Wednesday, September 20, 2023

We Must Be Doing Something Right

The Chicago Tribune ran an editorial Monday that started

What’s the best four-year college in Illinois? Not the University of Chicago, Northwestern University or the University of Illinois at Urbana-Champaign.

No, the best college in the state is the Illinois Institute of Technology, of course!

The editorial was referring to the Wall Street Journal that ranked Illinois Tech 23rd in the nation, tops in Illinois, up from 117 last year. Illinois Tech also cracked the top 100 in the latest US News rankings, up from 127. 

Did we just get that much better? Yes, yes we did! 

Or maybe it had to do with changes in methodology. The Wall Street Journal's rankings this year puts a heavy emphasis on "how much will it improve the salaries they earn after receiving their diplomas". As Illinois Tech caters to students who often are the first in their families to go to college, and focuses on technical degrees, we can really raise up students who might not have otherwise had such an education. It's one of the things that brought me to the university in the first place.

US News also "increased the emphasis on how often schools' students from all socioeconomic backgrounds earned degrees and took advantage of information on graduate outcomes that was not available until recently". 

All rankings should be taken with a grain of salt, and universities will always tout rankings where do they well while conveniently ignoring others. It's impossible to linearly order colleges--there are just too many different factors that make different schools better for different people. 

But as people start to question the value of college the rankings are starting to address their concerns. And if that bumps up my university, so be it.

Sunday, September 17, 2023

ACM to go paper-free! Good? Bad?

The ACM (Association of Computing Machinery) will soon stop having print versions of most its publications. Rather than list which ones are going paper free, I list all those that are not going paper free:
Communications of the ACM
ACM Inroads
XRDS: Crossroads

What are the PROS and CONS of this? What are the PROS and CONS of any publication or book being paper free?

1) I like getting SIGACT News on paper since 
a) It reminds me to read it
b) Reading on the screen is either on my phone which is awkward (especially for math) or my desktop (so I have to be AT my desktop). 
I DO NOT think this is my inner-Luddite talking. 
2) QUESTION:  Will SIGACT News and JACM and other ACM publications continue to have  page limit for articles? When I was SIGACT News Book Rev Col editor, and now as Open Problems Col editor, I have often had to ask the editor Can I have X pages this time? The answer was always yes,  so perhaps there never really was a page limit. But is having no page limit good? Not necessarily. Having a limit may force you to only write down the important parts.

3)  PRO: Its good for the ecology to not make so much paper.  While this is certainly true, I think the world  needs to rethink our entire consumer society to really make a difference for the ecology. In fact, I wonder if e-cars, carbon-offsets,  and paper free products make us feel good without really helping much.

4) CON but good timing: I recently had an open problems column with two co-authors. One of them is not in the ACM and is not in the community, but wanted to see a copy of the article. I have arranged to have a paper copy of that one issue sent to him.  If I had published this column in 2024, I could not do this. And saying Just go to link BLAH' does not have the same impact as PAPER. I could have printed it out for him, but that just does not seem like the same as having an official copy. 
I DO think this is my inner-Luddite talking. Or his.

5) For quite some time computer science  conference proceedings have not been on paper (there have been a variety of ways this is done). Before that time the following happened a lot: I am in Dave Mounts office talking about something (e.g., who should teach what). He gets a phone call but motions that it will be short so I should still hang out. While hanging out I pick up a RANDOM proceedings of the conference RANDOM  and find the one or two article in it about Ramsey Theory and read them, or at least note them and read them later. That kind of RANDOM knowledge SEEMS less common  in a paper-free age. But maybe not.  I HAVE clicked around the web and accidentally learned things. Like the facts I learned for my post on simulation theory here.

6) Similar to point 5- I used to go to the math library and RANDOMLY look at a volume of the American Math Monthly or some other similar journal and look at some articles in it.  But now that's harder since they have stopped getting journals on papers and only get them electronically. To be fair, paper versions of the journals are EXPENSIVE. 

