What is a natural question? Who should decide?

(Thanks to Timothy Chow for inspiring this post.)

My survey on Hilbert's Tenth Problem(see  here) is about variants of the problem. One of them is as follows: 

For which degrees d and number-of-vars n, is Hilbert's tenth problem decidable? undecidable? unknown? 

 I wondered why there was not a website with this information. More generally, the problem didn't seem to be getting much attention. (My survey does report on the attention it has gotten.) 

I got several emails telling me it was the wrong question. I didn't quite know what they meant until Timothy Chow emailed me the following eloquent explanation:

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One reason there isn't already a website of the type you envision is that from a number-theoretic (or decidability) point of view, parameterization by  degree and number of variables is not as natural as it might seem at first glance. The most fruitful lines of research have been geometric, and so geometric concepts such as smoothness, dimension, and genus are more natural than, say, degree. A nice survey by a number theorist is the book Rational Points on Varieties by Bjorn Poonen. Much of it is highly technical; however, reading the preface is very enlightening. Roughly speaking, the current state of the art is that there is really only one known way to prove that a system of Diophantine equations has no rational solution.

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AGAINST THE NUMBER THEORISTS VIEWPOINT:

1) ALICE: Why are you looking for your keys under the lamppost instead of where you dropped them?

   BOB: The light is better here.

2) I can imagine the following conversation:

BILL: I want to know about what happens with degree 3, and number of variables 3.

MATHPERSON: That's the wrong question you moron. The real question is what happens for fixed length of cohomology subchains.

BILL: Why is that more natural?

MATHPERSON: Because that is what we can solve. And besides, I've had 10 papers on it.

FOR THE NUMBER THEORISTS VIEWPOINT

1) They are working on really hard problems so it is natural to gravitate towards those that can be solved.

2) I suspect that the math that comes out of studying classes of equations based on smoothness, dimension, genus is more interesting than what comes out of degree and number of vars. Or at least it has been so far. 

META QUESTION

Who gets to decide what problems are natural?

People outside the field (me in this case) are asking the kind of questions that a layperson would ask and there is some merit to that.

People inside the field KNOW STUFF and hence their opinion of what's interesting to study has some merit. But they can also mistake `I cannot solve X' for `X is not interesting'

Does the university matter?

As we come out of a pandemic with online teaching and research collaborations, how much do we actually need the university?

Theoretical research in computer science, math and elsewhere it hardly slowed down with everyone hunkered down at home, and when it did it was more because of supervising kids with their on-line learning than being away from the university. Collaborating with those around the world was basically the same as collaborating with your colleagues on campus.

Many courses, especially the larger ones, worked about as well on-line as they do in person.

The pandemic is accelerating changes already in place. Before say 1960, the fastest travel for the masses was on train and boats and fast communication limited to expensive phone calls. You needed strong colleagues, a strong library and a strong support staff to be a successful academic. Traveling to meet colleagues and attend conferences was a luxury that few could do often.

The 60's gave us air travel though it wasn't until the 90's that we could readily send academic papers electronically. It's really only recently that we have the infrastructure to allow high-quality teaching and research collaboration online.

Suppose universities now just disappeared. Professors would be free agents, supported by grants and tuition from students taking their on-line courses and consulting. Students would pick and choose courses from the best instructors, their "transcript" being recorded on a blockchain-like database. PhD students would be more of an apprenticeship, a low salary in exchange for personalized mentoring from a professor. 

For the experimental sciences, there would be a set of national labs that professors could join.

Startups would create apps to enable all these things, professors would just become yet another piece of the gig economy. Superstar academics can pull in large salaries, the rest would struggle to make a decent wage--not that different from actors and musicians. 

Don't worry, universities aren't going anywhere soon. Or are they?

Emerging from the Pandemic

The City of Chicago yesterday agreed with the latest CDC guidelines that those of us fully vaccinated no longer have to wear masks in most settings. Lollapalooza, the big Chicago music festival, will be held in full capacity this summer. There are still some restrictions but barring any surprise setbacks should be mostly gone by the Fourth of July.

It's not appropriate to call the pandemic over. Too many countries are suffering and lack good access to vaccines or strong health care. But in most of the US vaccinations are readily available and it really feels like we are putting the pandemic in the past.

