A new dean has taken my place, and I have returned to the professoriate at Illinois Tech, ending thirteen years in administration, six as dean and seven as department chair at Georgia Tech. I won't rule out more administrative roles in the future, but only if the right role presents itself.
I'll teach intro theory in the fall, my first course since 2018, and take a sabbatical in the spring, mostly at Oxford. I plan to focus on writing, hoping to get out another book or books and other projects. It will be hard to go back to traditional computational complexity research, the field has changed considerably. I plan to spend some time understanding how AI changes the way we think about computation. Particularly why we see many of the benefits of P = NP while cryptography remains secure.
Also for the first time in 13 years I don't have a "boss". Technically I report to the department chair, who until a few days ago reported to me. But tenure protects my job, I choose my own research agenda, and teaching and service assignments are more of a negotiation than a top-down decision. Freedom!
For the blog, I have held back talking about the inner workings of universities while I had administrative roles. I'll now be more open in giving my thoughts, at least in general terms.
(I wrote this post a while back so its no longer NEW. More important--- if there has been a follow-up to the story that is not in my post, let me know.)
We have something NEW and something OLD about the Pythagorean Theorem. Now all we need is something BORROWED and something BLUE.
NEW
Two high school students have a new proof of the Pythagorean Theorem, see here. (An anonymous commenter provides a pointer to their published article in the American Math Monthly.)
They used trigonometry. Oh wait, that sounds circular since Trig is based on the Pythagorean Theorem. While this is a fair question to ask about the theorem, it has recently been accepted for publication and looked at carefully (those two are not equivalent) so it seems to be correct.
The Pythagorean Theorem is an often-proved-theorem. Often an often-proved-theorem has proofs that use hard math (e.g., proofs that primes are infinite using Ramsey Theory, see my post on that here). However, the new proof of PT seems to be elementary.
Kudos to them!
OLD:
Pythagorean theorem found on clay tablets 1000 years earlier than Pythagoras: see here.
My students would ask me
How come Pythagoras didn't go to arXiv to see if someone already had the theorem?
Alberto Fraile and Daniel Fernández guest post on random walks generated by the distribution of prime numbers.
In our recent papers, we explored the sequence of prime numbers by defining "random walks" governed by simple algorithms applied to their sequence.
We introduced a prime number visualization called Jacob’s Ladder. The algorithm plots numbers on a 2D graph that oscillates up and down based on the presence of prime numbers, creating a ladder-like structure. The path ascends or descends based on the primality of subsequent numbers. When a prime number is encountered, the path alters direction, leading to a zig-zag pattern. Number 2 is prime, so it flips and goes down. Now 3 is prime, so the next step changes direction and goes up again, so we move up. But 4 is not a prime, so it continues up, and on it goes.
Jacob’s Ladder from 1 to 100,000 (Top) and from 1 to 1,000,000 (Bottom). The blue line represents y=0, or sea level.
The x-axis can be imagined as sea level, the zig-zag above it as mountains, and those below as ocean chasms. Our intrepid navigator sails eastward, occasionally discovering new lands—sometimes tiny islands, other times vast continents.
As we chart this new world, it is natural to wonder about the location of the next continent (if any), the height of its highest mountain, or the depth of its deepest ocean. One thing we know for sure is that gaps between primes can become arbitrarily large. This suggests there may be no upper bound on the mountains’ heights or the chasms’ depths.
A natural question arises: if the voyage continues into infinity, would this world contain equal amounts of land and sea? Or, more formally, does the construction exhibit symmetry in the limit, with equal numbers of positive and negative points? The beauty of Jacob’s Ladder lies in its simplicity, yet it raises many questions that are surprisingly difficult to answer.
Prime Walk
In our second study, we examined the behavior of a 2D "random walk" determined by the sequence of prime numbers, known as the prime walk (PW), choosing a direction based on the last digit of the next prime (1 down, 3 up, 7 right, 9 left) ignoring the primes 2 and 5.
Graphical representation of three different PWs up to N=108. Color coding represents step progression.
Observing the PW in action raises numerous questions.
For instance, will this PW eventually cover the entire plane as N → ∞? Will the area explored continue expanding indefinitely, or will it reach a limit? Initially, we conjectured the area would be unbounded.
We thought this conjecture might remain unanswered indefinitely, so we challenged the community with a modest prize for anyone who could prove it within two years of publication. Surprisingly, we found the solution ourselves, detailed in our recent work.
