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.
You can write laws that are very specific, like the US tax code, or open to interpretation like the first amendment. In the literature these are known as rules and standards respectively.
In computational complexity, we generally think of complexity as bad. We want to solve problems quickly and simply. Sometimes complexity is good, if you want to hide information, generate randomness or need some friction. But mostly we want simplicity. How does simplicity guide us in setting guidance, either through rules or standards?
Rules are like a computer program. Feed in the input and get an output. Predictable and easy to compute. So why not always have tight rules?
Nobody ever gets a computer program right the first time, and the same goes for rules. Rules can be overly restrictive or have loopholes, leading to feelings of unfairness. Rules can require hoops to jump through to get things done. Rules don't engender trust to the ones the rules apply to, like very tight requirements on how grant funds can be spent. We know that in general we can't predict anything about how a computer program behaves, so why do we trust the rules?
A good example of a standard is that a PhD dissertation requires significant original research. Rules are things like the exact formatting requirements of a thesis, or statements like a CS thesis must contain three papers published in a specific given set of conferences.
As an administrator I like to focus on making decisions based on what's best for my unit, as opposed to ensuring I followed every letter of every rule. Because if you live by the rules, you'll die by the rules. People will try to use their interpretation of the rules to force your hand.
Sometimes we do need strict rules, like safety standards, especially for people unfamiliar with the equipment. Structured rules do give a complete clarity of when an action is allowed. But it also gives an excuse. Have you ever been satisfied by someone who did something you didn't like but said "I was just following the rules"?
Even strict rules tend to have an out, like a petition to take a set of courses that don't exactly match the requirements of a major. The petition is a standard, open to interpretation to capture what the rules don't.
As a complexity theorist I know what programs can't achieve, and as an administrator I see the same with rules. I prefer standards, principles over policies. Set your expectations, live by example, and trust the people, faculty, staff and students, to do the right thing and push back when they don't. People don't want strict rules, but they mostly act properly when they believe they are trusted and have wide latitude in their work.
(Thanks to David Marcus who sent me the video I point to in point 4 of this post. Tip for young bloggers (if there are any) you can have a half-baked idea for a post and then someone sends you something OR you later have an idea to make it a full-baked idea for a post. That's what happened here. So keep track of your half-baked ideas.)
1) When I was 10 years old I wanted to find out how many seconds were in a century. I didn't have a calculator (I might not have known what they were). I spend a few hours doing it and I got AN ANSWER. Was it correct? I didn't account for leap years. Oh well.
(An astute reader pointed to a website that does the centuries-to-seconds conversion as well as many other conversions. It is here. If such was around when I was a kid, what affect would it have on my interest in math? Impossible to know.)
2) Fast forward to 2024: I wanted to find the longest sequence of composites all \( \le 1000\). One long sequence I found by hand is the following (I also include the least prime factor):
I wanted to find the answer WITHOUT using a computer program or looking at list of primes online (though I do allow a calculator just for division and multiplication).
Of more interest mathematically is trying to prove that there is no sequence of length 14. (If there is, then perhaps the attempt will lead us to it.)
Assume there was a sequence of consecutive composites \(\le 1000\) of length 14.
Map each one to the least prime that divides it.
At most 7 of them map to 2
At most 3 of them map to 3
At most 2 of them map to 5
At most 1 of them map to 7.
At most 1 maps to 11. (Can look at 11*p for all primes \(11\le p \le 89\) and see any of them are in a sequence of length 14.)
I'll stop here. This is getting tedious and might be wrong. So I asked Claude. It gave me a sequence of composites of length 19. Here it is (I include the least prime factor):
Can one show by hand that there is no sequence of length 20?
3) The more general question: what is the complexity of finding the longest string of composites all \( \le n\) . This is actually many questions:
a) By hand: by which I mean only allowing multiplication and division and only of numbers \(\le n.\)
b) Theoretically. Use whatever fancy algorithms you want.
c) Theoretically but can assume some Number theory Conjectures that are widely believed. The Wikipedia page on prime gaps is here. (ADDED LATER- an astude commenter pointed out that we want LARGE gaps between primes, but the wikipedia article is about SHORT gaps between primes.)
d) Do a,b,c for the set version which is as follows: Given \(n\) and\( L\) determine if there a sequence of consecutive composites of length L that are all \(\le n\).
4) Does anyone else care about calculation-by-hand? Yes! There are people who want to compute\(\ pi\) to many places JUST BY HAND. Here is a video about them here. Spoiler alert: they did very well.
On Route 1 in Saugus, Massachusetts, about a twenty minute drive from Cambridge, stood the Hilltop Steak House. When I went to graduate school in the late 80's, Hilltop led all restaurants in the United States by sales (about $30 million in annual revenue) and volume (about 2.5 million customers).
I went there several times during graduate school. Despite the size, about 1500 seats, always a long wait but worth it to get a good steak at a price a grad student could afford.
But in 1994, Hilltop was bought by an investment company from the original Giuffrida family. The cost of labor went up and to keep costs reasonable, the new owners cut the quality of the meat. No longer a place to get good steak at a good price and with changing tastes, they lost customers and would finally close in 2013.
Computer Science as an academic discipline has had its Hilltop moment with tremendous growth pretty consistently from 2010 through about 2023. But with the growth of AI and an uncertain job market, if we don't maintain quality and adjust to the new needs and interests of our students, CS may become just a road sign on Route 1.
(Trivia: What song has the lyric Some are Mathematicians, some are Carpenters's wives ? It's not a parody song, though sometimes it's hard to tell a parody song from a so-called real song.)
In my post about Pope Leo XIV I made the following comments in different parts of the post:
Pope Leo XIV has a degree in Mathematics.
Prevost [his pre-Pope name] has a degree in mathematics from Villanova.
He is not the first Pope to know some mathematics.
I also wrote:
Since Pope Leo XIV was a mathematician, as Pope he won't only know about sin but also about cos.
Someone emailed me about this line, not to say it was a bad joke or even a good joke, but to say
Since Pope Leo XIV was a mathematician: What qualifies one to be considered a mathematician?
A few thoughts on this question.
1) I blogged about this topic here. Hence today I will discuss issues I did not discuss then.
