Sunday, July 12, 2020

Ronald Graham: A summary of blog Posts We had about his work

To Honor Ronald Graham I summarize the blog posts we had about his work.

1) Blog post New Ramsey Result that will be hard to verify but Ronald Graham thinks its right which is good enough for me.

Wikipedia (see here) says that in the early 1980's (can't Wikipedia be more precise than that?) Ronald Graham conjectured the following:

For all 2-colorings of N, there exists x,y,z all the same color such that (x,y,z) form a Pythagorean triple.

I cannot imagine he did not also conjecture this to be true for all finite colorings.

I suspect that when he conjectured it, the outcomes thought to be likely were:

a) A purely combinatorial (my spell check says that combinatorial  is not a word. Really? It gets 14,000,000 hits) proof. Perhaps a difficult one. (I think Szemeredi's proof of his density theorem is a rather difficult but purely combinatorial proof).

b) A proof that uses advanced mathematics, like Roth's proof of the k=3 case of Sz-density, or Furstenberg's proof of Sz theorem.

c) The question stays open though with some progress over the years, like R(5).

What actually happened was

d) A SAT Solver solves it AND gets exact bounds:

For all 2-colorings of {1,...,7285} there is a mono Pythag triple.

There exists a 2-coloring of {1,...,7284} with no mono Pythag triple.

I wonder if this would have been guessed as the outcome back in the early 1980's.

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2) Blog Post Ronald Graham's Other Large Number- well it was large in 1964 anyway

Let

a(n) = a(n-1) + a(n-2)

I have not given a(0) and a(1). Does there exists rel prime values of a(0) and a(1) such that for all n, a(n) is composite.

In 1964 Ronald Graham showed yes, though the numbers he found (with the help of 1964-style computing) were

a(0) = 1786772701928802632268715130455793

a(1) = 2059683225053915111058164141686995

I suspect it is open to get smaller numbers, though I do not know.


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3) Blog Post Solution to the reciprocals problem

Prove or disprove that there exists 10 natural numbers a,...,j such that

2011= a+ ... + j
1 = 1/a + ... + 1/j

I had pondered putting this on a HS math competition in 2011; however, the committee thought it was too hard. I blogged on the problem asking for solutions, seeing if there was one that a HS student could have gotten. The following post (this one) gave those solutions. My conclusion is that it could have been put on the competition, but its a close call.

All of the answers submitted had some number repeated.

So I wondered if there was a way to do this with distinct a,...,j.

 I was told about Ronald Grahams result:

For all n at least 78, n can be written as the sum of DISTINCT naturals, where the sum of
the reciprocals is 1.

This is tight: 77 cannot be so written.

Comment on that blog DID include solutions  to my original problem with all distinct numbers

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4) Blog Post A nice case of interdisplanary research tells the story of how the study of history lead to R(5) being determined (see here for the actual paper on the subject). One of the main players in the story is the mathematician

Alma Grand-Rho.

Note that this is an anagram of

Ronald Graham.

What is the probability that two mathematicians have names that are anagrams. I suspect very small. However, see this blog post to see why the probability is not as small as it might be.

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5) Blog Post Winner of Ramsey Meme Contest This post didn't mention Ronald Graham; however I think he would have liked it.








Thursday, July 09, 2020

Reflections on Ronald Graham by Steve Butler

Ronald Graham passed away on July 6 at the age of 84. We present reflections on Ronald Graham by Steve Butler.



Getting to work with Ron Graham

Ron Graham has helped transform the mathematics community and in particular been a leader in discrete mathematics for more than 50 years. It is impossible to fully appreciate the breadth of his work in one sitting, and I will not try to do so here. Ron has put his papers online and made them freely available, a valuable treasure; and there are still many a hidden gem inside of these papers that are waiting to be picked up, polished, and pushed further.

I want to share about how I got to know and work with Ron. To be fair I knew about Ron long before I ever knew Ron. He was that rare pop-star mathematician who had managed to reach out and become visible outside of the mathematical community. And so as a teenager I read about Ron in a book about Erdos. I thought to myself that this guy sounds really cool and someday I might even get to see him give a talk (if I was lucky).

I went to UC San Diego for graduate school and after a series of near-misses ended up studying under Fan Chung. I passed Ron in the stairwell once, and then also helped them move some furniture between their two adjoining homes (graduate students are great for manual labor). But I became determined to try and find a way to start a conversation with Ron and maybe work up to working on a problem. So I took the usual route: I erased the chalkboards for him.

Before his class on discrete mathematics would start, I would come in and clean the chalkboards making them pristine. It also gave me time to occasionally engage in some idle chat, and he mentioned that his papers list was far from complete. I jumped on it and got to work right away and put his papers online and have been maintaining that list for the last fifteen years. This turned out to be no small feat and required about six months of work.  Many papers had no previous online version, and there were even a few papers that Ron had written that he had forgotten about! But this gave me a reason to come to Ron and talk with him about his various papers and then he would mention some problems he was working on with others and where they were stuck and thought I might give them a try.

So I started to work on these problems and started to make progress. And Ron saw what I was able to do and would send me more problems that fit my abilities and interests, and I would come back and show him partial solutions, or computations, and then he would often times fill in the gaps. He was fun to work with, because we almost always made progress; even when we didn't make progress we still understood things more fully. Little by little our publications (and friendship) grew and we now have 25+ joint publications, and one more book that will be coming out in the next few years about the enumerating juggling patterns.

After all of that though, I discovered something. I could have just gone to Ron's door and knocked and he would have talked to me, and given me problems (though our friendship would not become so deep if I had chosen the forthright method). But almost no graduate students in math were brave enough to do it; they were scared off by his reputation. As a consequence, Ron had far fewer math graduate students than you would expect. (To any math graduate student out there, don't let fear stop you from talking with professors; many of them are much nicer than you think, and the ones that are not nice are probably not that great to work with.)

So one of the most important lessons I learned from Ron was the importance of kindness. Ron was generous and kind to everyone (and I really stress the word everyone) that he met. It didn't matter what walk of life you were in, what age you were, or what level of math (if any) that you knew, he was kind and willing to share his time and talents. He always had something in reach in his bag or pocket that he could pull out and show someone and give them an unexpected sense of wonder.

Richard Hamming once said "you can be a nice guy or you can be a great scientist", the implication being that you cannot do both. Ron showed that you can be a nice guy and a great scientist. And I believe that a significant portion of his success is owed to his being kind; all of us should learn from his examples and show more kindness towards others.

This is only one of many lessons I learned from Ron. Another thing I learned from Ron is the importance of data. I have seen multiple times when we would work on a problem and generate data resulting in what I thought were hopeless numbers to understand. But Ron looked at that same data and with a short bit of trial and error was able to make a guess of what the general form was. And almost inevitably he would be right! One way that Ron could do this was to start by factoring the values, and if all the prime factors were small he could guess that the expression was some combination of factorials and powers and then start to play with expressions until things worked out. Even when I knew what he did, I still am amazed that he was able to do it.

I will miss Ron, I will never have a collaboration as deep, as meaningful, and as personal. I am better for having worked with him, and learning from him about how to be a better mathematician and a better person.

Thank you, Ron.

Sunday, July 05, 2020

A table for Matrix Mortality- what I wanted for Hilbert's 10th problem

In this post I speculated on why I could not find anywhere a table of which cases of Hilbert's 10th problem were solvable, unsolvable, and unknown. (I then made such a table. It was very clunky,  which may answer the question.)

I told my formal lang theory class about Hilbert's 10th problem as a natural example of an undecidable question- that is, an example that had nothing to do with Turing Machines. On the final I asked

Give an example of an undecidable problem that has nothing to do with Turing Machines.

Because of the pandemic this was a 2-day take home final which was open-book, open-notes, open-web. So they could have looked at my slides.

And indeed, most of them did give Hilbert's 10 problem (more formally, the set of all polynomials in many vars over Z which have a Diophantine solution).