Moreover, within the explored region, certain points remain unvisited—small regions or isolated spots. Could some points remain unreachable forever? These straightforward questions, intriguingly, remain remarkably difficult to answer.
The Univ of MD at College Park holds a HS Math Competition every year. At the reception for the winners Professor Larry Washington points to many past people who did well on the exam. Two stand out for different reasons:
1) Serge Brin did well on the UMCP HS competition and went on to be a Stanford Math Grad Student Drop Out. Oh well. (I originally had Standard instead of Stanford. That sentence will makes sense.)
2) Sarah Manchester did well on the UMCP HS competition and went on to win $1,000,000 on Wheel of Fortune.
Is there a connection between doing well on the UMCP Math Competition and winning $1,000,000 on Wheel of Fortune?
Only 4 people have won the $1,000,000. Worse, if you don't win the $1,000,000 you will probably win less than $50,000. An article about those 4 is here. This is so few that while I am sure Sarah is good at Wheel of Fortune (a) she had to also be very lucky, and (b) I doubt being good at Math had much affect on her winning.
While we are here, lets look at two other game shows.
14 people have won $1,000,000 or more on Who wants to be a Millionaire?, see here. That show has the advantage that even if you don't win $1,000,000 it's not so unusual to get over $100,000.
(What kind of people do not want to be millionaires? I give two answers later.)
Deal or No Deal has different versions in different countries so it gets more complicated:
UK: 9 big winners, Turkey: 1 big winner, Australia: 4 big winners, America: 2 big winners.
ANYWAY back to Sarah and Wheel of Fortune: The statement
Sarah did well on the UMCP HS Math Competition and went on to win $1,000,000 on Wheel of Fortune
is technically true but conveys a causality that is not true.
Who does not want to be a millionaire?
Would be a terrible name for a quiz show. However, taking it as a question the answer is with one caveat: I define Millionaire as someone who has AROUND $1,000,000.
a) Billionaires. Actually, anyone who has much more than $1,000,000 would not want to come down to only $1,000,000.
b) People who think it would change their life in ways they don't want.
As the entire Fulbright board resigned last week and as the program that promotes international visits for US researchers, and vice-versa, may not survive the Trump administration, I thought I would recount some memories from my Fulbright scholarship to the Netherlands in 1996-97.
The program had considerable paperwork for a relatively small stipend, but it went beyond the compensation. I went to a meeting in Amsterdam with the other fellows, mostly grad students and postdocs. I was the old one as a recently tenured associate professor. The others had strong reasons for being in the Netherlands: An historian who wanted to research the Dutch army, and a piano player with hands too small who came to study with the world's leading teacher on a specialized narrow-keyboard grand.
And so they asked me why I would go to the Netherlands from the US to study computer science. But I spent the sabbatical year at the Centrum Wiskunde & Informatica (Center for Mathematics and Computer Science) in Amsterdam working with Harry Buhrman, Leen Torenvliet, Paul Vitányi and others. I also made several trips to universities in Germany, England, France and Israel, and had one of my best research years.
My coolest Fulbright experience was having Thanksgiving dinner at the US ambassador's house in The Hague, perhaps the most American thing I did during the year.
I also participated in celebrations marking the fiftieth anniversary of the Fulbright program, established in 1946 to encourage international educational and research collaborations, alongside the Marshall Plan and NATO, to draw the US closer to Europe and later the rest of the world. Too bad we seem to be moving away from those ideals today.
A year ago if I showed you a picture of The Pope wearing a Baseball cap for the Chicago White Sox (or any Amercan team) you would assume it was computer-generated. And you would likely be right.
Are there any real pictures of any Pope before Pope Leo XIV with a baseball cap on?
I doubt it
A real picture of Pope Leo wearing a Chicago White Sox baseball cap is here.
1) Having an American Pope is so incongruent with how we think of Popes that pictures like Pope Leo XIV in a baseball cap look very odd.
2) Pope Francis was a soccer fan (see here). Is there a real pictures of him wearing a soccer jersey?
3) Before the web we didn't know much about the personal lives of leaders. I don't just mean we didn't know about (say) their affairs; we didn't even know about non-controversial things like what sports team they root for.
4) Lance has tweeted that he never thought he would see the day where he and The Pope rooted for the same baseball team. I want to see a picture of The Pope and Lance together at a game, both wearing Chicago White Sox caps. A real picture may be hard to do, but I suspect that once The Pope sees this post he will create such a picture.