2) Robert W Prevost wrote a book that (just from the title) seems to use some math:
that was published in print in 1990 and online in 2023. I wonder if it will sell more copies now. I am tempted to ask for a free copy to read and do a review of, but I'm not sure I really want to read it.
One would think that if someone named Robert Prevost wrote a book that seems to use math and theology then it would be the Robert Prevost who is now called Pope Leo XIV. I thought that. A commenter on my blog thought that. But Lance read an earlier version of this post and pointed out that
Robert W Prevost NE Robert F Prevost AND
Robert F Prevost = Pope Leo XIV.
Hence, alas, the author of the book is NOT Pope Leo XIV. It's striking how plausible it would be that Pope Leo IS the author. The book STILL might sell more copies since people may think it's by the Pope.
3) If someone keeps LEARNING math but doesn't DO math I WOULD consider them a mathematician.
4) If someone is a math crank then the question of are they a mathematician will depend on how cranky they are.
5) If one KNOWS a lot of math but is neither learning anymore or doing any more (perhaps myself when I retire) can you consider them a mathematician?
6) If someone gets a PhD from MIT in Pure math but then goes to industry and programs would you consider them to be a mathematician?
7) If X is NOT a math crank and X considers themselves a mathematician, are they a mathematician?
8) I WOULD consider applied math to be math. This should not need to be said but there may be some pure-math-snobs reading this post. Computer scientists, statisticians, are more of a borderline case that, without being a snob, I might not consider mathematicians.
9) Someone posted on the blog where this came up Does Lance consider himself a mathematician? I asked Lance and he said:
For the next 37 days I consider myself a Dean. After that, who knows?
On Saturday, I had my last Illinois Tech graduation as dean before I step down at the end of June. The College of Computing had nearly 1600 graduates and I shook many, many hands that morning.
After graduation I caught a plane to Washington, DC to attend the wedding of my daughter's college friend. We were invited because we became good friends with the bride's parents when we lived in Atlanta. My last trip before Covid was to DC for an NSF panel and this was my first trip to Washington since.
The out-of-town guests were housed at the Hyatt Regency Bethesda, coincidentally the same hotel that hosted STOC 2009. My favorite STOC talk of all time took place in that building. But I remembered nothing about the venue except for the jogging path near the hotel.
No trip to the DMV would be complete without seeing my co-blogger Bill Gasarch in person for the first time in years. We chatted about many things, most notably his last few posts where he joked about the new pope's undergraduate math thesis, taken seriously by both our readers and ChatGPT. I reminded Bill that we are the blog of record, often used by wikipedia as a primary source for much in and out of complexity. Later Bill took out the fake thesis titles.
With graduation behind me, not much more for me to do as dean before my term ends. On the plane ride home, I thought about the question everyone asks me: What's next? I have no idea, but it's going to be fun.
There have been times when satire has been mistaken for reality. A list of Onion stories that were mistaken for reality (or was it a mistake?) is here. When I say mistaken for reality I mean that a large set of people were fooled.
My own Ramsey-History-Hoax (blog here, latest version of the paper here) has fooled some people; however the number of people is small since the number of people who know the underlying math is small.
In my last blog (see here) I said that the Pope Leo XIV majored in math (that is true) and then I gave a false title for his thesis (I HAVE SINCE REMOVED THE ENTIRE PASSAGE).
Later in the post I said that his ugrad thesis was not on that topic, but gave another false title (I HAVE RMEOVED THAT AS WELL.)
I thought the reader would know that it was false, but onecomment inquired about it so I left a comment admitting it was false.
This is all very minor: Not that many people read this blog and very few non-math people would care about what the topic of the Pope's undergraduate thesis.
The last part of the last sentence is false. Its the POPE! People Do care about his background.
But surely my blog post isn't so well read so as to make the fictional title of his thesis a hoax that fools a lot of people.
Even so, I left a comment wondering if LLM's might learn the incorrect title of the Pope's ugrad thesis.
A reader named E posted the following:
It might be too late. I did this search this evening: E: Did Pope Leo XIV study Ramsey Theory?
Gemini: Pope Leo XIV, whose given name is Robert Francis Prevost, earned a Bachelor of Science degree in mathematics from Villanova University in 1977. His undergraduate thesis focused on Rado's Theorem for Nonlinear Equations.
0) This may not be too bad- one would have to ask about The Pope and Ramsey Theory to get that answer. But in the future this answer might pop up on the question`What did the Pope Study as an Undergraduate' or similar questions.
1) Might future satires or April Fool's Day jokes be mistaken for reality in the future by AI and hence reach a much larger audience than this blog does?
2) If so, should we be careful with what we post (not sure how to do that)?
3) What about people who have a much larger following than complexityblog (yes, there are such people)?
4) In the past one had to be a celebrity or similar to change peoples perception of reality (see Stephen Colbert and Wikipedia here). Now a complexity blogger may be able to change people's perception of reality. Hence I ask
The National Science Foundation was founded on May 10, 1950, 75 years ago last Saturday. No doubt the NSF has seen better days, but first let's take a look back.
At the end of World War II, Vannevar Bush, Director of the Office of Scientific Development, wrote Science, The Endless Frontier, where he laid out the importance of scientific research and the need for the US government to foster that research.
A new agency should be established, therefore, by the Congress for the purpose. Such an agency, moreover, should be an independent agency devoted to the support of scientific research and advanced scientific education alone. Industry learned many years ago that basic research cannot often be fruitfully conducted as an adjunct to or a subdivision of an operating agency or department. Operating agencies have immediate operating goals and are under constant pressure to produce in a tangible way, for that is the test of their value. None of these conditions is favorable to basic research. Research is the exploration of the unknown and is necessarily speculative. It is inhibited by conventional approaches, traditions, and standards. It cannot be satisfactorily conducted in an atmosphere where it is gauged and tested by operating or production standards. Basic scientific research should not, therefore, be placed under an operating agency whose paramount concern is anything other than research. Research will always suffer when put in competition with operations.
The report laid out the National Research Foundation that would actually spread across three agencies, DARPA, NIH, and the NSF.
While Bush didn't significantly mention computing, given the time, Computing would become a central part of NSF's mission with the establishment of the Computing and Information Science and Engineering (CISE) directorate in 1986, placing Computing at the same level as the Math and Physical Sciences Directorate and the Engineering Directorate.