Variants that beat the vaccines could emerge. Vaccines could become ineffective over time. Too many still need to be vaccinated and too many don't want to do so. Typically problems start small and quickly get big by exponential growth but likely with enough warning if we need boosters or extra precautions. And all those who cut in line to get shots early are now my canaries for the effects wearing off.

What will this post-pandemic world look like? Continued virtual meetings from people who don't want to spend the 10-15 minutes to get across campus or to come to campus at all. A bigger move to casual dress, even in the corporate world. No more snow days. Changes in ways we won't expect. 

We all have our different attitudes but I'm ready to move on. Time to start the "after times".

Why do countries and companies invest their own money (or is it?) in Quantum Computing (Non-Rhetorical)

 There have been some recent blog posts by Scott (see here) and Lance (see here)  about the hype for SHORT TERM APPLICATIONS of Quantum Computing, which they both object to. 

I have a question that has been touched on but I want to get it more out there.

PREAMBLE TO QUESTION:  The following scenarios, while distasteful, do make sense: 

a) A researcher on their grants exaggerates or even out-right lies about the applications of their work. 

b) A journalist in their articles exaggerates or even out-right lies about the applications of the science they are covering.

c) A company exaggerates or even out-right lies about the applications of their project to a venture capitalist or other kind of investor. 

QUESTION: 

Why does a company or country invest THEIR OWN MONEY into Quantum Computing which is unlikely to have a short term profit or benefit? Presumably they hire honest scientists to tell them the limits of the applications in the short term. 

ANSWERS I HAVE GOTTEN FROM ASKING THIS

1) QC might be able to do something cool and profitable, like factoring, or simulating physics quantum experiments or something else, in the short term. Quantum Crypto is already happening, and that's a close cousin of Quantum Computing. 

2) QC might be able to do something cool and profitable (like in (1)) in the long term, and both companies and countries think they will be around for a long time. (For a list of America's 10 oldest companies see here.) 

3) The company or country is in this for the long term, not for a practical project, but because they realize that doing GOOD SCIENCE is of a general benefit (this might make more sense for a country than a company). And funding Quantum Computing is great for science. 

4) Bait and Switch: The company claims they are doing Quantum to attract very smart people to work with them, and then have those smart people do something else.

5) (this is a variant of 1) While Quantum Computing may not have any short term applications, there will be classical applications INSPIRED by it (this has already happened, see here).

6) Some of these companies make money by applying for GRANTS to do QC, so its NOT their money. In fact, they are using QC to  GET money.

7) For a country its not the leaders money, its that Taxpayer's money- though that still leaves the question of why spend Taxpayer money on this and not on something else.

8) For a company its not their money- its Venture Capitalists  and others (though for a big company like Google I would think it IS their money). 

9) The scientists advising the company or country are giving them bad (or at least self-serving) advice so that those scientists can profit- and do good science. So this is a variant of (3) but without the company or country knowing it. 

10) In some countries and companies group-think sets in, so if the leader (who perhaps is not a scientist) thinks intuitively that QC is good, the scientists who work for them, who know better, choose to not speak up, or else they would lose their jobs...or worse. 

11) For countries this could be like going to the moon: Country A wants to beat Country B to the moon for bragging rights. Scientists get to do good research even if they don't care about bragging rights. 

12) (similar to 11 but for a company) If a company does great work on QC then it is good publicity for that company. 

13) Some variant of the greater fool theory. If so, will there be a bubble? A bail-out? 

Cryptocurrency, Blockchains and NFTs

 I first wrote about bitcoin in this blog ten years ago after I gave a lecture in a cryptography class I taught at Northwestern. Two years later I had a follow-up post, noting the price moved from $3 to $1000 with a market cap of about $11 Billion. My brother who thought they were a scam back then has since become a cryptocurrency convert. The bitcoin market cap is now over a trillion dollars and other cryptocurrencies are not far behind. No longer can we view cryptocurrencies as simply a neat exercise in applied cryptography now that it has serious value.

The main uses of cryptocurrencies are for speculation or illegal activities, such as drug sales, ransoms, money laundering and tax evasion. Sure you can buy a Tesla with bitcoins but that's more of a gimmick. Cryptocurrency spending is simply too slow, expensive and volatile right now to replace other methods of electronic payment. 