7) In the year 1999 my grad student Evan Golub got his PhD and he had to bring a PAPER copy of it to some office where they measured margins and stuff of EVERY PAGE to make sure it was within university specs.  Why? Because in an earlier era this was important for when the thesis was put on microfilm.  Were they doing that in 1999? I doubt it.  Some of my younger readers are thinking OH, they didn't have LaTeX packages that take care of marginfor you.  Actually they DID have such packages but, to be fair, the requirement that the university literally measures margins on EVERY PAGE was completely idiotic. I am  happy to say that in 2007 when my student Carl Anderson  got his PhD nobody needed a paper version. I do not know when the rules changed but I am glad they did. 

8) The ACM should promote this paper free change by doing a rap song like Progressive Insurance did here

9) Recently I had a paper with 3 co-authors that all three of us, and some others, proofread (I thought) very carefully. The referee accepted it but with a somewhat nebulous this paper needs a better proofreading. I then PRINTED IT OUT and read it AWAY FROM MY COMPUTER (the paper is on overleaf) with a RED PEN and I found LOTS of stuff to fix that we all missed before. So there are some advantages to getting OFF of the computer, though that may require PRINTING. (I also blogged about this topic here.) 

Thursday, September 14, 2023

Mr. Jaeger and The Scroll

There were three major influencers in my educational journey: Mike Sipser, my PhD advisor at Berkeley and MIT, Juris Hartmanis who founded the field of computational complexity and led me to it at Cornell, and Philip Jaeger, a math teacher at Millburn High School in New Jersey.

Mr. Jaeger
Perhaps your "scroll" will outlast both of us.

I took two math courses with Philip Jaeger, Algebra II and Probability. Mr. Jaeger (I still can't call him Phil) also ran the computer room, a narrow room of three teletype machines where we saved programs on paper tape, where we would do simulations of poker hands for probability class among various other programming. He truly set me up for my future career. 

Mr. Jaeger was also the advisor of the computer club when I was president. 

That's me in the middle of the first photo, with Mr. Jaeger on my right.

Sometime during my high school years I took one of the rolls used in the teletype machine and wrote down a lengthy formula. According to Mr. Jaeger, the formula is for the area of a random triangle. I'm sure it made sense to me in high school.

The Scroll

The Scroll partially unscrolled

Another student Cheryl took the entire formula and put it entirely on an index card. Mr. Jaeger saved both the roll and the card.

The index card (actual size is 3x5 in)

Fast forward to 2023. Mr. Jaeger finds the roll and the card. His son, an academic in his own right, suggested that they track Cheryl and me down. I'm not hard to find. After some emails, they invited me to visit to pick up the roll. Last weekend I was in New Jersey visiting family so I did just that.

What great fun to talk to my old teacher, reminiscing about the high school people and times, and catching up after four decades. It was Mr. Jaeger who showed me the computer dating form that had me create a version for our high school.

I gave Mr. Jaeger a copy of my book and he gave me the scroll and the index card. (Cheryl, now a lawyer, wasn't interested in it). I safely brought it all home to Chicago along with all the memories. I dug up my old yearbook for first two photos above. The scroll will indeed outlast the both of us.

Mr. Jaeger, myself and the scroll.

Sunday, September 10, 2023

Asymptotics of R(4,k)- a new result!

 At the workshop 

Ramsey Theory: Yesterday, Today, and Tomorrow, Edited by Alexander Soifer, 2011. (There is a printed proceedings that you can find.)

 I saw Joel Spencer give a great talk titled 

                               80 years of R(3,k).

( Recall that  R(a,b) is the least n such that for all 2-colorings of the edges of K_n there is either a red K_a or a blue K_b).

 The talk was about improvements on both the upper and lower bound on R(3,k) and finally:

                                  R(3,k) = \(\Theta\biggl (\frac{k^2}{\ln^2 k}\biggr )\).

The obvious question was raised: What about R(4,k). The general sense I got was that this would be a much harder problem. However, there has been some progress. A recent paper, here, improved the best known lower bound, so it now stands at

                                   \( c_1\frac{k^3}{\log^4 k} \le r(4,k) \le c_2\frac{k^3}{\log^2 k} \)

How long before we see matching upper and lower bounds?

1) How long did it take to get matching upper and lower bounds for R(3,k). The name of the talk would make one think 80 years, but I would start at a paper of Erdos from 1961 which had the first non-trivial lower bounds. And it was solved in 1995, so that's 34 years. (Note, the talk of Spencer was also on later algorithmic aspects of the problem). 

