Centaur Math

In the past, new PhD students would ask how they could succeed when they had to compete with the likes of say, Richard Karp or Avi Wigderson. I would say Karp and Wigderson have limited bandwidth and you can work on problems they don't work on, or think deeper about a problem than Karp or Wigderson has time to.

Now we get the same question but with names like Claude and ChatGPT and it's hard to make the same bandwidth argument. What do we tell them as we get closer to Math AGI?

What even is Math AGI? It's not that every math problem gets solved. I don't expect P vs NP to be solved anytime soon. It would require a completely new approach, and AI doesn't (yet) think outside the box, though it has a very large box.

Math AGI means that with rare exceptions, if AI can't solve a math problem then no human could either. If you need a proof, you'd have to pay for more cycles, or wait for the next new and improved model. Like the Turing test, we'll only truly realize we've reached Math AGI once we've gone well past it.

We haven't reached Math AGI yet and we may never fully get there. We have entered the world of Centaur Math. Mathematicians can still prove theorems AI can't, AI can prove some theorems mathematicians haven't yet proven, but the real strength comes with mathematicians and AI working together. Working with AI today is like having a pretty good PhD student, who has a huge broad base knowledge of mathematics, is a whiz at coding, but still needs direction, encouragement and verification.

Chess had a short centaur moment when humans and AI working together could beat the best human players and AI programs. Now, any human would play worse not following what AI says. Nevertheless, we still enjoy watching two sub-AI humans play chess against each other. I doubt the same would hold for sub-AI mathematicians.

So what do we tell the students? If you love math, do math. Embrace AI, use it to go further, not as a crutch. Challenge yourself and remain agile so you can find success whatever the future might hand us. And remember, math is not ultimately about the theorems we prove but how we understand the principles behind them, and that's a human endeavor not a machine one.

IIT is the canary in the coalmine (Do our younger readers know what that means? Do we have younger readers?)

Lance has posted about his, and around 160 others, being laid off from IIT here.(IIT stands for Illinois Institute of Technology which is where Lance was employed.)

Hence I looked into what is happening at IIT to see if there is a lesson for us all.

I) I wondered why IIT had declining enrollment. I wondered which of the two reasons below  was the issue (and of course there are other reasons). 

--The enrollment cliff.  (See here)

--International students have declined in number. Why? (1) Having getting student visas, (2)  They think they are not welcome here (3) Schools in their home country getting better. Note that I am just guessing.

Rather than speculate I asked ChatGPT for the data on both enrollment and on international enrollment from 2015 to 2025. Here is the data: here.

a) 2015 had an enrollment of 7792. It went down in 2020 (COVID?) but then came up again and it was 8838 in 2024.  In all the years for which there is data international students were about half of the students.

b) 2024: 8838 students, of which 4596-International (52%), 4242-Americans

   2025: 7502 students, of which 2141-International (42%), 5361-Americans

So it looks like the enrollment cliff was not a problem since more Americans came, but the decline in international students is a problem.

I think that international students pay more, so their decline creates more of a financial pinch.

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II) I wondered if other schools had massive layoffs so  I asked Google AI what other schools had laid off more than 30 professors in the last two years. It had an issue with that since schools combine faculty and staff.  Even so, the main fact is that (a) it is happening at other schools, and (b) 160 is more than usual. 

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III) Challenges facing Universities

a) The enrollment cliff. This will be a real problem for small schools.  For big schools it may be an opportunity to have smaller classes.  This issue cannot be stopped.

b) Less international students means less money. This may change since this or a later administration may change the rules. However, changing the rules might not help much since other countries have fine schools and international students may feel they are not welcome here.

c) Grants are drying up. Again, this may change.

d) AI and cheating. This may make us rethink the entire point of education.

e) Does college exist to create an educated public who can make decisions and vote intelligently, or are colleges vocational? It's getting harder to do both.

f) Fads: The hot topics in CS now are Quantum, ML, and Quantum ML. What will they be in 10 years?How do we adjust? Do other fields have this problem? ML (or more broadly AI) is hot in that students want to take it because it both sounds interesting and sounds employable. Quantum is hot for grant money and some students think its going to be cool. (In this context `Hot' and `Cool' are not opposites.) 

g) Tuition keeps going up. The business model may be broken.

h) A while back online education seemed like it might be an alternative for some students (MOOCS were hot). That hasn't happened yet but it might.

i) This is a far bigger subject than the points above. Maybe Lance can write a book on the topic now that he has some free time.