5) I hope the next Pope is a fan of the San Diego Padres for two reasons
a) The name of the team.
b) They have never won a World Series. They need all the help they can get.
In the June CACM, Micah Beck writes an opinion piece Accept the Consequences where he is quite skeptical of the role of theory in real-world software development, concluding
It is important that we teach practical computer engineering as a field separate from formal computer science. The latter can help in the understanding and analysis of the former, but may never model it well enough to be predictive in the way the physical sciences are.
I certainly agree that theoretical results can't perfectly predict how algorithms work in practice, but neither does physics. The world is much more complex, both computationally and physically, to perfectly model. Physics gives us an approximation to reality that can help guide engineering decisions and theory can do the same for computation.
You need a basis to reason about computation, lest you are just flying blind. Theory gives you that basis.
Let's consider sorting. Beck complains that Quicksort runs in \(O(n^2)\) time in the worst case even though it is used commonly in practice, while the little-used Insertion sort runs in worst-case \(O(n\log n)\). Let's assume Beck meant an algorithm like heapsort that actually has \(O(n\log n)\) worst-case performance. But theorists do more than fixate on worst-case performance, Quicksort runs in \(O(n\log n)\) on average, and on worst-case if you use a random pivot, or a more complex deterministic pivoting algorithm. Introsort combines Quicksort efficiency and worst-case guarantees and is used in some standard libraries.
Beck worries about secondary storage and communication limitations but theorists have studied sorting in those regimes as well.
The other example he talks about is about a theoretical result that one cannot use an unreliable network to implement one that is completely reliable while textbooks consider TCP to be reliable. But in fact TCP was designed to allow failure because it took account of the theoretical result, not in spite of it.
Beck ends the article talking about Generative AI where theory hasn't kept up with practice at all. Beck calls for using classical AI tools based on formal logic as guardrails for generative AI. However, the lack of theoretical understanding suggests that such guardrails may significantly weaken generative AI's expressive power. Without adequate theory, we must instead rely more heavily on extensive testing, particularly for critical systems.
There are stronger examples Beck could have used, such as algorithms that solve many NP-complete problems efficiently in practice despite their theoretical intractability. Even here, understanding the theoretical limitations helps us focus on developing better heuristics and recognizing when problems might be computationally difficult.
I agree with Beck that relying solely on the theoretical models can cause some challenges but rather than have the students "unlearn" the theory, encourage them to use the theory appropriately to help guide the design of new systems.
Beck's call to separate software development from theory only underscores how important we need to keep them intertwined. Students should know the theoretical foundations, for they shape problem solving, but they should also understand the limitations of these models.
We (Bill and Lance) care about this result for two different reasons.
BILL: The result has applications to constructive Ramsey---
LANCE: Ramsey Theory? Really? This is a great result about
Eshan Chattopadhyay
pseudorandomness! In fact the only interesting thing to come out of Ramsey Theory is the Probabilistic Method (see our discussion of this here).
BILL: Can't it be BOTH a great result in derandomization AND have an application to Ramsey Theory. Like Miller Lite: Less Filling AND Tastes Great (see here)
LANCE: But you don't drink!
BILL: Which means I can give a sober description of their application to Ramsey Theory.
All statements are asymptotic.
Let \(R(k)\) be the least \(n\) so that for all 2-colorings of \(K_n\) there is a homog set of size \(k\).
Known and easy: \(R(k)\le 2^{2k}/\sqrt{k} \)
Known and hard: \(R(k) \le 3.993^k \). How do I know this is true? Either I believe the survey papers on these kinds of results (see here) or a former student of mine emailed me a picture of a T-shirt that has the result (see here) from (of course) Hungary.
Known and Easy and Non-Constructive: \(R(k)\ge k2^{k/2}\)
Can we get a constructive proof? There were some over the years; however, the paper by Eshan Chattopadhyay and David Zuckerman improves the constructive bound to exponential in \( 2^{(\log k)^\epsilon}.\)
SO Lance, why do you care?
LANCE: First of all when I chose this paper as one of my favorite theorems (a far bigger honor than the so-called Gödel Prize) I gave the post the clever title Extracting Ramsey Graphs that captures both the pseudorandomness and the Ramsey graphs. But of course the Ramsey result is just a minor corollary, the ability to get a near perfect random bit out of two independent sources of low min-entropy is the true beauty of this paper.