2) Argument for why matching bounds on R(4,k)  will be found  in \(\le 10\) years: There are more people using more sophisticated tools then were known for the R(3,k) search.

3) Argument for why matching bounds on R(4,k) will take (\ge 20\) years: This problem is dang hard! Triangles are much easier to deal with then  4-cliques.

This is a general problem math has: If a problem is not been solved, is it just dang hard, or are people one or two (or some small finite number) steps away from solving it?

Wednesday, September 06, 2023

Books to Inspire Math

Two of my colleagues and co-authors from my early days at the University of Chicago have released books over the past few months designed to excite people with math, Howard Karloff's Mathematical Thinking: Why Everyone Should Study Math and Lide Li's Math Outside the Classroom. Karloff was a fellow professor and Li was my PhD student. Neither are currently in academia but both still found the need to inspire young people in mathematics.

Both books aim to make math fun, away from the rote problem solving from high school and early calculus courses to concepts like prime and irrational numbers (Karloff) and sequences and geometric shapes (Li). The books have some overlap, both cover deriving e from interest rates and probability including the Monty Hall problem. Both books have lots of problems to work on. 

Between the two I would suggest Karloff's book for junior high/high school age kids and Li's book for older high school and early college students given the topics covered.

At a time that math plays a larger role in our society, especially dealing with data, finding ways to get more young people interested in mathematics is important. These books fill an important niche for the mathematically curious students to dive in topics they won't likely see in their math classes. Great to see my former colleagues taking the time to reach these students through these books. 

Sunday, September 03, 2023

The CONTRADICTION of Margaritaville and other songs

Jimmy Buffett passed away on Sept 1, 2023. His Wikipedia entry (see here) says his death was peaceful and he was surrounded by friends, family, and his dog, so it was likely expected and of natural causes. I later saw a report that he had a serious skin cancer. He was 76. 

He is not related to Warren Buffett--- they actually took a DNA test to find out, see here. They are friends. Buffett isn't that common a name, see here, so it was plausible they were related, but, alas, they are not.

His signature song is Margaritaville (My spellcheck thinks that I misspelled Margaritaville   but I checked it and it looks fine. OR it's one of those things where I keep misreading it.) It wasn't just his signature song---he made a career of it outside of music, see here.

Jimmy Buffett fans are called parrot heads.

There are songs where the lyrics are misheard. Margaritaville is not one of them. Instead, its lyrics are misunderstood. This raised the question:

There are  other songs whose LYRICS and WHAT PEOPLE THINK ABOUT THEM are in CONTRADICTION. What caused the contradiction? Could I make this into a HW assignment the next time I teach logic? Not if my students are looking for their lost shaker of salt.

This link here has 25 songs with misunderstood lyrics. Margaritaville comes in at the 24th. I think it should rank higher (lower index, higher ranking) but I can't complain since I am not an expert and they put in the work (unlike my ranking of satires of Bob Dylan, here, where I am an expert and I put in the work).

I list a few of the songs, plus two more,  and WHY the contradiction. I also listened to them with the following question: ONCE you know what the song is supposed to be about and you listen to it, do you say OF COURSE THAT'S WHAT ITS ABOUT or REALLY? I STILL DON"T SEE IT. This is similar to reading a math proof knowing where it is going so perhaps you say OF COURSE. Of course, you might also say REALLY? I STILL DON"T SEE IT.

 The name of the song in the list below is also a pointer to a video of it.

Imagine by John Lennon. People think it's about peace and love. The writer John Lennon (not be be confused with Vladimr Lenin) says it's a Communist manifesto.  I just listened to it and OF COURSE it's  a Communist Manifesto- but its sung with such an optimistic loving tone that one could miss that. This is John Lennon's best known post-beatles song. 

Total Eclipse of the Heart by Bonnie Tyler. People think it's a power ballad- about love and such. Its actually a vampire love song. REALLY? I STILL DON"T SEE IT. A love song is a love song. It could be about humans, vampires, or, in the case of The Klein Four, Math, but unless they put something Vampire-ish  into it, you can't tell its about Vampires. Two notes: (1) Its Bonnie Tyler's biggest hit, and
(2) it was released in 1983 but also had a large number of sales in 2017. Why? Either guess or see here.