In 1999, the President’s Information Technology Advisory Committee (PITAC) issued a report that led to the NSF Information Technology Research (ITR) program, which became one of the largest NSF research initiatives of the early 2000s. The report helped reframe computing not just as infrastructure but as a scientific discipline in its own right, deserving of the same kind of basic science funding as physics or biology.
CISE has led many initiatives through the years, for example the TRIPODS program established several centers devoted to the theoretical study of data science.
In recent weeks, the NSF director stepped down, hundreds of grants were canceled, new grants were put indefinitely on hold, indirect costs on new grants will be capped at 15%, and many staff members were pushed out. Divisions below the directorates are slated for elimination, advisory committees have been disbanded, and Trump's proposed budget cuts NSF’s allocation by about half. The CISE AD (Assistant to the NSF Director, or head of CISE), Greg Hager, stepped down last week and through the CRA sent a message to the community.
Under these circumstances, my ability to carry out my vision, to provide a voice for computing research, and to provide authentic leadership to the community are diminished to the point that I can have more impact outside NSF than within it. Echoing Dr. Nelson’s powerful article, leaving “allows me to speak more clearly in my own language,” and, in doing so, even more effectively amplify the work of the incredible, dedicated CISE leadership and staff who continue to strive to fulfill NSF’s mission.
As I move beyond NSF, I will continue to make the case for computing research. Computing is central to so much in today’s world and computing advances are now core assets to the Nation’s research enterprise. NSF’s support for the past 75 years has forcefully demonstrated the value of computing research for advancing national health, prosperity and welfare; enhancing national economic competitiveness; securing the national defense and helping promote all of science and engineering. NSF-funded work has continually catalyzed new innovations, created new industries, and made us the envy of the world.
We all need to join Greg in continuing the fight to ensure that Vannevar Bush's vision continues to survive another 75 years and beyond.
The New Pope is Pope Leo XIV (pre-Pope name is Robert Prevost).
1) Pope names are one of the few places we still use Roman Numerals. I saw an article that was probably satirical that Americans prefer Roman Numerals (the numbers Jesus used) over Arabic Numerals. Also note that Pope Francis did not have a Roman Numeral- that is because he is the first Pope Francis. They could call him Pope Francis I now, rather than later, to avoid having to rewrite things. (John Paul I took the name John Paul I.)
2) Over the years I have read the following and had the following thoughts (Spoiler- I was wrong on all of them)
a) The last non-Italian Pope was Pope Adrian VI who was Pope from Jan 9 1522 to Sept 14 1523. His Wikipedia entry: here. He was 80 years old when he became Pope and died of a heart attack.
BILL THOUGHT: We will never see a non-Italian Pope again.
REALITY: John Paul II from Poland was Pope from 1978 until 2005. His Wikipedia page is here
MORE REALITY: Since then we've had Pope Benedict XVI (Germany), Pope Francis I (Argentina),and Pope Leo XIV (America). I now wonder if we will ever have an Italian Pope again but I make no predictions.
b) There will never be an American Pope because people think that America already has too much power and if there ever was an American Pope then people would think it was engineered by the CIA.
BILL THOUGHT: That sounded correct to me. Not that the election would be engineered by the CIA, but that people would think that.
REALITY: Pope Leo XIV is American. Some MAGA people are calling Pope Leo a Woke Marxist Pope (see here). Not the type the CIA would install.
QUESTION: How much power does the Pope really have? I ask non-rhetorically as always.
c) The shortest Pope Reign was Pope Urban VII (1590) who reigned for 13 days. The tenth shortest was Pope Benedict V (964) who reigned for 33 days. I originally thought the short reigns were from assassinations, but I looked it up and there were two that may have been murdered, but the rest died of natural causes. Having looked it up I wrote it up here.
BILL THOUGHT: The 10th shortest reign was 33 days. With better health care and less skullduggery in the Papacy that won't happen again.
REALITY: Pope John-Paul I in 1978 died of a heart attack after being Pope for 33 days.
d) The last Pope to resign was Pope Gregory XII in 1415 (his Wikipedia page is here). He resigned to heal a schism in the church (its more complicated than that, see his Wikipedia page).
BILL THOUGHT: We will never see a Pope resign again.
REALITY: Pope Benedict XVI resigned (see here for the Wikipedia page on the resignation) in 2013. He said it was for health reasons.
BILL THOUGHT: Now that Pope Benedict has resigned it will be easier for later popes who feel they are not healthy enough for the job to resign. But I make no predictions.
3) Pope Leo XIV has a degree in Mathematics. I emailed the following to my Ramsey Theory class which is an excerpt from his Wikipedia Page with one incorrect sentence. (NOTE- I USED TO HAVE A TITLE OF THE POPE'S UGRAD MATH THESIS, WHICH WAS FAKE, BUT SINCE AI'S PICKED IT UP AS REAL I HAVE DELETED THAT, AND ALSO DELETED A LATER PART OF THIS POST WHERE I GIVE THE REAL TITLE, WHICH IS ALSO FAKE.)
Prevost earned a Bachelor of Science (BS) degree in mathematics from Villanova University, an Augustinian college, in 1977. He obtained a Master of Divinity (MDiv) from Catholic Theological Union in Chicago in 1982, also serving as a physics and math teacher at St. Rita of Cascia High School in Chicago during his studies. He earned a Licentiate of Canon Law in 1984, followed by a Doctor of Canon Law degree in 1987 from the Pontifical University of Saint Thomas Aquinas in Rome. His doctoral thesis was titled The Role of the Local Prior in the Order of Saint Augustine. Villanova University awarded him an honorary Doctor of Humanities degree in 2014
4) He is not the first Pope who knew some mathematics. In a general sense people used to be more well-rounded before fields got so specialized. So in that sense I am sure that some prior Popes knew some math. But more concretely Pope Sylvester II was, according to the articleWhen the Pope was a Mathematician Europe's leading mathematician, (at the time a modest distinction) reigning as Pope Sylvester from 997 to1003. His Wikipedia page is here.
5) Since Pope Leo XIV was a mathematician, as Pope he won't only know about sin but also about cos.
