit of stay-at-home parenthood at about $160,000/year. Moreover, children with a stay-at-home parent show long term benefits such as better school performance — something that most academic women would want for their children. But a lot of people only choose it out of necessity — about half of stay-at-home moms would prefer to be working (Ciciolla, Curlee, & Luthar 2017). When the reality is that your salary is barely more than the cost of daycare, then a lot of people wind up giving up and staying home with their kids rather than paying for daycare. In a heterosexual couple it will usually be the woman that winds up staying home since she is the one who needs to do things like breast feed anyways. And so we lose these women from the workforce.
And yet, somehow, during this informal research adventure of mine, most scholars and policy makers seem to be advising that we try to encourage young girls to be interested in STEM, and to address sexism in the workplace, with the implication that this will fix the high attrition rate in STEM women. But from what I’ve found, the stats don’t back up sexism as the main reason women leave. There is sexism, and that is a problem, and women do leave STEM because of it — but it’s a problem that we’re already dealing with pretty successfully, and it’s not why the majority of women who have already obtained STEM PhDs opt to leave the field. The whole family planning thing is huge and for some reason, almost totally swept under the rug — mostly because we’re too shy to talk about it, I think.
In fact, I think that the plethora of articles suggesting that the problem is sexism actually contribute to our unwillingness to talk about the family planning problem, because it reinforces the perception that that men in power will not hire a woman for fear that she’ll get pregnant and take time off. Why would anyone talk about how they want to have a family when they keep hearing that even the mere suggestion of such a thing will limit their chances of being hired? I personally know women who have avoided bringing up the topic with colleagues or supervisors for fear of professional repercussions. So we spend all this time and energy talking about how sexism is really bad, and very little time trying to address the family planning challenge, because, I guess, as the stats show, if women are serious enough about science then they just give up on the family (except for the really, really exceptional ones who can handle the stresses of both simultaneously).
To be very clear, I’m not saying that sexism is not a problem. What I am saying is that, thanks to the sustained efforts of a large number of people over a long period of time, we’ve reduced the sexism problem to the point where, at least at the graduate level, it is no longer the largest major barrier to women’s advancement in STEM. Hurray! That does not mean that we should stop paying attention to the issue of sexism, but does mean that it’s time to start paying more attention to other issues, like how to properly support women who want to raise a family while also maintaining a career in STEM.
So what can we do to better support STEM women who want families?
A couple of solutions have been tentatively tested. From a study mentioned above, it’s clear that providing free and conveniently located childcare makes a colossal difference to women’s choices of whether or not to stay in STEM, alongside extended and paid maternity leave. Another popular and successful strategy was implemented by a leading woman in STEM, Laurie Glimcher, a past Harvard Professor in Immunology and now CEO of Dana-Farber Cancer Institute. While working at NIH, Dr. Glimcher designed a program to provide primary caregivers (usually women) with an assistant or lab technician to help manage their laboratories while they cared for children. Now, at Dana-Farber Cancer Institute, she has created a similar program to pay for a technician or postdoctoral researcher for assistant professors. In the academic setting, Dr. Glimcher’s strategies are key for helping to alleviate the challenges associated with the individualistic culture of academia without compromising women’s research and leadership potential.
For me personally, I’m in the ideal situation for an academic woman. I graduated my BSc with high honours in four years, and with many awards. I’ve already had success in research and have published several peer reviewed papers. I’ve faced some mild sexism from peers and a couple of TAs, but nothing that’s seriously held me back. My supervisors have all been extremely supportive and feminist, and all of the people that I work with on a daily basis are equally wonderful. Despite all of this support, I’m looking at the timelines of an academic career, and the time constraints of female reproduction, and honestly, I don’t see how I can feasible expect to stay in academia and have the family life I want. And since I’m in the privileged position of being surrounded by supportive and feminist colleagues, I can say it: I’m considering leaving academia, if something doesn’t change, because even though I love it, I don’t see how it can fit in to my family plans.
But wait! All of these interventions are really expensive. Money doesn’t just grow on trees, you know!
It doesn’t in general, but in this case it kind of does — well, actually, we already grew it. We spend billions of dollars training women in STEM. By not making full use of their skills, if we look at only the american economy, we are wasting about $1.5 billion USD per year in economic benefits they would have produced if they stayed in STEM. So here’s a business proposal: let’s spend half of that on better family support and scientific assistants for primary caregivers, and keep the other half in profit. Heck, let’s spend 99% — $1.485 billion (in the states alone) on better support. That should put a dent in the support bill, and I’d sure pick up $15 million if I saw it lying around. Wouldn’t you?