But some did not. Some said there could never be such a problem (this is an incorrect answer), Some were incoherent. One just wrote ``Kruskal Trees''  (not sure if he was referring to MSTs or WQOs or to something that Clyde Kruskal did in class one day).

One student said that the problem of, given a CFG G, is the complement of L(G) also CFG.
This is indeed undecidable and does not have to do with TMs. I doubt the student could have proven that. I doubt I could have proven that. I do not doubt that my advisor Harry Lewis could have proven that, and indeed I emailed him asking for a proof and he emailed me a sketch, which I wrote out in more detail here.

The most interesting answer was given by some students who apparently looked at the web (rather than at my slides) for lists of problems and found the following called Matrix Mortality:

{ (M_1,...,M_L) : such that some product of these matrices (you are allowed to use a matrix many times) is the 0 matrix}

Why was this the most interesting? The TA did not know this problem was undecidable until he saw it on the exams and looked it up. I did not know it was undecidable until my TA told me.

I then raised the question: How many matrices to you need and how big do their dimensions have to be?

Unlike H10, there IS a table of this. In this paper they have such a table. I state some results:

Undecidable:
6 matrices, 3x3
4 matrices, 5x5
3 matrices 9x9
2 matrices 15x15

Decidable
2 matrices 2x2

So there are some gaps to fill, but there is not the vast gulf that exists between dec and undec for Hilberts 10th problem. I also note that the paper was about UNDEC but mentioned the DEC results, where as the papers on H10 about UNDEC seem to never mention the DEC.

I am glad to know another natural Undec problem and I will likely tell my students about it next spring. And much like H10, I won't be proving it.

An open problem in education: how come some of my students got it wrong? gave an answer that was not in my notes or slides? One student told me it was easier to google

Natural Undecidable Questions

then look through my slides. Another one said:

In class you said `this is a natural undecidable problem'.

On the exam you said `a problem that does not mention Turing Machines'

I did not know they were the same. 

That student submitted the Matrix problem stated above. It IS a fair point that `natural' is an
undefined term.  But the problem on the final used the well defined concept `does not mention Turing Machines'






Monday, June 29, 2020

Can you name a famous living Chemist? Can anyone?


I was re-watching the  Greatest-of-all-time Jeopardy championship and the following happen (I paraphrase)

----------------------
Alex Trebek: The category is Chemistry and we have a special guest reading the clues.

Darling: I wonder who that will be.

Bill: Hmm. I assume some famous chemist.
------------------------

So who was it? Bryan Cranston, the actor who PLAYED chemist Walter White on Breaking Bad.

Why couldn't they get a famous living chemist to read the clues?

My guess: there are no famous living chemists.

The number of famous living scientists is fairly short and they are often known for things that are not quite their science. Some are famous because the popularize science (deGrasse Tyson, Dawkins) or because of something unusual about their life (Hawkings when he was alive) or for something else entirely that they did (Ted Kaczynski).  Are any famous for the actual work that they do in the science?

Andrew Wiles was famous for a brief time, and even made People Magazine's 25 most intriguing people of the year list in the early 1990's (after he solved Fermat's Last Theorem). So he was famous but it was short lived.

Terry Tao was on the Colbert Report (see here) after he won the Fields Medal, the MacAuthor Genius award, and the Breakthrough prize. And even that fame was short lived.

I looked at the web page of Nobel Prize winners, here.

The only Chemistry Nobel's I recognized were Marie Curie,  Irene Joilet-Curie (Marie's Daughter), and Erst Rutherford.

The only Physics Nobel's I recognized were

Richard Feynman,

 Eugene Wigner (for writing about The unreasonable effectiveness of mathematics in the natural sciences),

Richard Hofstadter (since he was the father of Douglas H and an uncle of Leonard H)

 Andrew Geim (since he won  both an Ig-Noble prize and a Nobel prize, see  here)

Wolfgang Pauli (I've heard the term `Pauli Principle" though I did not know what it was until I looked it up while preparing this blog. I prob still don't really know what it means.)

Enrico Fermi

Erwin Schrodinger

Paul Dirac

Robert Millikan

Albert Einstein

Max Karl Ernest Ludwig Planck (I thought his last name was `Institute')

Johannes Diderik van der Waals

Pierre Curie

Marie Curie


So, some questions:

a) Am I wrong? Are there famous living chemists I never heard of? Are there any famous living scientists who are famous for their work in science?

b) If I am right then was there ever a time when there were famous scientists?

c) If there was such a time, what changed?

(I ask all of this non-rhetorically and with no agenda to push.)





Monday, June 22, 2020

Winner of Ramsey Meme Contest

My REU program had a Ramsey Meme Contest.

The winner was Saadiq Shaik with this entry:

I Don't Always...

I challenge my readers to come up with other Ramsey Memes! or Complexity Memes! or point me to some that are already out there.




Thursday, June 18, 2020

On Chain Letters and Pandemics

Guest post by Varsha Dani.

My 11-year-old child received a letter in the mail. "Send a book to the first person named," it said, "then move everyone's name up the list, add your own name and send copies of the letter to six friends. In a few weeks you will receive 36 books from all over the world!". Wow. When I first encountered chain letters in the mid eighties, it was postcards, but even then it hadn't taken me in. Since then I hadn't seen one of these in a long time, but I guess with a lot of people suddenly at home for extended periods, people crave both entertainment and a connection to others. 

What's wrong with chain letters? Well quite apart from the fact that they are illegal, even a child can comprehend that the number of books (or postcards or other gifts) received must equal the number sent, and that for every participant who does get a rich reward, there will be many who get nothing. 

But there is another kind of chain communication going around. It is an email, asking the recipient to send a poem or meditation to somebody, and later they will receive many communications of the same sort. How endearing. Poetry. Sweetness and Light. No get-rich-quick pyramid schemes here. What's wrong with that? 

Of course, it depends on what one means by "wrong". Maybe you like exchanging poetry with strangers. Maybe you don't find it onerous or wish that your spam filter would weed it out. But let's leave aside those issues and look at the math alone. You send the email to two friends, each of whom forwards it to two of their friends and so on. So the number of people the email reaches ostensibly doubles every step. Exponential growth. But in fact that is not what the graph of human connections looks like. Instead, what happens is that the sets of friends overlap, so that after a while the growth stops being exponential and tapers down. 

Where else have we seen something like that? Oh, right. The pandemic. The virus jumps from infected people to the people they meet, and from them to the people they meet and so on. Initially, that's exponential growth fof new cases, but after a while  it tapers off, forms a peak and then starts to decrease. Why? Because eventually there is overlap in the sets of people that each infected person is "trying" (unintentionally) to infect, and a newly infected person who got the virus from one or many previously infected people is still just one newly infected person.  

So the chain letter spreads just like a virus. Indeed if one were to, somewhat fancifully, think of the chain letter as an independent entity whose goal is to self-replicate, then it looks even more like a virus, and, like a virus, it can only achieve its self-replication goal through the help of a host. But here's a way in which it is not like a virus. Once one has got the virus and recovered, one (hopefully) does not get it again. Not so the chain letter, of which one may get many copies over time! So maybe you will get some gifts or poetry, but you will likely also get more requests for them!

So what's wrong with the poetry chain email? It depends on your perspective.  To those of you who are wistfully waiting for that Poem from a Stranger, I dedicate the following to you.



An open letter to my 2n dearest friends:

A letter came for me today
It promised wondrous ends
If only I would forward it 
To just two other friends.

If they in turn should send it on
to two more that they know,
the goodwill that we're sending out
would grow and grow and grow.

Is this as pleasant as it seems?
Alas, dear friends, it's not.
This exponential growth can lead
To quite a sticky spot.

Friends of friends of friends of mine
May very well be linked
The further that the letters go.
These folks are not distinct!

Ensuring there's no overlap
Is a logistic* pain.
As you will see, when you receive
That letter yet again. 

So while you're stuck at home this year
And pacing in your room.
Pick up the phone and make a call
Or see your friends on Zoom.

Your real thoughts would make me smile.
Chain letters are a con.
Do everyone a favor and
Don't send that letter on!