6) The name Leo struck me since one of my TAs is named Leo. I asked him, if he became Pope, would he change his name. He said
Hmm, after careful consideration, I think I would take another name. I like being Leo, but I think I would want to try out a different name. I looked up Papal names, and I would probably pick something cool like Boniface, Honorius, or Valentine. But I would do the name change after the semester ends so as not to screw up the payroll office.
7) Popes did not always change their names. For a long article on that see here. For a short version here are some points:
a) The first Pope to change his name was born Mercurious, a Pagan God Name, and changed it to be Pope John II. That was in 533.
b) The name-change did not become standard for a while. Before the year 1000 only 3 Popes changed their names, all to John. The other two had given name Peter and felt they should not take the name Peter since Peter was the first Pope and an apostle. Kind of like having a jersey number retired by a sports team.
c) After the year 1000 some changed,some didn't, but the last one to not change was Pope Marcellus II in 1555. His reign was 22 days, though I doubt that is related to not changing his name.
8) Math question: What is the average length of a Papacy and what is the average length of a presidency (of America)?
The first Pope was Peter, began in 30AD.
The 266th Pope was Francis whose reign ended in 2025.
SO we have 266 Popes in 1995 years, so the average reign is 7.5 years.
The first president was George Washington whose presidency began in 1789.
The 46th president was Joe Biden and it ended in 2025.
SO we have 46 presidents (we ignore the Grover C thing) in 236 years, so the avg reign is 5.1 years.
The 7.5 and 5.1 are more different than they look since the length of presidents is usually 4 or 8 years,while the length of a Papal reign has had a min of 13 days and a max of 31 years (Pope Pius IX).
I'l be curious what the standard deviation and deviance are for both Papal Reigns and Presidents. I suspect that it's much bigger for Papal reigns, and not just because the presidency is at most 8 years (with one exception-FDR was president for 12 years).
9) There was betting and betting markets on the next Pope. This raises the question of how often someone NOT on the short list (so probably not bet on much) becomes Pope. Lets look at the last few:
Pope Leo XIV- not on the short list
Pope Francis- not on the short list
Pope Benedict XVI- a favorite
Pope John Paul II- not on the short list
Pope John Paul I- I don't know and I will stop here.
Upshot: it may be foolish to bet on the next Pope. Even more so than betting on the Vice Prez nominee which I commented on here.
10) Art imitates life: Some of the cardinals at the conclave watched the movie Conclave to get an idea of what a conclave is like. I suspect the real conclave was much less dramatic than the movie Conclave.
11) Trump thinks that since the Pope is American, America should annex the Vatican. Or does he? See this article here.
12) Pope Leo has an opinion about AI (I wonder if his math background helps); see here. This is a good example of the limits to the Pope's power-does anyone who can do anything care what Pope Leo XIV thinks? I ask non-rhetorically as always.
I reviewed a paper recently and I had to agree not to use AI in any aspect of the reviewing process. So I didn't but it felt strange, like I wouldn't be able to use a calculator to check calculations in a paper. Large language models aren't perfect but they've gotten very good and while we shouldn't trust them to find issues in a paper, they are certainly worth listening to. What we shouldn't do is have AI just write the review with little or no human oversight, and the journal wanted me to check the box probably to ensure I wouldn't just do that, though I'm sure some do and check the box anyway.
I've been playing with OpenAI's o3 model and color me impressed especially when it comes to complexity. It solves all my old homework problems and cuts through purported P v NP proofs like butter. I've tried it on some of my favorite open problems where it doesn't make new progress but it doesn't create fake proofs and does a good job giving the state of the art, some of which I didn't even know about beforehand.
We now have AI at the level of new graduate students. We should treat them as such. Sometimes we give grad students papers to review for conferences but we need to look over what they say afterwards, the same way we should treat these new AI systems. Just because o3 can't find a bug doesn't mean there isn't one. The analogy isn't perfect, we give students papers to review so they can learn the state of the art and become better critical thinkers, in addition to getting help in our reviews.
We do have a privacy issue. Most papers under review are not for public consumption and if uploaded into a large-language model they could become training data and be revealed if someone asks a relevant question. Ideally we should use a system that doesn't train on our inputs if we use AI for reviewing but both the probability of leakage and amount of damage is low, so I wouldn't worry too much about it.
If you are an author, have AI review your paper before you submit it. Make sure you ask AI to give you a critical review and make suggestions. Maybe in the future we'd required all submitted papers to be AI-certified. It would make the conference reviewers jobs less onerous.
For now, humans alone or AI alone is just not the best way to do conference reviews. For now when you do a review, working with an AI system as an assistant will lead to a stronger review. I suspect in the future, perhaps not that far, AI alone might do a better job. We're not there yet, but we're further than you'd probably expect.
In a recent post by Scott (see here or just read my post which includes his post) he listed topis that he conjectured would NOT cause an outrage.
I was going to write a long comment in his comments section, which would only be read by people who got to comment 100 or so. OR I could comment on it in my blog.
SO, here is his blog post and my comments on it.
------------------------------
A friend and I were discussing whether there’s anything I could possibly say, on this blog, in 2025, that wouldn’t provoke an outraged reaction from my commenters. So I started jotting down ideas. Let’s see how I did.
1) Pancakes are a delicious breakfast, especially with blueberries and maple syrup.
BILL: Pancakes have a terrible ratio of health to enjoyment.
2) Since it’s now Passover, and no pancakes for me this week, let me add: I think matzoh has been somewhat unfairly maligned. Of course it tastes like cardboard if you eat it plain, but it’s pretty tasty with butter, fruit preserves, tuna salad, egg salad, or chopped liver.
BILL: UNFAIRLY MALIGNED. That's an interesting concept in itself since there are so many opinions on the internet there is not really a consensus on.... anything. My 2 cents: I like the taste of cardboard and hence I like the taste of matzoh.
3) Central Texas is actually really nice in the springtime, with lush foliage and good weather for being outside.
BILL: I WILL DEFER to Scott, who is now a Texan, on this one.
4) Kittens are cute. So are puppies, although I’d go for kittens given the choice.
BILL: PETS are a waste of time and energy. My opinion shows something more important: Scott and I disagree on this but we are not OUTRAGED at each other.