By demonstrating that we will support women in STEM who choose to have a family, we will encourage more women with PhDs to apply for the academic positions that they are eminently qualified for. Our institutions will benefit from the wider applicant pool, and our whole society will benefit from having the skills of these highly trained and intelligent women put to use innovating new solutions to our modern day challenges.
The Theory Group in the Dept. of Computing Science at U. of Alberta invites applications for a postdoc position.
The successful applicant is expected to work closely with Zachary Friggstad and Mohammad Salavatipour in the areas of: approximation algorithms, hardness of approximation, combinatorial optimization. For details see https://webdocs.cs.ualberta.ca/~mreza/pdf-ad7.pdf
Website: https://webdocs.cs.ualberta.ca/~mreza/
Email: mrs@ualberta.ca
Recently read: Robert Bosch’s Opt Art: From Mathematical Optimization to Visual Design, as reviewed in The Math Less Traveled (). Some others that are less mathematical: Susan Phillips The City Beneath (the rare book about graffiti where the words are more interesting than the photos); Kelly & Zach Weinersmith’s Soonish; Spectrum 26: The Best in Contemporary Fantastic Art.
I’m sad that Cambridge Zero is not a name for the convention that, in writing a decimal number between 0 and 1, the leading 1’s digit is included (e.g. 0.618034, not .618034). (.)
John Wallis and the Roof of the Sheldonian Theatre (, via). It’s an elegant way to build a wide roof out of short beams with no joinery. But the history is somewhat lacking: Similar structures were known much earlier to Leonardo Da Vinci, Villard de Honnecourt, and Sebastiano Serlio. See Sylvie Duvernoy, “An introduction to Leonardo’s lattices”.
The 100 worst ed-tech debacles of the decade ().
Rhapsody in Blue (1924) just reached the public domain, showing the insanity of U.S. copyright law ().
Geometric collages by Augustine Kofie (). More at the artist’s web site.
How few -gons can make a polyhedron, for different choices of ? (, via, via.) The answers include an amazing high-genus polyhedron with 12 faces, each of which is an 11-gon, posted Nov 2018 by Ivan Neretin (sadly, with multiple adjacencies for some pairs of faces, dubious by some definitions of polyhedra, rather than having one edge per face pair).
Information about ACM’s opposition to mandatory open access of publicly-funded research ().
Russian Academy of Science cleans house (, via). Their investigation finds 2528 plagiarized papers in 541 Russian-language journals, gets roughly 1/3 of them retracted, and threatens uncooperative journals with de-listing from their indexes. They also recommended blackballing 56 candidates for academy membership over plagiarism and other misbehavior.
The New York subway system thinks it has copyright on any stylized geometric map of its system and is sending takedown notices to the artist of the unofficial map used by Wikipedia (, via). As the article clearly explains, none of the underlying data of the map, the approximate geographic locations of its stations, or the idea of geometric stylization are copyrightable.
Paris Musées releases 100,000 images of artworks for unrestricted public use ().
How simple math can cover even the most complex holes (). Quanta on covering all diameter-one shapes with the smallest possible convex region
MIP*=RE or, less technically, “two entangled provers could convince a polynomial-time verifier than an arbitrary Turing machine halts” (, see also, see also). This appears to be a major breakthrough in quantum complexity and its applications in von Neumann algebra. See also some background from an author.
Computational Geometry Media Exposition coming in Zurich in late June (). It’s one of the events associated with the annual Symposium on Computational Geometry, and was formerly called the video review of computational geometry. This year it’s expanding to a much wider range of media. Submission deadline February 21; see link for details.
How does this intersect David Deutsch’s thoughts in 1985?
Composite crop from homepages |
Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen (JNVWY) have just posted a paper titled . The title means that multiple provers sharing quantum entanglement, given any Turing machine and string accepted by , can convince a polynomial-time bounded verifier with high probability that . The time is polynomial in regardless of how long takes to halt on .
Today we applaud this work and try to convey some short way of apprehending it.