--------------


Monday, June 15, 2020

Presentations of Diffie-Helman leave out how to find g

When I first taught Diffie Helman I read the following
1) Alice and Bob agree on p a prime and g a generator
2) Alice picks a, sends g^a to Bob, Bob picks b, sends g^b to Alice
3) Alice computes (g^b)^a and Bob computes (g^a)^b so they both have g^{ab}

I knew how to find a prime- pick a number of length n (perhaps make sure the last digit is not even) and test for primality, if not then try again, you'll get one soon enough. I did not know how to find g. I had thought you first find p, and then given p you find g. I then figured out that you make actually pick  p to be a  safe prime, so q=(p-1)/2 is a prime, and then just pick random g and test them via computing  g^2  and g^q: if neither is 1 then g is a generator. You will find a generator soon enough.

That was all fine. But how come my source didn't say how to find g.?You need to know that to run the algorithm. That was years ago. Then I wondered how common it is for an explanation to not say how to find g. So I Googled ``Diffie-Helman'' I only record those that had some technical content to them, and were not about other DH such as Elliptic Curves.

0) The Original DH paper Page 34: alpha is a fixed primitive element of GF(alpha). No mention of how to find either the prime q or the prim root alpha.

1) Wikiepdia: ... protocol uses the mult group of integers mod p, where p is a prime and g is a prim root mod p. NO mention of how they find p or g.

2) Wolfram's MathWorld: They agree on two prime numbers g and p, where p is large and g is a prim root mod p. In practice it is good to choose p such that (p-1)/2 is also prime. They mention (p-1)/2 but not for the reason I give. (There are algorithms for Discrete Log that do well if (p-1)/2 has many factors.)

3) Comparatech: Alice and Bob start out by deciding two numbers p and g. No mention of how to find p or g.

4) Searchsecurity Won't bother quoting, but more of the same, no mention of how to find p or g.

5) The Secret Security Wiki Alice and Bob agree on p and g.

6) Science Direct More of the same.

7) Notes from a UCLA Crypto Course YEAH! They say how to find g.

8) Brilliant (yes that really is the name of this site) Brilliant? Not brilliant enough to realize you need to say how to find p and g.

9) OpenSSL Hard to tell. Their intuitive explanation leaves it out, but they have details below and code that might have it.


I looked at a few more but it was the same story.

This is NOT a RANT or even a complaint, but its a question:

Why do so few expositions of DH mention how to find p,g? You really need to do that if you really want to DO DH.

Speculation

1) Some of the above are for the laymen and hence can not get into that. But some are not.

2) Some of them are for advanced audiences who would know how to do it. Even so, how to find the generator really needs to be mentioned.

3) Goldilocks: Some papers are for the layman who would not notice the gap, and some papers are for the expert who can fill in the gap themselves, so no paper in between. I do not believe that.

4) The oddest of the above is that the original paper did not say how to find g.









Monday, June 08, 2020

The Committee for the Adv. of TCS- workshop coming up SOON!


(Posted by request from Jelani Nelson.)

The Committee for the Advancement of Theoretical Computer Science (CATCS)
is organizing a Visioning workshop.  The primary objective of the workshop
is for TCS participants to brainstorm directions and talking points for TCS
program managers at funding agencies to advocate for theory funding.

There was some question of whether or not it would run this summer, but
YES, it is going to run.

If you are interested then reply (at the link below) by June 15.
This is SOON so click that link SOON.

The time commitment is 4-5 hours during the week of July 20-July 24 for
most participants, or roughly 10 hours for those who are willing to
volunteer to be group leaders.

The link to sign up is:



Wednesday, June 03, 2020

How to handle grades during the Pandemic

In March many Colleges sent students home and the rest of the semester was online. This was quite disruptive for the students. Schools, quite reasonably, wanted to make it less traumatic for students.

So what to do about grades? There are two issues. I state the options I have heard.


ISSUE ONE  If P/F How to Got About it?

1) Grade as usual.

2) Make all classes P/F.

PRO: Much less pressure on students.

CON: Might be demoralizing for the good students.

3) Make all classes P/F but allow students to opt for letter grades BUT they must decide before the last day of class. Hence teachers must post cutoffs before the final is graded

CON: Complicated and puts (a) teachers in an awkward position of having to post cutoffs before the final, and (b) puts students in an awkward position of having to predict how well they would do.

CON: A student can blow off a final knowing they will still get a D (passing) in the course.

PRO: Good students can still get their A's

CAVEAT: A transcript might look very strange. Say I was looking at a graduate school applicant and I see

Operating Systems: A

Theory of Computation: P

I would likely assume that the Theory course the student got a C. And that might be unfair.

3) Make all classes P/F but allow students to opt for letter grades AFTER seeing their letter grades. 

PRO: Less complicated an awkard

PRO: A students blah blah

CAVEAT above still applies.

ISSUE TWO If P/F what about a D in the major

At UMCP COMP SCI (and I expect other depts)

a D is a passing grade for the University

but

a D is not a passing grade for the Major.

So if a s CS Major gets a D in Discrete Math that does not count for the major--- they have to take it over again.

But if classes are P/F what do do about that.

Options

1) Students have to take classes in their major for a letter grade.

CON: The whole point of the P/F is to relieve pressure on the students in these hard times.

PRO: None.

2) Students taking a course in their major who get a D will still get a P on the transcript but will be told that they have to take the class over again.

3) Do nothing, but tell the students

IF you got a D in a course in your major and you are taking a sequel, STUDY HARD OVER THE SUMMER!

4) Do nothing, but tell the teachers

Students in the Fall may have a weak background. Just teach the bare minimum of what they need for the major.

(Could do both 3 and 4)

SO- what is your school doing and how is it working?

Monday, May 25, 2020

Oldest Living Baseball Players- can you estimate...

(The Baseball season is delayed or cancelled, so I post about baseball instead.)

This post is going to ask a question that you could look up on the web. But what fun with that be?

The following statements are true

1) Don Larsen, a professional baseball player who played from 1953 to 1967, is still alive. He is 90 years old (or perhaps 90 years young---I don't know the state of his health).  He was born Aug 7, 1929. He is best know for pitching a perfect game in the World Series in 1956, pitching for the Yankees. He played for several other teams as well, via trades (this was before free agency).
(CORRECTION- I wrote this post a while back, and Don Larsen has died since then.)


2) Whitey Ford, a professional baseball player who played from 1950 to 1967, is still alive. He is 91 years old (or perhaps 91 years young---I don't know the state of his health).  He was born Oct 21,  1928. He had many great seasons and is in the hall of fame. He played for the New York Yankees and no other team.

3) From 1900 (or so) until 1962 there were 16 professional baseball teams which had 25 people each. From 1962 until 1969 there were 20 teams which had 25 people each. There were also many minor league teams.

4) The youngest ballplayers are usually around 20. The oldest around 35. These are not exact numbers

SO here is my question: Try to estimate

1) How many LIVING  retired major league baseball players are there now who are older than Don Larsen?

2) How many LIVING retired major league baseball players are of an age between Don and Whitey?

3) How  many LIVING retired major league baseball players are older than Whitey Ford?

Give your REASONING for your answer.

Tuesday, May 19, 2020

Obit for Richard Dudley

Richard M. (Dick) Dudley died on Jan. 19, 2020 (NOT from Coronavirus).You can find obituaries for him  herehere, and here and an interview with him from 2019  here.


Professor Dudley worked in Probability and Statistics. His work is now
being used in Machine Learning. Here is a guest-post-obit by
David Marcus who had Prof. Dudley as his PhD Thesis Advisor.

-----------------------------------

Guest Blog Obit by David Marcus:

Dick was my thesis advisor at M.I.T. After I got my Ph.D. in 1983, I went
to work in industry, so did not work closely with him, as some of his other
students did. But, I enjoyed working with him very much in graduate school.

Dick was very precise. His lecture notes and articles (and later his books)
said exactly what needed to be said and didn't waste words. In his classes,
he always handed out complete lecture notes, thus letting you concentrate
on the material rather than having to take a lot of notes.