Non-fungible tokens (NFTs) truly puzzle me. They are just a digital certificate of authentication. What could you do with them you couldn't do with docusign? Collectibles of publicly available digital goods is a fad already fading.

I'm not a fan of a fiat currency governed by strict rules not under governmental control. Bad things could happen. However thinking of cryptocurrencies and the blockchain technology that underlies them have brought up real needs for our digital world.

• An easy way to pay online without significant fees, expenses or energy consumption.
• An easy and cheap way to transfer money between different countries.
• A distributed database to allow tracking of supply chains, credentials and financial transactions for example. I see less a need to make these databases decentralized.
• A need, for some, to have a digital replacement for the anonymity of cash.
• People need something to believe in once they have given up believing in religion and a functioning democracy. 

Don't change your investing habits based on anything I write in this post. Speculation and illegal activities are powerful forces. Or it could all collapse. Make your bets, or don't.

Note: Since I wrote this post yesterday, Elon Musk tweeted that Tesla will no longer accept bitcoins, and the bitcoin market cap has dropped below a trillion.

Trump, Facebook, and ComplexityBlog

 I care about the Facebook decision to ban Trump, but I do not have a strong opinion about it. I have heard arguments on both sides now, from up and down, and still somehow... I don't know how I feel. So instead of posting my opinion I post other opinions and my opinion of them.

1) Facebook is a private company. If they want to have liberal bias or a free for all or whatever then  it is not the governments place to interfere. If enough people don't like what they see then they will lose customers. The invisible hand of the market will regulate it enough. Libertarians and honest small-gov republicans might believe this. On a personal level, I don't want someone else telling Lance and I that we can't block some comment; however, for now, more people use Facebook then read Complexity Blog. 

2) Facebook is a private company but they need to follow standard business practices of having their uses sign an agreement and stick to it. Since the user signed the agreement, Facebook need only stick to that agreement. This is problematic in that (1) if the agreement is not that rigorous then Facebook can be arbitrary and capricious, but (2) if the agreement is to rigorous then people can game the system. Imagine if Lance and me had  rule that you could not use profanity in the comments. Then someone could comment 

People who think P vs NP is ind of ZFC can go Fortnow themselves. They are so full of Gasarch.

 (Something like this was the subplot of an episode of The Good Fight)

3) Facebook is so big that it has an obligation to let many voices be heard, within reason. This could lead to contradictions and confusions:

a) Facebook cannot ban political actors. What is a political actor? (Jon Voight is pro-trump and Dwayne ``The Rock'' Johnson is anti-trump, but that's not what I mean.) High level people in the two main parties qualify (how high level?). IMHO third parties (Libertarian and Green come to mind) need the most protection since they don't have as many other ways to get out their message and they are serious. (I wonder if Libertarians would object to the Government  forcing Facebook to not ban them). What about the Surprise Party or the Birthday Party (which did have a platform see here). And what about people running for Mayors of small towns (much easier to do now with Facebook)? Should just running be enough to ban banning? 

b) Facebook can ban posts that are a threat to public health and safety. I am thinking of anti-vaxers and insurrectionists, though I am always wary of making them free speech martyrs. 

c) Fortunately a and b above have never conflicted. But they could. I can imagine a president who has lost an election urging his followers to storm the capitol. Then what should Facebook do?  (ADDED LATER- A commenter points to a case where a and b conflicted that is not the obvious case.) 

4) Facebook is so big that it has an obligation to block posts that put people in danger. This may have some of the same problems as point 3---who decides? 

5)  Facebook is so big and controls so much of the discourse that it should be heavily regulated (perhaps like a utility).  This has some of the same problems as above- who decides how to regulate it and how?

6) As a country we want to encourage free speech and a diversity of viewpoints. There are times when blocking someone from posting may be better for free speech then letting them talk. When? When that person is advocating nonsensical views that stifle the public discussion. But I am talking about what the country should want. What do they want? What does Facebook want? Does either entity even know what they want? These are all ill defined questions. 