Blackbird by the Beatles. People think its just about a blackbird with a broken wing. Its about civil rights for blacks (or all countries- so I can't use the term African American) REALLY? I STILL DON"T SEE IT. I believe the Beatles intended that meaning.  They also would not play to audiences that had rules about Whites only, or were segregated. So YEAH for them, but I still don't see it. Or hear it. Why the contradiction? Perhaps if I heard it in 1968 I would have understood what it was about. Perhaps they really weren't that clear about it. Perhaps they had to avoid it being censored.

Born in the USA by Bruce Springsteen. This is a well-known misunderstood song, so better to say People USED TO think it was a Pride-in-America song but it was really about the plight of lower class Americans, especially Vietnam War Veterans, after the war. OF COURSE ITS ABOUT THAT. Why was there the contradiction? (1) the chorus is loud and understandable and belts out BORN IN THE USA! as if that's a good thing, (2) the other lyrics are somewhat mumbled (I had to listen to it on a you tube video with closed caption to understand the song), (3) People hear what they want to hear. 

Notes: I was GOING to look up what The Boss's top hit ever was, expecting it to be Born to Run, but that's only his 18th biggest hit. Born in the USA is 8th, and Secret Garden is 1st. Even so, I think of Born to Run as his best known song. Why? (1) It was sung as the opening number of the 2010 Emmy awards (not by him, but done really well- Jimmy Fallon does a GREAT Bruce Springsteen), see here (2) there are several parodies of it, see born to Run (COVID), Born to Run (Bridgegate), Meant To Be (a best man's song), Jedi are Done. Having went to the effort to find parodies of Born to Run I then found parodies of Born in the USA: Bored in the USA (COVID)Touched by the TSAConned in the USABorune'D in the USA (cryptocurr) And there are more. Upshot: trying to find out what someone's best-known song is can be a quagmire, but at least I  got to find some cool parodies.

Who Let the Dogs Out by the Bah  Men. People thought this song was about ... Hmmm, I don't know what people thought. Perhaps it was about someone who let the dogs out. Its actually about how BAD it is when men cat-call women. OF COURSE ITS ABOUT THAT once you see the lyrics.  Why the contradiction? Its really hard to understand anything except the chorus. Their biggest hit.

The Macarena by Los Del Rio. People tend to not listen to music that they dance to. So people really did not think it had a meaning. Also some of it is in Spanish. I can't write what it's about here since I may violate community standards as we did with a prior post (see here). See here for what the lyrics mean OR the list I pointed to above. If you listen to it or read the lyrics OF COURSE IT MEANS THAT! HOW DID I MISS IT? TOO BUSY DANCING! Why the contradiction- as I said above, its really a dance song. Their biggest hit. 

Note: Dance Songs usually don't have that many words. Knuth (see here for the original article and  here for the Wikipedia page about it which has later results) noted that the complexity of  That's the way uh-uh I like it is O(1).

Margaritaville by Jimmy Buffett (I'd be curious to see a version by Warren Buffett). This is a well-known misunderstood song, so better to say that people USED TO think it celebrated a relaxed lifestyle but its actually a sad son about a drunk. OF COURSE THE SONG IS ABOUT BEING DRUNK AND DEPRESSED. So why the contradiction? The tune is so happy-go-lucky, and Jimmy Buffett (and others) talk POSITIVELY about The Margaritaville lifestyle. Whats really odd is that the real meaning of the song IS WELL KNOWN, yet is ignored.

Note: A more realistic take on this topic, to the same tune, is  here.  A Marijuana version of the song is here. A crystal meth version  of the song is here. There are FOUR parodies that are NOT about being drunk, high, or on Crystal-meth, but about... COVID: hereherehere, and here

99 Red Balloons or  99 Luft ballons (the original German Version) To quote the original link: Whether in its original German language or in English, the happy-pop New Wave jam is easily the most danceable  song about a nuclear holocaust caused by balloons. When I listened to it and read the lyrics OF COURSE ITS ABOUT A NUCLEAR HOLOCAUST CAUSED BY BALLOONS. Why the contradiction?  The more popular version is in German, the English version is a bit mumbled (but not much), but most importantly, if there is going to be a nuclear holocaust caused by balloons  I will get up and dance!