Dick was very organized, but his office had piles of papers and journal
articles everywhere. There is a picture here.

Before Dick was my advisor, I took his probability course. My orals were
going to be towards the end of the term, and I was going to use probability
as one of my two minor areas. So, I spent a lot of time studying the
material. Dick gave a final exam in the course. The final exam was unlike
any other final exam I ever took: The exam listed twelve areas that had
been covered in the course. The instructions said to pick ten and for each
area give the main definitions and theorems and, if you had time, prove the
theorems. Since I had been studying the material for my orals, I didn't
have much trouble, but if I hadn't been studying it for my orals, it would
have been quite a shock!(COMMENT FROM BILL: Sounds like a lazy way to make up an exam, though on this
level of may it works. I know of a prof whose final was

Make up 4 good questions for the final. Now Solve them.

)

Once Dick became my advisor, Dick and I had a regular weekly meeting. I'd
tell him what I'd figured out or what I'd found in a book or journal
article over the last week and we'd discuss it and he'd make suggestions.
At some point, I'd say I needed to think about it, and I'd leave. I never
did find out how long these meetings were supposed to last because I was
always the one to end them.(COMMENT FROM BILL: It's good someone ended them! Or else you might never
had graduated :-) )

When I began working with Dick, he said he already had a full
load of students, but he would see if he had something I could work on. The
problem Dick came up with for me to work on was to construct a
counterexample to a theorem that Dick had published. Dick knew his
published proof was wrong, and had an idea of what a counterexample might
look like, so suggested I might be able to prove it was a counterexample.
In retrospect, this was perhaps a risky thesis problem for me since if the
student gets stuck, the professor can spend time figuring out how to do it.
But, in this case, presumably Dick had already put some effort into it
without success. Regardless, with Dick's guidance, I was able to prove it,
and soon after got my Ph.D.(COMMENT FROM BILL: Sounds risky since if Dick could not do it, maybe it's too hard.)

In 2003 there was a conference in honor of Dick's 65th birthday. All of his
ex-students were invited, and many of them attended. There was a day of
talks, and we all went out to dinner (Chinese food, if I recall correctly)
in the evening. At dinner, I asked Dick if any of his other students had
written a thesis that disproved one of his published theorems. He said I
was the only one.(COMMENT FROM BILL: Really good that not only was he okay with you disproving
his theorem, he encouraged you to!)


Thursday, May 14, 2020

Awesome Video from Women In Theory!

Below is an awesome video made by WIT (Women In Theory) on May 10, 2020 to celebrate the women in our field and in place of the Women in Theory Workshop that was supposed to take place
@Simons in June. ENJOY:



Monday, May 11, 2020

And the winners are ....

The Computational Complexity Conference has announced the accepted papers for the 2020 now virtual conference. Check them out!

Speaking of the complexity conference, my former PhD student Dieter van Melkebeek will receive the ACM SIGACT Distinguished Service award for his leadership in taking the conference independent. They grow up so fast! 

Robin Moser and Gábor Tardos will receive the Gödel Prize for their work giving a constructive proof of the Lovász Local Lemma, one of my truly favorite theorems as it gave a far stronger bound, a shockingly simple and efficient algorithm and an incredibly beautiful proof. Back in 2009 Moser gave my all-time favorite STOC talk on an early version of the paper. I (and others) sat amazed as his algorithm and proof came alive. During the talk I asked Eric Allender sitting next to me "Are we really seeing a Kolmogorov complexity proof of the Lovász Local Lemma?" Yes, we did.

Cynthia Dwork will receive the Knuth prize given for her life's work. The prize would be justified by her work on distributed computing alone but it is her leadership in formalizing Differential Privacy, one of the coolest concepts to come out of the theoretical computer science community this century, that will leave her mark in theory history. 

Thursday, May 07, 2020

Vidcast on Conferences

Bill and Lance have another socially-distanced vidcast, this time with Lance telling the story of two conferences (ACM Economics and Computation and the Game Theory Congress). As mentioned in the video the Game Theory Congress has been postponed to next year. Also mentioned in the video, for a limited time you can read Lance's book on P v NP on Project Muse.


Tuesday, May 05, 2020

Why is there no (d,n) grid for Hilbert's Tenth Problem?


Hilbert's 10th problem, in modern language is:

Find an algorithm that will, given a poly over Z in many variables, determine if it has a solution in Z.

This problem was proven undecidable through the work of Davis, Putnam, Robinson and then
Matiyasevich supplied the last crucial part of the proof.

Let H10(d,n) be the problem with degree d and n variables.

I had assumed that somewhere on the web would be a grid where the dth row, nth col has

U if  H10(d,n) is undecidable

D if H10(d,n) is decidable

? if the status of H10(d,n) was unknown.

I found no grid. I then collected up all the results I could find here

This lead to the (non-math) question: Why is there no grid out there? Here are my speculations.

1) Logicians worked on proving particular (d,n) are undecidable. They sought solutions in N. By contrast number theorists worked on proving particular (d,n) decidable. They sought solutions in Z.. Hence a grid would need to reconcile these two related problems.

2) Logicians and number theorists didn't talk to each other. Websites and books on Hilbert's Tenth problem do not mention any solvable cases of it.

3) There is a real dearth of positive results, so a grid would not be that interesting. Note that we do not even know if the following is decidable: given k in Z does there exists x,y,z in Z such that

x^3 +y^3+ z^3 = k. I blogged about that here

4) For an undecidable result for (d,n) if you make n small then all of the results make d very large.

For example

n=9, d= 1.6 x 10^{45}  is undecidable. The status of n=9, d=1.6 x 10^{45} -1 is unknown.

Hence the grid would be hard to draw.

Frankly I don't really want a grid. I really want a sense of what open problems might be solved. I think progress has gone in other directions- H10 over other domains. Oh well, I want to know about

n=9 and d=1.6 x 10^{45}-1. (parenthesis ambiguous but either way would be an advance.)



Friday, May 01, 2020

Predicting the Virus

As a complexity theorist I often find myself far more intrigued in what we cannot compute than what we can. 

In 2009 I posted on some predictions of the spread of the H1N1 virus which turned out to be off by two orders of magnitude. I wrote "I always worry that bad predictions from scientists make it harder to have the public trust us when we really need them to." Now we need them to.

We find ourselves bombarded with predictions from a variety of experts and even larger variety of mathematicians, computer scientists, physicists, engineers, economists and others who try to make their own predictions with no earlier experience in epidemiology. Many of these models give different predictions and even the best have proven significantly different than reality. We keep coming back to the George Box quote "All models are wrong, but some are useful."

So why do these models have so much trouble? The standard complaint of inaccurate and inconsistently collected data certainly holds. And if a prediction changes our behavior, we cannot fault the predictor for not continuing to be accurate.

There's another issue. You often here of a single event having a dramatic effect in a region--a soccer game in Italy, a funeral in Georgia, a Bar Mitzvah in New York. These events ricocheted, people infected attended other events that infected others. This becomes a complex process that simple network models can never get right. Plenty of soccer games, funerals and Bar Mitzvahs didn't spread the virus. If a region has hadn't a large number of cases and deaths is it because they did the right thing or just got lucky. Probably something in between but that makes it hard to generalize and learn from experience. We do know that less events means less infection but beyond that is less clear.

As countries and states decide how to open up and universities decide how to handle the fall semester, we need to rely on some sort of predictive models and the public's trust in them to move forward. We can't count on the accuracy of any model but which models are useful? We don't have much time to figure it out.

Wednesday, April 29, 2020

A Guest Blog on the Pandemic's affect on disability students

I asked my Grad Ramsey Theory class to email me about whatever thoughts they have on the pandemic that they want to share with the world, with the intend of making some of them into a blog post. I thought there would be several short thoughts for one post. And I may still do that post. But I got a FANTASTIC long answer from one Emily Mae Kaplitz. Normally I would ask to shorten or edit a guest post, but I didn't do that here since that might make it less authentic.