5) Hamilton is a great musical—so much so that it’s become hard to think about the American Founding except as Lin-Manuel Miranda reimagined it, with rap battles in Washington’s cabinet and so forth. I’m glad I got to take my kids to see it last week, when it was in Austin (I hadn’t seen it since its pre-Broadway previews a decade ago). Two-hundred fifty years on, I hope America remembers its founding promise, and that Hamilton doesn’t turn out to be America’s eulogy.
BILL: Agree. Also lead to the best math novelty song of all time, See here.
6) The Simpsons and Futurama are hilarious.
BILL: The cliche The Simpsons was better in its first X seasons is true, but it can still crank out an excellent episode once in a while. The episode Treehouse of Horrors: Simpsons Wicked This Way Comes (from 2024--Wikipedia entry here) is a microcosm of the series: Two okay satires of two okay stories by Ray Bradbury and then a BRILLIANT satire of Fahrenheit 451. (Spell check thinks Treehouse is not a word .I looked it up to see what the geat God Google would say. The Treehouse of Horror episodes of the Simpsons use Treehouse. I googled Is Treehouse One word and got a YES. This is a rare time when spellcheck is just wrong.)
BILL: I think Futurama benefited from being on the air, then off, then on, then off, then on (is it on now?) since it came back with new stories.
7) Young Sheldon and The Big Bang Theory are unjustly maligned. They were about as good as any sitcoms can possibly be.
BILL: AGREE though again, some malign, some praise, some just watch it and laugh. I've had 5 blog posts inspired by these shows, and a few more that mention them in passing. I recently saw TBBT on a list of OVERRATED shows so someone must be liking it to cause it to be on that list.
BILL: In an earlier era it would be hard to watch every episode of a TV show since they were on once, and then maybe some reruns but maybe not. I've seen every episode of both TBBT and YS without even trying to.
BILL: (Added later inspired by a comment) For the entire run of the series YS the actress who played Missy, Raegan Revod did not have a Wikipedia page. I noted this in two prior blog posts. I am happy to say that she finally does, see here. It is a long overdue honor. (Is it an honor?)
8) For the most part, people should be free to live lives of their choosing, as long as they’re not harming others.
BILL: TRICKY- the For the most part causes arguments and outrage. Example: Helmet laws for motorcyclists. Should they be free to get brain injuries that the rest of society must pay for? I ask non-rhetorically as always.
9) The rapid progress of AI might be the most important thing that’s happened in my lifetime. There’s a huge range of plausible outcomes, from “merely another technological transformation like computing or the Internet” to “biggest thing since the appearance of multicellular life,” but in any case, we ought to proceed with caution and with the wider interests of humanity foremost in our minds.
BILL: I doubt it's the biggest thing since the appearance of multicellular life. My blog on AI here. I agree that caution is needed, though in two ways:
a) Programs are written that we don't understand and might be wrong in serious ways. (You can replace programs with other things.)
b) The shift in the job market may be disruptive. People point to that farmers stopped farming and moved to factory work, but there was an awful transition time. And the AI-shift might be much faster. Fortunately for me, ChatGPT is terrible at solving problems in Ramsey Theory. For now.
10) Research into curing cancer is great and should continue to be supported.
BILL: This one seems obvious but one has to ask the broader question: Which medical things should be funded and why? More generally, what should the government fund and why? These require careful thought.
11) The discoveries of NP-completeness, public-key encryption, zero-knowledge and probabilistically checkable proofs, and quantum computational speedups were milestones in the history of theoretical computer science, worthy of celebration.
BILL: Of course I agree. But the following questions haunt me:
a) What is a natural problem and do we spend too much time on unnatural ones. Even Graph Isom which seems like a natural problem does not have any applications (see my blog posts here and a ChatGPT generated post on this topic here).
b) Darling has asked me IF WE PROVE P NE NP THEN HOW WILL THAT HELP SOCIETY? Good question.
12) Katalin Karikó, who pioneered mRNA vaccines, is a heroine of humanity. We should figure out how to create more Katalin Karikós.
BILL: Cloning?
BILL: This raises the general question of how much ONE PERSON is responsible for great scientific discoveries.
13) Scientists spend too much of their time writing grant proposals, and not enough doing actual science. We should experiment with new institutions to fix this.
BILL: Also writing up papers and waiting for referees reports. A paper I submitted with students 3 years ago was accepted (Yeah) with many helpful comments (Yeah) but way too late to help those students get into grad school (they did anyway- Yeah). We had forgotten what we wrote and why we cared. (Boo) We did get the corrections done and resubmitted it. So I could say it will be out soon. But that's the weird thing-we posted it to arxivs three years ago so its been out for a while.
14) I wish California could build high-speed rail from LA to San Francisco. If California’s Democrats could show they could do this, it would be an electoral boon to Democrats nationally.
BILL: This seems fine but seems like an arbitrary thing to want as opposed to other pairs of cities and other achievement.
15) I wish the US could build clean energy, including wind, solar, and nuclear. Actually, more generally, we should do everything recommended in Derek Thompson and Ezra Klein’s phenomenal new book Abundance, which I just finished.
BILL: You inspired me to recommend the book Abundance to my book club. This is the second time that's happened- I also had them read the Stephen Pinker Book Enlightenment Now based on your blogs recommendation.
BILL: Some of the problem is political and some is technical. I don't know how much of each.
16) The great questions of philosophy—why does the universe exist? how does consciousness relate to the physical world? what grounds morality?—are worthy of respect, as primary drivers of human curiosity for millennia. Scientists and engineers should never sneer at these questions. All the same, I personally couldn’t spend my life on such questions: I also need small problems, ones where I can make definite progress.
BILL: Indeed-I like well defined questions that have answers, even if they are hard to answer. The questions you raise are above my pay grade.
17) Quantum physics, which turns 100 this year, is arguably the most metaphysical of all empirical discoveries. It’s worthy of returning to again and again in life, asking: but how could the world be that way? Is there a different angle that we missed?
BILL: I can either have an opinion on this or defer to one of the worlds leading authorities on the topic.
18) If I knew for sure that I could achieve Enlightenment, but only by meditating on a mountaintop for a decade, a further question would arise: is it worth it? Or would I rather spend that decade engaged with the world, with scientific problems and with other people?
BILL: If that enlightenment includes obtaining a proof that P NE NP then sign me up!