7) Facebook is a monopoly so use Anti-Trust laws on it. Anti-Trust was originally intended to protect the consumer from price-gouging. Since Facebook is free this would require a new interpretation of antitrust. Judicial activism? The Justices solving a problem that the elected branches of government are currently unable to solve? Is that a bad precedent? What does it mean to break up Facebook anyway--- its a network and hence breaking it up probably doesn't make sense (maybe use MaxCut). 

(ADDED LATER- a commenter pointed out that anti-trust is NOT just for consumer protection, but also about market manipulation to kill small innovators.) 

8) Lets say that Facebook and Society and the Government and... whoever, finally agree on some sort of standards. Then we're done! Not so fast. Facebook is so vast that it would be hard to monitor everything. 

9) As a side note- because Facebook and Twitter have banned or tagged some kinds of speech or even some people, there have been some alternative platforms set up. They always claim that they are PRO FREE SPEECH. Do liberals post on those sites? Do those sights  ban anyone? Do they have SOME rules of discourse? I ask non rhetorically. 

Negotiations

So you got an offer to be an assistant professor in the computer science department at Prestigious U. Congratulations! 

Time to negotiate your offer with the chair. Don't be nervous. This shouldn't be adversarial. Both of you have the same goal in mind--for you to come to Prestigious and be successful. 

Let's discuss the different aspects of each package.

Research 

Funds for supporting your research such as equipment, graduate student support, travel and postdocs. Here you should explain what you need to be successful. This will vary by subdiscipline, a systems researcher will need more equipment and students than a theorist. Keep in mind the university is giving you funds for 2-4 years to start your research, after which you are expected to fund your own research via grants.

I don't recommend taking on a postdoc right at the start of your first academic appointment. Postdocs require good mentoring while you need to spend the first year getting your research up and running. If you do ask for postdoc money, ask to have a flexible start time.

Many departments give course reductions to get your research going. I'd suggest asking to spend your first semester teaching a graduate topics course based on your thesis research to pick up some PhD students followed by a semester with no classes to get you research program going.

Salary

This includes actual salary, which is also the base for future raises, and summer salary in the first couple of years. Feel free to ask for more salary, but often these numbers are fixed for new assistant professors. There is more give if you take an academic job later in your career. You could also say something like, "Well if you can't give me more salary maybe you could give me another semester of grad student support?"

Partner

It seems 80% of the time, a job candidate has a partner that needs accommodating. Don't wait until the end of negotiations, bring it up early. The more time we have, the better we can help. Doesn't matter what job they want--we know people and we know people who know people.

Thesis

Many schools won't hire you as an assistant professor if you haven't finished your thesis. Has to do with college rankings work. Don't worry--they will generally give you some other role with the same package until you finish. This might delay your tenure clock though.

Delayed start time

A January start is usually fine with good reason but if you weren't planning to start until the fall of 2022 why are you on the market this year? If you do get the department to hold a position for you, remember you are also making a commitment--this is not an opportunity to try again for something better.

Overall

You may not get all that you want after a negotiation--don't take it personally. You shouldn't necessarily choose the place that gives you the biggest package. It's far more important in the long run that you pick a place where you can best succeed both professionally and personally, and the package is just a small piece of that puzzle.

The Mythical Man-Month, Hen-Day, and Cat-Minute (Fred Brooks Turned 90)

 (Added later: Fred Brooks passed away on November 7, 2022, at the age of 90.)

The Mythical Man-Month is a great book which talks about the (obvious in retrospect) fact that putting more people on a project may slow it down. It was by Fred Brooks who turned 90 in April (he is still alive). It's a good read. I actually read it many years ago when I exchanged books with a Software Engineer I was dating- She lent me The Mythical Man Month which I found interesting, and I lent her What is the name of this book by Smullyan which she found amusing. Did this exchange of books help our relationship? We have now been married for many years, though its not clear if we can trace this to the exchange of books OR to the fact that she had KNUTH Volumes 1 and 3, and I had KNUTH Volume 2. 

 Fred Brooks: You have my thanks and of course Happy Birthday!

When I read The Mythical Man-Month  I was reminded of a math problem I heard as a kid: 

If a hen-and-half lays an egg-and-a-half in a day-and-a-half then how many eggs can seven hen lay in seven days? 

My answer: if (3/2) hens lay (3/2) eggs in (3/2) days then that's 2/3 of an egg per hen-day, so the answer is 

49* 2/3 = 32 and 2/3 eggs.