Note:  I knew of one parody 99 Dead Baboons, but through the wonders of search and you tube I found more: the social media song, 99 90s shows, 99 unused balloons, 99 Steins of Beer 

For two more, though they are not on the list, see  'The Pina Colada' Song is Really Messed up and Why the Beastie Boys Hate `Fight for your right to Party'

TO SUM UP: songs that get misunderstood may (1) have some  hard to understand lyrics, (2) be dance songs, (3) have the melody and instruments be at odds with the lyrics, (4) have lyrics that people want to hear and others that they don't, (5) be partly or wholly in a foreign language.  I am sure there are other reasons.  

Jimmy Buffett: You will be missed!

Wednesday, August 30, 2023

What Makes a Constructive Proof?

In this weblog, we've used constructive in different ways. Often we talk about constructive as something we can create in polynomial time, like an expander. But how about constructive as in logic, when you don't get to assume the "excluded middle" where you get to assume some statement is either true or false?

The simplest well-known example is the theorem: There exists irrational \(a\) and \(b\) such that \(a^b\) is rational. 

  1. \(\sqrt{2}^\sqrt{2}\) is rational. Let \(a = b = \sqrt{2}\).
  2. \(\sqrt{2}^\sqrt{2}\) is irrational. Let \(a = \sqrt{2}^\sqrt{2}\) and \(b = \sqrt{2}\).
You don't know which \(a\) is correct. You just know it exists. (A far more complicated argument shows \(\sqrt{2}^\sqrt{2}\) is in fact irrational.) 

When I teach intro theory, my first proof that there are non-computable sets is by claiming the computable sets are countable but their are an uncountable number of sets over \(\Sigma^*\) so there must be an non-computable sets. I claim this is a non-constructive proof because I didn't give you the set and do an aside on constructive proofs using the example above. But that's not correct--the proof that there are uncountable number of sets over \(\Sigma^*\) is a constructive diagonalization. Give me an enumeration of the computable sets and I can easily construct a set not on that list.

In complexity, a well-known non-constructive theorem is by Kannan, showing that \(\Sigma^P_2\) does not have \(n^2\)-size circuits.

  1. SAT doesn't have n2-size circuits. Since SAT is in Σ2 we are done.
  2. SAT has n2-size circuits. Then by Karp-Lipton Σ4 = Σ2 so L is in Σ2 and we are done.
Jin-Yi Cai and Osamu Watanabe, and independently Sunny Daniels, gave a constructive \(\Sigma^2_P\) machine and thus a single language in \(\Sigma^P_2\) that doesn't have \(n^2\)-size circuits. But it is not a constructive proof, as the argument the machine works requires the two cases as to whether SAT has small circuits. As far as I know, a true constructive proof of Kannan's theorem remains open.

I have no problem with non-constructive proofs—I'm in a firm believer in \(P\vee\neg P\). But if you do talk about constructivity be sure and use it appropriately. 

Sunday, August 27, 2023

Theorems and Lemmas and Proofs, Oh My!

I was recently asked by a non-mathematician about the difference between the terms Theorem, Lemma, etc. My first reaction was I probably have a blog post on that. Actually, I looked and I don't seem to. Since I have, according to Ken Reagan, over 1000 posts (see here and here) I can easily confuse things I meant to write a post on with things I wrote a post on. My next thought was Lance probably has a post on that. I asked him, and he also thought he had, but also had not. So now I will!

Open Question: A well defined question that you don't know the answer to and may not even have a guess as to which way it goes. The above is not quite right: sometimes an open question is not that well defined (e.g., Hilbert's 6th problem: Makes Physics Rigorous) and sometimes you have some idea, or a rooting interest, in how it goes. I tried to find some open questions in Mathematics where people in the know don't have a consensus opinion.  I can think of two off hand: Is Graph Isom in P? and is Factoring in P. Maybe the Unique Game Conjecture, though I think people in the know think it's true. Here is a website of open questions, but I think for all of them people in the know  think we know how they go: here.