Yoking a classic undecidable problem to a polynomial-time task is not the only surprise. The proof refutes a conjecture in classical functional analysis that had apparently been widely believed. Thus this story continues the theme of surprises and possibly working the wrong way on conjectures, as we also just mentioned in our previous post. The new work subsumes a major paper last year showing that contains nondeterministic double exponential time (), which proves it different from its classical counterpart , which László Babai, Lance Fortnow, and Carsten Lund proved equal to .
The developments have been covered by Scott Aaronson here, Lance here, Boaz Barak here, and in a personal way by Vidick here. The new paper weights in at 165 pages. We will give our own snap-summary and try to add a little from the side.
The refuted conjecture was made by the Fields Medalist Alain Connes in a context having no overt involvement of quantum mechanics. In these 2013 eight–lecture course notes on the conjecture, the word “quantum” appears only once, to say on page 2 of lecture 1:
Other very recent discoveries include the fact that Connes’ embedding conjecture is related to an important problem in Quantum Information Theory, the so-called Tsirelson’s problem…
The problem of Boris Tsirelson ultimately harks back to the theorem of John Bell about correlations that are physically realizable using quantum entanglement but not by any classical physical system. In the CHSH game form of Bell’s theorem, our old friends Alice and Bob can win the game over 85% of the time using quantum, only 75% otherwise. They can get this with just one pair of entangled qubits per trial. Tsirelson proved that the 85% (to wit, ) is optimal. In extensions of these games to larger-size cases, the question becomes: what are the gaps between quantum and classical?
Whether there is a gap of more than a fixed then feeds into interactive protocols. We can have parties trying to prove these gaps using their own entanglement. Where it gets tricky is when you allow Alice and Bob to use larger and larger quantum states and ask, can they achieve the gap with some large enough state? The limiting behavior of the gaps is complex. What JNVWY proved is that this becomes like a halting problem. Not just a halting problem, the Halting Problem. Yet two quantum provers, working for a given gap that is achievable, can prove this to a polynomial-time classical verifier. This is the magic of the theorem.
The reduction from halting to the problem about limits and gaps comes before introducing two-prover systems, as is reflected by JNVWY and also in the wonderful introduction of a 2017 paper by William Slofstra which they reference. In advance of saying more about it, we’ll remark that the new work may provide a new dictionary for translating between (i) issues of finite/infinite precision and other continuous matters, and (ii) possible evolutions of a system of finite size in discrete steps of time and size, where both are unbounded but (in positive cases) finite.
The results strikes Dick and me as shedding new light on a principle stated by David Deutsch in a 1985 paper:
Every finitely realisable physical system can be perfectly simulated by a universal model computing machine operating by finite means.
I was a student at Oxford alongside Deutsch in 1984–1985, and I remember more going on than the searchable record seems to reflect. Deutsch believed that his model of a quantum computer could solve the Halting problem in finite time. He gave at least one talk at the Oxford Mathematical Institute on that claim. As far as I know the claim stayed local to Oxford and generated intense discussion led by Robin Gandy, Roger Penrose, and (if I recall) Angus Macintyre and at least one other person who was versed in random physical processes.
My recollection is that the nub of the technical argument turned on a property of infinite random sequences that, when hashed out, made some associated predicates decidable, so that Deutsch’s functions were classically total computable after all. Thus the hypercomputation claim was withdrawn.
Now, however, I wonder whether the two-prover system constitutes the kind of “machine” that Deutsch was intuitively thinking of. As I recall, his claim was not deemed wrong from first principles but from how theorems about random sequences interacted with machine model definitions. The theory of interactive provers as computational systems was then in its infancy. Could Deutsch have had some inkling of it?
Again we congratulate JNVWY on this achievement of a long-term research goal. Looking at the past, does it relate to the discussion of hypercomputation stemming from the 1980s? We mean a stronger connection than treated here or in this 2018 paper. Is it much different from ones where “the mystery … vanishes when the level of precision is explicitly taken into account” (quoting this). Looking ahead, are there any connection to the physical issues of infinity in finite time that we recently discussed here?
Updates 1/17: Gil Kalai has a post with background on further conjectures impacted by (the failure of) Connes’s conjecture and on quantum prover systems, plus a plethora of links.
A new article in Nature includes the following quotations:
The article and comments on Scott’s blog include interpretations that seem to oppose rather than support Deutsch’s principle on the finiteness of nature. The tandem of two unlimited provers may not qualify as a “finite machine.”
There are comments below querying whether the theorem is in first-order arithmetic or how strong a choice axiom it may need.
[added to first paragraph in second section, added updates]