19) I, too, vote for political parties, and have sectarian allegiances. But I’m most moved by human creative effort, in science or literature or anything else, that transcends time and place and circumstance and speaks to the eternal.
BILL: I find myself less interested in politics and more interested in math. Non-partisan example: I read many articles about who Trump will pick for his VP. Then he picked one. I then read many articles about who Harris will pick for her VP.Then she picked one. I WISH I HAD SPENT THAT TIME ON THE POLYNOMIAL-HALES-JEWITT THEOREM INSTEAD!
20) As I was writing this post, a bird died by flying straight into the window of my home office. As little sense as it might make from a utilitarian standpoint, I am sad for that bird.
BILL: If we could ,without too much effort, make this not happen in the future, that would be good. There were some suggestions for that in your blog comments.
As someone who has literally written a book on the topic, I have had many people over the years send me their attempts at P v NP proofs. On average, I receive about one a month, but I've had seven in the last two weeks. And not just the usual email with a PDF attachment. A DM in X. A phone call with a baby in the background. Via Linkedin, in Spanish. One with forty-one follow up emails.
P v NP is still the most important problem in theoretical computer science and perhaps all of mathematics, and not that difficult to understand, at least on an intuitive level. I can see the fascination in wanting to solve it, much the way my teenage self dreamed of being the first to prove Fermat's last theorem (damn you Wiles).
But why the spate of "proofs" right now? Maybe it's an artifact of the crazy world we live in.
In one sense, I appreciate the continued interest in this great question, even though these papers rarely (really never) provide new insights into the P v NP problem. Most of my colleagues just ignore these emails, I usually try to respond once, but most authors will come back and I just don't have time for those continued conversations.
These papers come in three categories.
Claiming to prove P ≠ NP by arguing that a polynomial-time machine must search a large space of solutions. To truly prove P ≠ NP, one cannot make assumptions about how the machine might work.
Giving an algorithm for an NP-complete problem which works on some small test case. I'd usually ask for them to solve SAT competition problems, solve RSA challenge problems or mine themselves some bitcoin.
Giving a new philosophical or physical spin on the P v NP problem and claiming that tells us about whether P = NP. But P v NP is at its heart a mathematical question, and no amount of philosophy or physics will help settle it.
I have a new suggestion for those who think they've settled P v NP: Run your paper through an AI system, preferably a reasoning model, using the prompt "Give me a critical review of this paper". If you can't convince AI, you're not likely to convince me.
I wrote this about a month ago but wanted to wait until after the REU PI conference (which was April 21-22-23) to post it. I add a few comments based on what has happened since, which I preface with ADDED.
I agree with their posts and do not have anything to add about the general situation.
Hence I give you a personal view. While not as important as the general problem, what is happening to me may be considered one of many canaries in the coal mines. (Do my readers know that expression and where it came from? If not then see here.)
Random points about my NSF-REU grant.
1) I got my REU grant, REU-CAAR (Combinatorics and AI for Real problems--that's not what it stood for then but its what it stands for now) in 2013 for 2013-14-15. (To see what REU grants are either go to my post about them here or goto my current website about it here.)
2) It has been renewed for 2016-17-18 and 2019-2020-2021 and 2022-2023-2024 and 2025-26-27. The last one with a caveat.
3) In all but the last one, being recommended for a grant was equivalent to getting the funding. But for the 2025-2026-2027 I have not seen a dime and I assume I will not get funding in time for a Summer 2025 program to be run normally. (ADDED: I was correct on this.)
4) I am running the program anyway- mostly local students (don't need housing) who don't need stipends. There may be a little (not much) money for a few stipends, from other sources.
ADDED: Here is a list of approaches people who have been promised money but haven't gotten it are doing
a) Run a program with very little money and have the students come WITHOUT stipend, WITHOUT housing, WITHOUT transportation, WITHOUT food. Mostly local students. The program can still run but is against the whole point of REU grants which is broadening of students and giving students from non-research schools a chance to do research. (Only 3 of my students are from Non Research Schools. Another 4 are from High Schools, so not sure how that counts for this.) One odd pro: In my case I have 25 students- I can make more offers since I am not paying anything.
b) Run a program with some money you have lying around. You may decide to give stipends but NOT housing.You may (like approach (a)) hence take mostly local students. But all students get stipends.
c) You assume you will get money by (say) May 15). So you take applicants, accept and reject as appropriate. PRO-if the money comes in, you run a normal program. CON- you may end up cancelling in (say) Mid May leaving students in the lurch.
d) Do not begin trying to run a program UNTIL you get funding. If you get funded late then run a small program of mostly local students.
e) There are probably other approaches or combinations of the above.
5) I've heard that the reason I won't see money in time is NOT that REU grants are DEI but because of the staff cuts at NSF make it harder to get funds out the door.
6) Will I get funded in time for 2026? I'd be surprised either way. Is it possible for both A and NOT(A) to be surprising? I'll make that an REU project in 2026 if I get funded by then.
7) One of the original motivations for the REU program was to give students at non-research schools a chance to do research. Hence I use comes from a non-research school as a non-merit criteria. Is using that criteria DEI? I doubt Elon has thought that through. (HE MIGHT HAVE- see comment 9.5 that I added after I posted this.)
8) Another non-merit criteria I use is how many students want to work on which projects. For example, if 10 very qualified applicants want to work on Ramsey Theory, I can't take all of them. I urge the applicants to specify at least 3 projects they are happy to work on, though many do not do that. Is using the distribution of projects, a non-merit criteria, DEI? I doubt Elon has thought that through. (HE MIGHT HAVE- see comment 9.5 that I added after I posted this.)
9) Another non-merit criteria I use is veterans. It is rare that a veteran applies to my program, but it does happen and they get a preference (we've had 3 veterans). Is that non-merit criteria DEI? I doubt Elon has thought that through. (HE MIGHT HAVE- see comment 9.5 that I added after I posted this.)
9.5) An astute reader emailed me that comments 7-8-9 may be unfair.
The guidelines on NSF grants are here. I quote an FAQ question
4. Can I still propose broadening participation activities (e.g.,
outreach) in fulfillment of the Broader Impacts criterion?