Here is Emily Kaplitz's email (with her enthusiastic permission)

--------------------------------------------------------------------------------------

Ok so this might be super ranty, (It definitely is.) but I think it is super important to bring up in a blog post written by an academic that will be probably read by other academics. 

The students that are being most affected by this pandemic with online learning are disability students. As a disability student, we carefully cultivate the way that we learn best based off of years of trial and error. This is harder than anything else, we have to face in our lifetime. Most of the time disability students are left on the back burner and that statement is so much more prevalent right now. My friends brother is autistic. He is struggling so much right now because he is at home. Disability students learn what environment works best for them and at home is usually not the best place. We have to split our lives into different boxes that each have different tools to help us get our brains to focus and work well when we need them too. Disability students will rely on everything being planned out, so that they can succeed. Teachers and professors cannot understand the stress and strain that having to work at home puts on the student. Every time I go to another school, it is a struggle to figure out what new thing I need to add into the mix and what old thing I need to throw away. It's exhausting, but when I go from one school to another I at least know that the basics are the same. I sit in a classroom, the professors lecturer, and then I do work at home that is assigned to me. Changing to online changes that dynamic so much. A professor cannot see when a student is visibly struggling with a topic because we'll all behind computers. A neurotypical person might ask, "well why don't you just ask a question? Why don't you just let the professor know that you don't understand". Let me answer that simply. If all your life you've been silenced because of something that you cannot control, is your first reaction to speak out or to stay silent. It is so hard for disability students to ask a question after we've been labeled the dumb kid. Every time we ask a question, we always have the thought of: is this going to make me sound stupid. We've worked so hard to eliminate that word from our vocabulary and from others who will throw that word back at us. Disability students are being left in the hands of their parents and teachers/professors who do not understand us and our needs even if they try to or want to. It is so hard for us to explain what our normal is because we don't live your normal and therefore don't know the difference. Many disability students have their confidence slashed the moment they enter a classroom and realize that they are not like the other kids. Even more so because they don't understand why they aren't. Disability students are one of the most hard-working individuals when we have a cheerleader to cheer us on because it's hard. It's harder than anything anyone has to do. Because no one listens to you when you are stupid and no one cares for you if you're not easy to care for unless they are given a specific reason to. Fighting a losing battle every day is awful. Now imagine all of your weapons that you have carefully crafted over the years have been taken away and you are left defenseless. While we have things like ADS that are supposed to help support us, it's not enough. Just like putting a Band-Aid on an open infected wound will not be enough. Now more than ever we need to learn from this as academics. We need to learn that helping disability students does not only help disability students. It helps all students because all students learn differently. All students if given the chance can excel at any field that we put them in. We just have to figure out the best way to get that student to shine. That is one of the reasons why I am a PhD student right now. I saw in the tutoring center at my undergrad how many students came to me with so much frustration about something they are doing in class. Both students with disabilities and without. These students are constantly apologizing because they don't understand something. In one session by just changing the way that we talk about a subject the student was able to get it in less time than the professor taught it. I've had students come to me after an exam and tell me that the only reason they got the grade that they did was because in their head was my voice coaching them on a subject. We are not teaching optimally. We are teaching the way that it has been done for years and years and years and that is not the best way to teach. It might be the best way to teach the strongest links but really the link that matters the most is the weakest link that will snap under pressure because you can't pull a tractor with a broken link. Disability students think differently. Imagine how many impossible problems we can solve when we have people that think differently. But that's just my two cents as a disability student who is struggling and sees other disability students struggling every day. And really just wants to help all students succeed.

I blame any misspellings, grammar errors, and run on sentences on my speech to text and text to speech. This was a long email and if we were on tumblr, I would post a potato at the end. Since we aren't, I will leave this email with this. Thank you for taking the time to read this rant. Even if you don't include this in your blog post, I believe one person reading this has made the difference.

Thanks!

Emily Mae Kaplitz

Monday, April 20, 2020

The Summer Virtual Conference Season

Both STOC and Complexity have announced they will go virtual for the summer. ICALP moved from Beijing to Saarbrücken to online. I expect every major summer conference and workshop will be cancelled, postponed or virtualized.

Most CS conferences serve as publication venues and can't be cancelled or postponed. So how do we virtualize a conference? The ACM has an evolving virtual conferences best practices guide. Putting the talks and poster sessions online is not trivial, but relatively straightforward. Personally I go to conferences mostly not for the talks but for the interactions with other participants--the receptions, meal time and just hanging in the hallways. The ACM document describes some approaches like Dagstuhl-style randomized virtual dinner tables. The IEEE VR conference tried virtual reality through Mozilla hubs. None of these can truly replicate the on-site experience.

Let me mention two other meetings the Game Theory Congress held every four years due to be held in Budapest and the CRA Snowbird Conference, a meeting of CS department chairs and computing leadership, held every other summer in Utah. Both meetings are not archival publications venues though have several talks and panels. But the main purpose of both is mostly to bring people together, game theorists and CS leaders. I hope they postpone rather than virtualize these meetings. Rather get together a year late than pretend to get together now.

Wednesday, April 15, 2020

Theoretical Computer Science for the Future

Guest post by the TCS4F initiative (Antoine Amarilli, Thomas Colcombet, Hugo Férée, Thomas Schwentick) 

TCS4F is an initiative by theoretical computer scientists who are concerned about that other major crisis of our time: climate change. We anticipate that the climate crisis will be a major challenge of the decades to come, that it will require major changes at all levels of society to mitigate the harm that it will cause, and that researchers in theoretical computer science, like all other actors, must be part of the solution and not part of the problem.

Our initiative is to propose a manifesto to commit to a reduction of greenhouse gas emissions: following IPCC goals, the objective is to reduce by at least 50% before 2030 relative to pre-2020 levels. The manifesto is more than a simple expression of concern, because it is a pledge with concrete objectives. However, it does not prescribe specific measures, as we believe this discussion is not settled yet and the right steps to take can vary depending on everyone's practices. 

The manifesto can be signed by individual researchers (like you, dear reader!), by research groups, and by organizers of conferences and workshops. Currently, over 50 researchers have signed it. The goal of TCS4F is also to start organizing a community of concerned researchers, across theoretical computer science, to think about the issue of climate change and how to adjust what we do, in particular our travel habits. 

We need your help to make this initiative a success and help theoretical CS lead the way towards a sustainable, carbon-neutral future:
  • If you agree with our concerns and are ready to commit to reducing your carbon footprint, consider signing the manifesto. Signing is open to all researchers in theoretical CS in the broadest possible sense.
  • Advertise your support of the manifesto (e.g., by putting one of our badges on your webpage). Talk in your research teams and departments about the manifesto, and see if you can gather support for signing the manifesto collectively as a research group.
  • If you are involved in conferences and workshops, start a discussion about the carbon footprint of the event, and whether the event could commit to the manifesto's goal. Indeed, now that conferences across the globe are moving online because of the COVID-19 pandemic, it is a good time to discuss how conferences could evolve towards more sustainable models.
  • Spread the word about the issue of climate change and the TCS4F initiative, and encourage discussion of this important challenge in our communities. 
As theoretical researchers, we are not used to discussing uncomfortable non-scientific questions like the effects of our activities on the world. However, we believe that the magnitude of the climate crisis obliges us to act now as a community. We are confident that great changes can be achieved if we do not limit our creativity to our specific research areas and also use it to re-think our way to do research.

Sunday, April 12, 2020

John Conway Dies of Coronvirus

John Conway passed away on April 11, 2020 of the Coronovirus. He is the first person I knew (for some definition of `know') who has died of it. I suspect this is true of many readers of this blog.
(Fellow bloggers Scott Aaronson and Terry Tao have already posted about John Conway,
here and here. I suspect there will be others and when they do I will add it here.
ADDED LATER: nice xkcd here

John Conway is a great example of how the line between recreational math and serious math is .... non-existent? not important? Take our pick.