Conjecture: A statement that you think is true, and may even have some evidence that its true, but have not proven yet. I am used to using this term in math, and hence I hope someone will PROVE the conjecture. Are there conjectures in empirical sciences? If so, then how do they finally decide it's true? Also note- I blogged about conjectures and how once they are proven the conjecturer is forgotten here. EXAMPLES OF CONJECTURES: The same link as in open problems above. 

True story but I will leave out names: There was a conjecture which I will call B's Conjecture. C & S solved it AS WRITTEN but clearly NOT AS INTENDED.  Even so, C & S got a published paper out of it. This paper made M so mad that he wrote a GREAT paper that solved the conjecture as intended (and in the opposite direction). That paper also got published. So one conjecture lead to two opposite solutions and two papers. 

Wild-Ass Guess: You can take a wild-ass guess what this is. 

Hypothesis: An assumption that you may not think is true but are curious what may be derived from it. The Continuum Hypothesis is one. For some reason Riemann's problem is called The Riemann Hypothesis even thought it's really a conjecture.  So is my notion of Hypothesis wrong? In any case, if you know other things that are called Hypothesis then please leave a comment. 

Lemma: A statement that is proven but only of interest in the service of proving a theorem. There are exceptions where a Lemma ends up being very important, see here. The word is also used in English, see here, but I've never heard of the word being used that way.

Theorem: A statement that has been proven. Usually it is somewhat general. There are a few exceptions: Fermat's Last Theorem was called that before it was a Theorem. If you know other things that were called theorems but weren't,  please comment.  EXAMPLES OF THEOREMS: The Fundamental Theorem of X (fill in the X), Ramsey's Theorem, VDW's theorem, Cook-Levin Theorem, The Governor's theorem (see here). There are many more theorems that have names and many that do not. 

Corollary: A statement that follows directly from a Theorem. Perhaps an interesting subcase of a Theorem. Often this is what you really care about. When trying to find a famous corollary I instead found The Roosevelt Corollary to the Monroe Doctrine,  Corollaries of the Pythagorean theorem, and Uses of the word Corollary in English. Are there any famous corollaries in mathematics that have names?

Claim: I do the following though I do not know if its common: During a proof I have something that I need for it, but it is  tied-to-the-proof-of-the-theorem so it would be hard to make a lemma. So I prove it inside the proof of the theorem and call it a claim. I use Claim, Proof of Claim, End of Proof of Claim to delimit it. 

Porism: A statement that you can get from a theorem by a minor adjustment of the proof. I've also heard the phrase Corollary of the proof of Theorem X. I first saw this in Jefferey Hirst's Phd Thesis which is here, on Reverse Mathematics. I liked the notion so much I've used it a few times. It does not seem to have caught on; however,  there is a Wikipedia entry for the term here which also gives two examples of its use, which are not from Hirst's  thesis or my papers. 

Proposition: I see this so rarely that I looked up what the difference is between a Proposition and a Theorem. From what I read a Proposition is either of lesser importance, or is easy enough to not need to give a prove, as opposed to a Theorem which is important and needs a proof.

Axiom: A statement that one assume is true and usually they are self-evident and true. Exceptions are The Axiom of Choice which some people reject since it is non-constructive. Also some people do not thing The Axiom of Determinacy is self-evident. Same for Large Cardinal Axioms. But really, most axioms are self-evident. Note that all branches of math use Axioms.

Postulate: Euclid used the term Postulate instead of Axiom. Actually, Euclid wrote in Ancient Greek so to say he used the term Postulate is probably not quite right. However, the term Postulate seems to mean an axiom of Euclid's, or perhaps an axiom in Geometry. One exception: Bertrand's Postulate which was a conjecture but is now a theorem. The link is to a math-stacks where there is some explanation for the weird name. 

Paradox:  A Paradox is a statement that is  paradoxical. Hmmm. that last sentence is self-referential, so its not enlightening. A paradox is supposed to be a statement that seems absurd or self contradictory, though under closer examination may make sense. Russell's Paradox shows that Frege's definition of a set does not work. The Monty Hall paradox, and the Banach-Tarski Paradox are just theorems that at first glance seem to be absurd. The Monty Hall Paradox is not absurd. Darling thinks the BT-paradox means that math is broken, see this post here for more on that.