Investigators should prioritize the first six broader impacts goals as
defined by the America COMPETES Reauthorization Act of 2010 (see here)
Investigators wishing to address goal seven — expanding participation
in STEM for women and underrepresented groups — must ensure that all
outreach, recruitment, or participatory activities in NSF projects are
open and available to all Americans. Investigators may conduct these
types of engagement activities to individuals, institutions, groups,
or communities based on protected characteristics only as part of
broad engagement activities. Investigators may also expand
participation in STEM based on non-protected characteristics,
including but not limited to institutional type, geography,
socioeconomic status, and career stage. However, engagement activities
aimed at these characteristics cannot indirectly preference or exclude
individuals or groups based on protected characteristics.
Instutitional type can be interpreted as non-research schools being okay. Career stage can be interpreted as returning students. Not sure about veterans of too-many-ramsey-theorists.
10) One of the things that made America great in the past was our scientific achievement. Hence we need a president who wants to Make America Great Again. We can abbreviate that to MAGA.
11) A good Popperian scientist would STUDY the NSF (and other programs) SEE what is wasteful and what is not and ACCEPT what they find. Had they done this it would have lead to some minimal changes at the NSF and the NIH and other organizations. But instead they just asserted that the NSF and NIH waste money. (Spellcheck thinks that Popperian is not a word. Oh well. For those who don't know what that means, see Karl Popper's Wikipedia entry here.)
12) Is the overhead on grants too high? That is a fair question to ask. But cutting overhead from 50% to 15% overnight is disruptive and does not get into the issue of how high overhead should be.
13) Will industry step in and fund research? I doubt it will be enough.
Right after the election I wrote a post predicting what would happen to higher education under Trump, most of which is coming true, but I had a massive failure of imagination missing the direct attacks on major universities. I won't detail all the challenges to higher ed, which change daily with every new executive order and court ruling. The Chronicle has a good tracker of these changes.
But often lost in all the news are the actual people who aren't making the news hurt by these actions, through no fault of their own: A student who had his visa cancelled while out of the country so he can't get back in. PhD students who just lost their funding when their advisor's grant was cancelled. A postdoc in a similar situation just months before he starts an academic job. A young faculty member who had to hold off submitting a Career award proposal until a TRO restored the indirect cost. A recently tenured professor at a good US university interviewing outside of the country. Potential students in other countries trying to decide if they should still go to school and build a life in the US.
The number of people, grants and universities affected is still on the low end, nearly all students continue to study, graduate and work without any problems, and many universities have offered legal and financial support to affected students and faculty. In my humble opinion, the strong educational opportunities inside the US still greatly exceed those outside. Universities have weathered many challenges before: McCarthyism in the 50s, campus occupations and protests in the 60s, and budget challenges from the great depression, to the fiscal crisis and covid. Trump's time as president has an end date and we'll get through all of this, but it requires all of us to push back and remind the administration and the public about the important role our universities play in the health of the country.
And as much as it pains me to say this as a Cornell alum, I'm rooting for Harvard.
I'm short on time time this week so I thought it would be good to look back, some 64 years ago, to Dwight Eisenhower's farewell address. It calls for balance between the industrial-military Complex and the scientific-technological elite. While written for a different time, it's well worth taking the time to watch or read the full speech and think what is says about today's complex world.
Dwight D. Eisenhower — Farewell
Address
The White House,
January 17, 1961
I. The American Experience
Good evening, my fellow Americans.
First, I should like to express my gratitude to the radio
and television networks for the opportunities they have given me over the years
to bring reports and messages to our nation.
My special thanks go to them for the opportunity of
addressing you this evening.
Three days from now, after half a century in the service of
our country, I shall lay down the responsibilities of office as, in traditional
and solemn ceremony, the authority of the Presidency is vested in my successor.
This evening I come to you with a message of leave-taking
and farewell, and to share a few final thoughts with you, my countrymen.
Like every other citizen, I wish the new President, and all
who will labor with him, Godspeed. I pray that the coming years will be blessed
with peace and prosperity for all.
Our people expect their President and the Congress to find
essential agreement on issues of great moment — the wise resolution of which
will better shape the future of the nation.
My own relations with the Congress, which began on a basis
of mutual confidence and ended on the same note, have been throughout marked by
a spirit of co-operation. In the councils of government, we must guard against
the acquisition of unwarranted influence, whether sought or unsought, by the
military-industrial complex.
The potential for the disastrous rise of misplaced power
exists and will persist.
We must never let the weight of this combination endanger
our liberties or democratic processes.
We should take nothing for granted.
Only an alert and knowledgeable citizenry can compel the
proper meshing of the huge industrial and military machinery of defense with
our peaceful methods and goals, so that security and liberty may prosper
together.
II. The Balance in Government
Akin to, and largely responsible for the sweeping changes in
our industrial-military posture, has been the technological revolution during
recent decades.
In this revolution, research has become central; it also
becomes more formalized, complex, and costly.
A steadily increasing share is conducted for, by, or at the
direction of the Federal government.
Today, the solitary inventor, tinkering in his shop, has
been overshadowed by task forces of scientists in laboratories and testing
fields.
In the same fashion, the free university, historically the
fountainhead of free ideas and scientific discovery, has experienced a
revolution in the conduct of research.
Partly because of the huge costs involved, a government
contract becomes virtually a substitute for intellectual curiosity.
The prospect of domination of the nation’s scholars by
Federal employment, project allocations, and the power of money is ever present
— and is gravely to be regarded.
Yet, in holding scientific research and discovery in
respect, as we should, we must also be alert to the equal and opposite danger
that public policy could itself become the captive of a
scientific-technological elite.
It is the task of statesmanship to mold, to balance, and to
integrate these and other forces, new and old, within the principles of our
democratic system — ever aiming toward the supreme goals of our free society.
III. Peace and Responsibility
Another factor in maintaining balance involves the element
of time.
As we peer into society’s future, we — you and I, and our
government — must avoid the impulse to live only for today, plundering for our
own ease and convenience the precious resources of tomorrow.
We cannot mortgage the material assets of our grandchildren
without risking the loss also of their political and spiritual heritage.
We want democracy to survive for all generations to come,
not to become the insolvent phantom of tomorrow.
Down the long lane of the history yet to be written, America
knows that this world of ours, ever growing smaller, must avoid becoming a
community of dreadful fear and hate, and be, instead, a proud confederation of
mutual trust and respect.