Examples

1) Conway invented Surreal Numbers. These can be used to express infinitely big and infinitely small numbers. One can even make sense of things like square root of infinity.  Conway's book is called On Numbers and Games (see here and here) Two free sources: here and here.

Note that Conway defined surreals in terms of games. Are they fun games? Probably not, but they are games!

2)  Conway's Game of Life (you really do need to use his name, note the contrast between The game of life here and Conway's Game of Life here

The game is simple (and this one IS fun). You begin with some set of dots placed at lattice points, and a set of rules to tell how they live, die, or reproduce.  The rules are always the same. Different initial patterns form all kinds of patterns.  Sounds fun! Is it easy to tell, given pattern P1 and P2 whether, starting with P1 you can get to P2. No. Its undecidable.

So this simple fun game leads to very complicated patterns.

And nice to have an undecidable problem that does not mention Turing Machines. (I will tell the students it is undecidable this semester, though I won't be proving it.)

3)  Berlekamp, Conway, and Guy wrote Winning Ways for your Mathematical Plays  See here and here

This is the ultimate book on NIM games.

4) The above is probably what the readers of this blog are familiar with; however, according to his Wikipedia page (see here) he worked in Combinatorial Game Theory, Geometry, Geometric Topology, Group Theory, Number Theory, Algebra, Analysis, Algorithmics and Theoretical Physics.

He will be missed.

Thursday, April 02, 2020

Let's Hear It for the Cloud

Since March 19th I have worked out of home. I've had virtual meetings, sometimes seven or eight a day, on Zoom, Bluejeans, Google Hangouts, Google Meet, Blackboard Collaborate Ultra and Microsoft Teams. I take notes on my iPad using Penultimate which syncs with Evernote. I store my files in Dropbox and collaborate in Google Drive. I communicate by Google Chat, Gmail, Facebook messenger and a dozen other platforms. I continue to tweet and occasionally post in this blog. 

A billion of my closest friends around the world are also working out of home and using the same and similar tools. Yet outside of some pretty minor issues, all of these services continue to work and work well. Little of this would have been possible fifteen years ago. 

As Amazon scaled up their web operations to handle their growing business in the early 2000's they realized they could sell computing services. AWS, Amazon Web Services, started in 2006. Microsoft Azure, Google and others followed. These sites powered smartphones and their apps that push heavy processing to the cloud, small startups who don't need to run their own servers, and companies like Zoom when they need to scale up quickly and scale down like Expedia when they don't need as much use. Amazon and Microsoft makes most of their profit on cloud services. Amazon can't get me toilet paper but they can make sure Blackboard continues to work when all of our classes move online. 

Just for fun I like to occasionally look over the large collection of Amazon Cloud Products. Transcribe an audio recording and translate to Portuguese, not a problem. 

The cloud can't allow all of us to work from home. We have many who still go to work including front-line health care workers putting their lives on the line. Many have lost their jobs. Then of course there are those sick with the virus, many of whom will never recover. We can't forget about the reason we stay indoors.

But every now and then it's good to look back and see how a technology has changed our world in a very short time. If we had this virus in the 90's we'd still be having to go to work, or simply stop teaching and other activities all together.

And how will our universities and other work spaces look like in the future now that we find we can work reasonably well from home and even better technologies develop? Only time will tell.



Tuesday, March 31, 2020

Length of Descriptions for DFA, NFA, CFG


We will be looking at the size of descriptions

 For DFAs and NFAs this is the number of states.

For CFG's we will assume they are in Chomsky Normal Form. So for this post CFG means CFG in Chomsky normal form. The length of a Chomsky Normal Form CFL is the number of rules.

1) It is known there is a family of languages L_n such that

DFA for L_n requires roughly 2^n states.

NFA for L_n can be done with roughly n states.

L_n =  (a,b)^* a (a,b)^n

Also note that there is a CFG for L_n with roughly n rules. (one can show this directly or by some theorem that goes from an NFA of size s to a CFG of size roughly s).

So L_n shows there is an exp blowup between DFAs and NFA's

2) It is known that there is a family of languages L_n such that

DFA for L_n requires roughly 2^n states

NFA for L_n requires roughly 2^n states

CFG for L_n can be done with roughly n rules

L_n = { a^{2^n}  }

So L_n shows there is an exp blowup between NFAs and CFGs.


3) Is there a family of languages L_n such that

NFA for L_n requires 2^{2^n} states

CFG for L_n can be done with roughly n rules.

The answer is not quite- and perhaps open.  There is a set of family of languages L_n such that for infinitely many n he above holds. These languages have to do with Turing Machines. In fact, you can replace

2^{2^n}} with any function f  \le_T  INF (so second level of undecidability).

For this blog this is NOT what we are looking for. (For more on this angle see here


4) OPEN (I think) Is there a family of langs L_n such that for ALL n

NFA for L_n requires 2^{2^n} (or some other fast growing function

CFG for L_n can be done with roughly n states (we'll take n^{O(1)})

5) OPEN (I think) Is there a family of langs L_n such that for ALL n
(or even for just inf many n)

DFA for L_n requires 2^{2^n}} states

NFA for L_n requires 2^n states and can be done in 2^n

CFG for L_n can be done with n rules.

(we'll settle for not quite as drastic, but still want to see DFA, NFA, CFG all
far apart).







Saturday, March 28, 2020

Robin Thomas

Graph Theorist and Georgia Tech Math Professor Robin Thomas passed away Thursday after his long battle with ALS. He was one of the giants of the field and a rare double winner of the Fulkerson Prize, for the six-color case of the Hadwiger Conjecture and the proof of the strong perfect graph theorem.

If you start with a graph G and either delete some vertices or merge vertices connected by an edge, you get a minor of G. The Hadwiger conjecture asks whether every graph that is not (k+1)-colorable graph has a clique of size k as a minor. Neil Robertson, Paul Seymour and Thomas proved the k=6 case in 1993 and still the k>6 cases remain open.

A graph G is perfect if for G and all its induced subgraphs, the maximum clique size is equal to its chromatic number. In 2002 Maria Chudnovsky, Robertson, Seymour and Thomas showed that a graph G is not perfect if and only if either G or the complement of G has an induced odd cycle of length greater than 3.

Robin Thomas was already confined to a wheelchair when I arrived at Georgia Tech in 2012. He was incredibly inspiring as he continued to teach and lead the Algorithms, Combinatorics and Optimization PhD program until quite recently. Our department did the ALS challenge for him. In 2016 he received the Class of 1934 Distinguished Professor Award, the highest honor for a professor at Georgia Tech.  He'll be terribly missed.

Tuesday, March 24, 2020

What to do while ``stuck'' at home/Other thoughts on the virus

Lance had a great post on what to do while you are stuck at home, which is of course relevant to whats happening now. Lance's post is here.

I will add to it, and then have other comments.

1) In our current electronic society we can do a lot from home. Don't think of it as being `stuck at home'

2) Lance points out that you should read a paper, read a textbook, etc. I of course agree and add some advice. Be Goldlocks!

This paper is too hard (e.g., a text on quantum gravity)

This paper is too easy (e.g., a discrete  math textbook for a freshman course)

This paper is just right (e.g., working out the large canonical Ramsey theorem)

3) If you catch up on your TV viewing on your DVR then beware: you will see commercials for Bloomberg.

4) DO NOT binge watch TV.  You will hate yourself in the morning.

5) Simons Inst Theory talks:

https://simons.berkeley.edu/videos

TCS+ talks

https://sites.google.com/site/plustcs/past-talks

or
https://sites.google.com/site/plustcs/

The Gathering for Gardner records all of their talks and puts the on you-tube
so goto youtube and search for Gathering for Gardners. These are Goldilocks talks since they
are easy but on stuff you prob don't know.

6) Keep fit. I used to go on treadmill for 45 minutes a day, now I am doing an hour.

7) Go for walks with a person who already shares your house, but avoid other people.

8) Book reviews, surveys, orig articles, that you were putting off writing- now write them.
but see next item.

10) Catch up on your blog reading. My favorite was Scott Aaronson's blog post about Davos:here. I also read every single comment. I hated myself in the morning. So that part may have been a mistake.