It did not bother me one whit that (1) you can't have 2/3 of an egg, and (2) Just like adding more people might slow down a project, adding more hens might end up being a bad idea-- especially if they are all crowded into the same chicken-coop and hence don't feel much like laying eggs.

Who was the first person to note that adding more people or hens might be a bad idea? I do not know, but here is an amusing, yet realistic, article by Mark Twain on what I would call The mythical cat-minute. My advisor Harry Lewis send it to me in the midst of an email exchange about The Mythical Man-Month. He got it from a student of his, Larry Denenberg. Here it is: 

CATS AND RATS

The following piece first appeared in ``The Monthly Packet'' of February
1880 and is reprinted in _The_Magic_of_Lewis_Carroll_, edited by John
Fisher, Bramhall House, 1973.


   If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill
   100 rats in 50 minutes?

   This is a good example of a phenomenon that often occurs in working
   problems in double proportion; the answer looks all right at first, but,
   when we come to test it, we find that, owing to peculiar circumstances in
   the case, the solution is either impossible or else indefinite, and needing
   further data.  The 'peculiar circumstance' here is that fractional cats or
   rats are excluded from consideration, and in consequence of this the
   solution is, as we shall see, indefinite.

   The solution, by the ordinary rules of Double Proportion, is 12 cats.
   [Steps of Carroll's solution, in the notation of his time, omitted.]

   But when we come to trace the history of this sanguinary scene through all
   its horrid details, we find that at the end of 48 minutes 96 rats are dead,
   and that there remain 4 live rats and 2 minutes to kill them in: the
   question is, can this be done?

   Now there are at least *four* different ways in which the original feat,
   of 6 cats killing 6 rats in 6 minutes, may be achieved.  For the sake of
   clearness let us tabulate them:
      A.  All 6 cats are needed to kill a rat; and this they do in one minute,
          the other rats standing meekly by, waiting for their turn.
      B.  3 cats are needed to kill a rat, and they do it in 2 minutes.
      C.  2 cats are needed, and do it in 3 minutes.
      D.  Each cat kills a rat all by itself, and takes 6 minutes to do it.

   In cases A and B it is clear that the 12 cats (who are assumed to come
   quite fresh from their 48 minutes of slaughter) can finish the affair in
   the required time; but, in case C, it can only be done by supposing that 2
   cats could kill two-thirds of a rat in 2 minutes; and in case D, by
   supposing that a cat could kill one-third of a rat in two minutes.  Neither
   supposition is warranted by the data; nor could the fractional rats (even
   if endowed with equal vitality) be fairly assigned to the different cats.
   For my part, if I were a cat in case D, and did not find my claws in good
   working order, I should certainly prefer to have my one-third-rat cut off
   from the tail end.

   In cases C and D, then, it is clear that we must provide extra cat-power.
   In case C *less* than 2 extra cats would be of no use.  If 2 were supplied,
   and if they began killing their 4 rats at the beginning of the time, they
   would finish them in 12 minutes, and have 36 minutes to spare, during which
   they might weep, like Alexander, because there were not 12 more rats to
   kill.  In case D, one extra cat would suffice; it would kill its 4 rats in
   24 minutes, and have 26 minutes to spare, during which it could have killed
   another 4.  But in neither case could any use be made of the last 2
   minutes, except to half-kill rats---a barbarity we need not take into
   consideration.

   To sum up our results.  If the 6 cats kill the 6 rats by method A or B,
   the answer is 12; if by method C, 14; if by method D, 13.

   This, then, is an instance of a solution made `indefinite' by the
   circumstances of the case.  If an instance of the `impossible' be desired,
   take the following: `If a cat can kill a rat in a minute, how many would be
   needed to kill it in the thousandth part of a second?'  The *mathematical*
   answer, of course, is `60,000,' and no doubt less than this would *not*
   suffice; but would 60,000 suffice?  I doubt it very much.  I fancy that at
   least 50,000 of the cats would never even see the rat, or have any idea of
   what was going on.

   Or take this: `If a cat can kill a rat in a minute, how long would it be
   killing 60,000 rats?'  Ah, how long, indeed!  My private opinion is that
   the rats would kill the cat.