Such a confederation must be one of equals. The weakest must
come to the conference table with the same confidence as do we, protected as we
are by our moral, economic, and military strength.
That table, though scarred by many past frustrations, cannot
be abandoned for the certain agony of the battlefield.
Disarmament, with mutual honor and confidence, is a
continuing imperative.
Together we must learn how to compose differences, not with
arms, but with intellect and decent purpose.
Because this need is so sharp and apparent, I confess that I
lay down my official responsibilities in this field with a definite sense of
disappointment.
As one who has witnessed the horror and the lingering
sadness of war — as one who knows that another war could utterly destroy this
civilization which has been so slowly and painfully built over thousands of
years — I wish I could say tonight that a lasting peace is in sight.
Happily, I can say that war has been avoided.
Steady progress toward our ultimate goal has been made.
But so much remains to be done.
As a private citizen, I shall never cease to do what little
I can to help the world advance along that road.
IV. Final Thoughts
So — in this my last good night to you as your President — I
thank you for the many opportunities you have given me for public service in
war and peace.
I trust that in that service you find some things worthy; as
for the rest of it, I know you will find ways to improve performance in the
future.
You and I — my fellow citizens — need to be strong in our
faith that all nations, under God, will reach the goal of peace with justice.
May we be ever unswerving in devotion to principle,
confident but humble with power, diligent in pursuit of the nation’s great
goals.
To all the peoples of the world, I once more give expression
to America’s prayerful and continuing aspiration:
We pray that peoples of all faiths, all races, all nations,
may have their great human needs satisfied — that those now denied opportunity
shall come to enjoy it to the full — that all who yearn for freedom may
experience its spiritual blessings.
Those who have freedom will understand, also, its heavy
responsibilities — that all who are insensitive to the needs of others will
learn charity — that the scourges of poverty, disease and ignorance will be
made to disappear from the earth, and that, in the goodness of time, all
peoples will come to live together in a peace guaranteed by the binding force
of mutual respect and love.
Now, on Friday noon, I am to become a private citizen. I am
proud to do so. I look forward to it.
Def: A semigroup is a pair \((G,*)\) where \(G\) is a set and * is a binary operation on \(G\) such that * is associative. NOTE: we do not require an identity element, we do not require inverses, we do not require commutative. We DO require that G is closed under *.
Theorem: Let (G,*) be a finite semigroup. There exists x in G such that \(x*x=x\).
Proof: Let \(x_1,x_2,\ldots,x_r\) be a sequence of elements of G (repetition is allowed---indeed required since we will need \( r >|G| \).) Let \(r\) be such that any |G|-coloring of \(K_r\) has a mono triangle.
Such an \( r\) exists by Ramsey's Theorem \((|G| \) colors, seek mono \(K_3\)).
Consider the following coloring: for \(i<j\), \(COL(i,j) = x_i* \cdots* x_{j-1} \).
By the choice of \(r\) there exists \(i<j<k\) such that
\(x_i* \cdots * x_{j-1} = x_j *\cdots *x_{k-1} = x_i* \cdots *x_{k-1} \). We call this \(x\).
Since \(x_i *\cdots * x_{k-1} = x_i *\cdots *x_{j-1} * x_j \cdots *x_{k-1}\) we have \(x*x=x\).
End of Proof
Great! Lets find some semigroups to apply this to.
1) If G has an identity element \(e\) then the Theorem is trivial, take \(x=e\). So we seek a semigroup without identity.
2) Can't we just take a group and remove its identity element? No- then it won't be closed under *.
3) Can't we just take the set of N that are \ge 1, under addition. No good- that's infinite. Note that the theorem does not hold there.
4) Can't we just google. I kept getting infinite examples or being told that I can ADD the identity to a semigroup to get an identity.
5) Can't we just ask AI. I used Claude which gave me a trivial 2-element example. I then asked for an example with more than 10 elements. It DID give me one:
\(G=\{1,\ldots,12\} \)
\(x*y=\min\{x,y,10\}\)
For this semigroup (and similar one) the theorem is trivial since \(\forall x\le 10, x*x=x\).
I asked Claude for an example with more than 10 elements that does not use MIN and it said
Due to capacity constraints NO CAN DO.
6) SO what I really want is the following:
Give me a FINITE semigroup G WITHOUT identity for which the statement
is there an \(x\) with \(x*x=x\)
is not obviously true- so that the Theorem above is interesting.
In the recent Signalgate scandal, several senior Trump administration appointees used the Signal app on their phones to discuss an attack on the Houthis. People discussed the risk of the phones being compromised or human factors, such as adding a prominent journalist to the group chat by mistake. But mostly no one had concerns about the cryptography itself on this widely-available app.
It wasn't always this way--cryptography used to be a cat and mouse game, most notably the breaking of the German Enigma machine dramatized in the movie The Imitation Game. Then Diffie and Hellman in their classic 1976 paper stated
Theoretical developments in information theory and computer science show promise of providing provably secure cryptosystems, changing this ancient art into a science.
And in recent memory we've had several successful cybersecurity attacks, but never because the encryption was broken.
We've made incredible progress in solving problems once thought unsolvable, including language translation, chess, go, protein folding and traveling salesman. We have great SAT and Mixed-Integer Programming algorithms. We've blown through the Turing test. None of these algorithms work all of the time but no longer do hard problems seem so hard. Yet cryptography remains secure. Why? How did we get to this Optiland, where only the problems we want to be hard are hard? Quantum computers, if we have them can attack some cryptographic protocols, but we're a very long way from having those capabilities.
My latest theory involves compression. Machine learning works by finding a representation of a function in a neural net or other form that gives an imperfect compressed version of that function, removing the random components to reveal the structure inside. You get a smaller representation that, through Occam's Razor, is a hopefully mostly accurate version of that data. For example, we learn the representation of a Go player by training a neural net by having the computer play itself over and over again.
Cryptography is designed to look completely random. No compression. If you remove the randomness you have nothing left. And thus modern algorithms have no hook to attack it.
This is just the beginnings of a theory. I don't have a good formalization and certainly not even the right questions to ask of it.
So to me it's still a big mystery and one that deserves more thought, if we really want to understand computational complexity in the world we actually live in.