OTHER THOUGHTS

1) Do you really have more free time? No commuting, no teaching, but you still have the rest of your job, and perhaps it is harder if some things are easier at work. And calling relatives and friends to make sure they are okay, and just to talk, is a great thing to do, but its time consuming.

2) I'm beginning to lose track of what day-of-the-week it is since I don't have school to keep me on track, and I only watch TV shows on DVR so I watching a show on a day does not mean I know what day it is.

3) Avoid being price-gouged. The first few days that I tried to buy TP for my mom on amazon (I do this in normal times--- I order lots for my mom on amazon--- she is tech shy. She is also over 90.) her usual brand was out of stock, and the other brands were either higher quality so higher prices or just
absurdly priced. She wisely said to wait a week. She was right- it was easy to get at the usual price.

4) More generally, it seems like the shortages are people-created. For example, if in a store you see they are low on X, then you buy LOTS of X, and everyone does that, so then their really is a shortage of X. But I think thats calmed down some.

5) It important to have a `we will recover from this, life will go on' attitude (while following the things ALL experts say- wash your hands a lot, drink lots of water, get lots of sleep, which is prob
good advice anyway) and hence I will try to, for the next few weeks, blog on NORMAL things----Hilberts's 10th problem, Large Ramsey, etc.

ADDED LATER- there is a very nice contrarian view in the comment by Steve, the first comment. You should read that!







Thursday, March 19, 2020

What to do while stuck at home Part I

First of all both the Turing award and Abel Prize announced yesterday.

As we start moving from the panic phase of the coronavirus to the boring phase, what kinds of things should you do or not do while stuck at home for the next two weeks to eighteen months.

First of all still do your job. Teach your online classes. Try to do some research. Meet with your colleagues/students/advisor virtually (best with Zoom or something similar). Submit to conferences. What else? Use the situation for your advantage.

Attend virtual conferences: Really attend. Pretend that you flew there and devote the entire day to going to virtual talks or chatting with other attendees in virtual hallways. I said it wouldn't happen this way last week so prove me wrong.

Create a Virtual Workshop: Because you can. Invite people to give online talks. Open it up for all to listen. Find ways to discuss together.

Connect: Make a virtual get-together with an old colleague or someone you've always wanted to meet. Researchers around the world will be holed up and happy to get some interactions.

Learn Something New: Read a textbook. Take an online course in CS or something completely different. There are plenty.

Help Others Learn: Start that book you've always wanted to write. Or just write a short survey article giving your view of a slice of the theory world. Create some videos or a podcast to explain stuff.

Pick up a hobby: Something outside computer science just to keep your sanity.

Watch some fun computer-related movies: Her, Sneakers, The Computer wore Tennis Shoes, 2001, The Imitation Game, Hidden Figures, Colossus: The Forbin Project, Ex Machina. Add your own favorites in the comments.

And on the other hand don't

Become an epidemiologist: As a computer scientist you are an expert in networks, graph theory and exponential growth so you can create models that show we are grossly under preparing and/or overreacting to the virus and want to tell the world how you are right and the so-called "experts" are wrong. Please don't.

Prove P ≠ NP: Trying to settle P v NP and failing is instructive. Trying to settle P v NP and thinking you succeeded is delusional.

Freak Out: We will get past this virus and the world will recover.

Bill will follow up with his own ideas in part II next week.

Tuesday, March 17, 2020

Richard Guy passed away at the age of 103

Richard Guy passed away on March 9, 2020 at the age of 103. Before he died he was the worlds oldest living mathematician (see here for a list of centenarians who are famous scientists or mathematicians). He was also the oldest active mathematician-- he had a paper on arxiv (see here) in October of 2019.  (ADDED later since a commenter pointed it out to me--- a paper by Berlekamp and Guy posted in 2020: here)

I met him twice- once at a Gathering for Gardner, and once at an AMS meeting. I told him that Berlekamp-Conway-Guy had a great influence on me. He asked if it was a positive or negative influence. He also seemed to like my talk on The Muffin Problem, though he might have been being polite.


I did a blog about Richard Guy on his 103rd birthday, so I recommend readers to go there
for more about him.  One point I want to re-iterate:

Richard Guy thought of himself of an amateur mathematician. If he means someone who does it for love of the subject then this is clearly true.  If it is a measure of how good he is (the term `amateur' is sometimes used as an insult) then it is clearly false. If it means someone who does not have formal training than it is partially true.




Thursday, March 12, 2020

The Importance of Networking

People skip conferences because of the coronavirus or for global warming or just because conferences are too expensive and time consuming. I'm certainly no fan of the current conference structure but I would never want to virtualize all of them. Even if we could completely recreate the conference experience in virtual reality, people would not hang out in the halls without the commitment of having made the physical trip. I made this point in a tweet with a depressing response.

I don't disagree with anything Mahdi says except for the "crucial importance". Great ideas come from chance encounters and random conversations. Many of my research papers would never have happened if not for a conversation had at a conference or on the plane or train rides that took me there. Harken Gilles Brassard's origin story of quantum cryptography.
One fine afternoon in late October 1979, I was swimming at the beach of a posh hotel in San Juan, Puerto Rico. Imagine my surprise when this complete stranger swims up to me and starts telling me, without apparent provocation on my part, about Wiesner’s quantum banknotes! This was probably the most bizarre, and certainly the most magical, moment in my professional life6. Within hours, we had found ways to mesh Wiesner’s coding scheme with some of the then-new concepts of public-key cryptography.... The ideas that Bennett and I tossed around on the beach that day resulted in the first paper ever published on quantum cryptography, indeed the paper in which the term “Quantum Cryptography” was coined. 
And Footnote 6 read as follows.
At the risk of taking some of the magic away, I must confess that it was not by accident that Bennett and I were swimming at the same beach in Puerto Rico. We were both there for the 20th Annual IEEE Symposium on the Foundations of Computer Science. Bennett approached me because I was scheduled to give a talk on relativized cryptography on the last day of the Symposium and he thought I might be interested in Wiesner’s ideas. By an amazing coincidence, on my way to San Juan, I had read Martin Gardner’s account of Bennett’s report on Chaitin’s Omega, which had just appeared in the November 1979 “Mathematical Games” column of Scientific American—so, I knew the name but I could not recognize Bennett in that swimmer because I did not know what he looked like.
After we see a slate of conferences held virtually due to the virus, networking may indeed become a thing of the past. But we'll never know the research not done because of people who never connected.

Tuesday, March 10, 2020

Theorist Paul R Young passed away

In the early days of theoretical computer science, say 1960-1990 the main tools used were logic.
This made sense since, early on:

a) Some of the basic notions like DTIME(T(n)), P, NP used Turing Machines in their definitions

b) Some of the basic notions like reductions were modeled after similar concepts in
computability theory.

One of the people who did much work in the interface between Logic and TCS was Paul Young.
He passed away in December. Here are some highlights of his work:

1) One of the first books that covered both computability and complexity:

An Introduction to the general theory of Algorithms
by Machtey and Young.

2) In Computability theory all many-one complete sets are computably isomorphic. Berman and
Hartmanis conjectured that the poly-many-one degree of the NP-complete sets was the same. This
would mean that all NP-complete sets were poly-isom (all of the known ones are).

Mahaney and Young in the paper

Reductions Among Polynomial Isomorphism Types

showed that every many-one poly degree either has one degree or has an infinite number of
degrees in a very complicated way.

3) Recall that a Cook Reduction from A to B allows many queries to B, whereas a Karp Reduction
only allows one query and your answer must be the same sense as the query.
Are there  cases where a Cook reduction is faster? Yes, from the paper

Cook reducibility is faster than Karp Reducibility

by Longpre and Young

(The original title was going to be Cook is Faster than Karp, but it was changed since it invoked
images of Cook and Karp in a footrace.  Hmmm. Which one would be faster?)


4) The Boolean Hierarchy is a hierarchy of iterations of NP sets. What if instead of starting with P
one started with RP (Randomized Poly time). What an intriguing notion! To find out read


Generalized Boolean Hierarchies over RP

by  Alberto Bertoni, Danilo Bruschi, Deborah Joseph, Meera Sitharam, Paul Young

5) There are many more, mostly on the theme of the interaction of logic and computer science.

                       I saw him speak on some of these topics and was inspired by how much one could
take notions of computability and translate them into complexity theory.  The field has gone in a
different direction since then (more combinatorial) but we still use many of the basic concepts
like reducibility.  As such we all owe a debit to Paul Young.

Thursday, March 05, 2020

A New College of Computing at Illinois Tech


In 1890, Chicago South Side pastor Frank Gunsaulus gave a sermon where he said that with a million dollars he could build a school where students of all backgrounds could prepare for meaningful roles in a changing industrial society. One of the congregants, Philip Armour, came up to him after the service and told Gunsaulus that "if you give me five years of your time, I will give you the money." Thus was born the Armour Institute of Technology, the forerunner of the Illinois Institute of Technology.

Today Illinois Tech enters a new chapter, announcing a College of Computing, and I am honored to have been asked to serve as its inaugural dean. The college will take on a horizontal mission, to infuse computation and data science thinking throughout the curriculum in every discipline, while understanding the power, limitations and social implications of the technologies they create. We will significantly grow computing to produce the talent needed for a growing Chicago tech community. The college will develop an agile curriculum to continually reevaluate our offerings as computing technology continues to advance, and develop education as a life-long process where our alumni can always count on Illinois Tech to continually reskill to advance their careers.

We will do it all by keeping the core principle of the original "million-dollar sermon," as important as ever, to prepare students of all backgrounds for meaningful roles in a changing technological society.

Monday, March 02, 2020

Logic examples for your Discrete Math class



(I injured my hand about a month ago so I have had a hard time typing. That is why
I have not blogged for a while. I'm better now but still slow. This is a post I prepared
a while back.)



Here are some examples of English and logic for your discrete math class. Or for mine anyway.


1) A computer programmer leaves work and heads for home. Being the good spouse that he is, he calls  his partner and asks if there's anything that needs to be picked up on the way.

Yes, a gallon of milk and, oh, if they have eggs, get a dozen.

Later he arrives home and stumbles into the kitchen burdened with a dozen gallons of milk. His partner  perplexed, asks him ``why in the world did you buy 12 gallons of milk?''

What did he answer?

When I told this to my class one student said that he should answer:

                                                      I love you too Darling

while that is always a good thing to tell Darling, it is not the answer I had in mind.

The  answer is  here.

2) I saw a headline:

                                                  Rise in faux-incest porn alarming

Give two different interpretations of this sentence. (Note- One you might agree with, the other you will likely disagree with.)

My answer is  here

3) Recently someone was describing what I work on to someone else and he said the following wonderfully ambiguous sentence

                           Bill works on puzzles and games. He also work on cake cutting, to be fair.

Give two different interpretations of this sentence.  My answer is here


4) A common saying is

                           All that glitters is not gold

What does this mean literally? What did they really mean to say? My answer is here.

(I had originally thought this was a quote from the Led Zeppelin song Stairway to Heaven;
however, an astute reader left a comment reminding me that, in that song, they actually
say that there is a lady who believes All that Glitters is Gold. The song implies that she is incorrect, so really
NOT(All that Glitters is Gold) which means (exists x)[x glitters but x is not gold] which actually
IS what they meant to say. Yeah!)

5)When the chess player Bobby Fisher died I saw in one article about him the sentence

                                Bobby Fisher was a terrible anti-semite.

This can be interpreted two ways. What are they? Which one did the writer probably mean? My answer is here

6) When Donald Trump broke the Nuclear Treaty with Iran he said

                              Iran is the worse enabler of terrorist in the mideast

This can be interpreted two ways. What are they? Which one did Trump mean? My answer is here.


7) I saw the headline (see here)

There was actually good news in the War on Women in 2019, news we have to build on in 2020.

This can be interepreted in two ways. This one I leave to you, or read the article.




Sunday, February 16, 2020

Pre-(Publish and Perish)

Guest post by Evangelos Georgiadis

Quite a few posts have recently focused on papers,publications and venues; "optimal" venues for papers under different objective functions,e.g. minimizing carbon footprint while maximizing community building, networking as well as information sharing, see Moshe Vardi.

Here we would like to take a closer look at one of the key assumptions -- the paper. In order to generate a paper, one needs to come up with a result, something novel, fresh or interesting to say. The question that has baffled this author is what represents a conducive or perhaps even optimal setting for generating papers. Since papers come in different flavors ranging from "solid technical papers to risky innovative ones" the settings may vary; but ultimately, what would be interesting to investigate (or for that matter crowdsource) is whether there is a common denominator in terms of setting or environment, a necessary but not sufficient condition (so to speak).

Here are some accounts of others which may be helpful as reference points.

Knuth's papers entitled "Semantics of context free grammar" along with "The analysis of algorithms" represent two instances that suggest research institutes might not provide an optimal environment for idea generation.

As Knuth points out in "Selected Papers on Computer Languages" (Chapter 18, p. 431):
Perhaps new ideas emerge most often from hectic, disorganized activity, when a great many sources of stimulation are present at once -- when numerous deadlines need to be met, and when other miscellaneous activities like child-rearing are also mixed into the agenda.
Knuth goes on to say, that it was challenging to do creative work in office and that finding a few hideaways provided some form of solution -- aka sitting under 'that' oak tree near Lake Lagunita. That said, the inspirational setting for getting into the zone for the aforementioned two papers were provided by (Californian) beaches. Hold that observation. Is this not something we have come across somewhere else ? Fields medalist Stephen Smale in "Chaos: Finding a Horseshoe on the Beaches of Rio" suggests that some of his best work happened at his "beach office". Whether beaches do provide for a good setting remains to be shown; perhaps for very innovative ideas, oceanic freedom is necessary. That said, the author recalls (hopefully accurately enough) an account by the young James H Simons, who attended a conference in Japan in the early days. Instead of choosing a spacious accommodation (which he was able to afford), he restricted himself to the typically confined room type -- not only confined by space, but also pressured by time, young Simons was able to generate an interesting result for that conference. (This probably demonstrates that technical results don't necessarily require 'oceanic freedom'.)

Some meaningful probabilistic advice comes from the fat-tails department, in "The Black Swan" by Nassim Taleb (on page 209) : "Go to parties! If you're a scientist, you will chance upon a remark that might spark a new research. "

Murray Gell-Mann provides an interesting collective account in his Google Tech Talk entitled "On Getting Creative Ideas." He recollects a workshop he attended in 1969 in Aspen that focused on the experience of getting creative ideas, not just among mathematicians and theoretical physicists but also poets and artists. This account seems to neglect the actual setting that might nurture creative thought process, but provides interesting references to people such as Hermann von Helmholtz, who happened to have thought about this topic and partitioned the process in terms of "saturation, incubation and illumination".

For those interested in an account that focuses on the Eureka moments of exclusively mathematicians/theoretical physicists see Jacques Hadamard's book "The Mathematician's Mind". Hadamard iterated on Helmholtz's 3 stage process and it's worth taking a look at what he came up.

At last, what are good venues or workshops for generating papers ? Or let's rephrase that a bit, what type of atmosphere at venues fosters creativity -- what food for thought to provide participants and how to distribute that food for thought over a given day ? Ryan R Williams proposed (as practiced by 34th Bellairs Winter Workshop on Computational Geometry) "... easy problems, informal atmosphere focusing exclusively on thinking about problems in a cycle of down-time where one meets in two intense sessions and have free time otherwise." (This type of setting seems to resonate with the 3 stages of "saturation, incubation and illumination".)

That said, most workshops including the Simons workshops don't seem to follow such a recipe. They are more geared towards the follow-up step, namely, communicating what people have found, rather than collaborating with them to tackle open problems. Perhaps some re-evaluation might be required in how workshops are run.