Unexpected Unemployment

Enjoying Idaho while ignoring Illinois

Today is the first day of my life that I am unemployed. And not by choice.

As I mentioned on LinkedIn last week, me and about 160 of my colleagues, staff and faculty, untenured and tenured, lost our positions at Illinois Tech after they declared "financial exigency" which allows them to eliminate tenured positions. I'll use this post to tell my story, but keep in mind there are 160 other ones.

The president announced that he would be asking the board to declare financial exigency in mid-July so we knew layoffs were coming but not when. On July 25th, we left for a planned 12-day vacation to Idaho. Why Idaho? It's my fiftieth state, so my wife and I decided to make a vacation of it. We saw Boise, canyons, craters, mountains, lakes. We were in Twin Falls three days before a mass shooting but that's a different story.

Usually I avoid reading work emails on vacation but decided I probably should this time. And on day four of vacation, I got an email invite to a meeting with the Vice-Provost of Faculty Affairs and an HR representative titled "Organizational Update" and I knew my fate was sealed. By the next day I was tired thinking about it and just decided to enjoy the rest of the vacation and deal with everything when I got back last Thursday. It might have been better if I simply didn't read email like usual.

It really hit me as I started to pack up my office Monday, for the first time with no office on the other side. Monday was also the first day of orientation week and a group of new students walked by as I was packing boxes into my car, though I don't think they noticed.

Illinois Tech got hit hard by a large drop in foreign graduate enrollment due to changing visa requirements, general anti-US sentiment, more opportunities in other countries and a weaker job market for graduating Masters students partly due to artificial intelligence. Universities face challenges beyond international students including Baumol's disease, administrative bloat to meet expanding regulations, the demographic cliff, reduced grant funding, and less support of universities by the public and both political parties, and AI changing how and why we teach. Illinois Tech is one of the first tech research schools to eliminate tenured roles, but it won't be the last.

I'll be okay but many of the other faculty could really use another position, in some cases so they can stay in the US. If you have opportunities for faculty in any discipline, let me know and I'll pass it along.

Math Concepts With Funny Names

(Some of this came from a Reddit post  I read, and some of  the comments on it.) 

Here are theorems with names that I think are funny or unusual. The names are also pointers to the Wikipedia entry on them or some other source.

The Chicken McNugget Theorem

The Ham Sandwich Theorem

The Hairy Ball Theorem

Freshman Dream

The law of the unconscious statistician

The Homicidal Chauffeur Problem

The Pigeonhole Principle (We are so used to this we no longer find it funny, but it is.) 

Eventown and Oddtown 

I am sure there are more and I invite you to comment BUT there are some issues

1) What is funny?

2) I want math concepts that are known. As a counterexample, if Lance proved a theorem and called it the Funkytown Theorem just to get on this list then I'd be surprised. It also would not count. 

3) The name has to be some relevance to the concept. 

And I now list two titles of papers that I find funny. Note that what I write IS the title, even though the first one is an odd title and the second one really does not look like a title.

A minus sign that used to annoy me but now I know why it is there (two constructions of the Jones polynomial)

Some title containing the words "homotopy" and "symplectic", e.g., this one

A problematic category on Jeopardy Raises a Good Question

I was watching a Jeopardy from 2004 (The Game Show Channel is rerunning Ken Jennings streak) and the following question raises a good question.

The contestants where Ken, Jerry, Jennifer.

In Double Jeopardy there was a category  Biblical Name The Same.

The clue was three last names (e.g., Driver, Sandler, West) and the correct response is a biblical name that is the first name of people with that last name (Adam: Adam Driver, Adam Sandler, Adam West).

What happened with Biblical Name The Same for 1200 shows an issue with the question.

Clue: Stewart, Graham, Grimes.

Jerry said  James. This was ruled incorrect.

Ken said Martha. This is correct.

Why is Martha Correct:

Martha Stewart is a well known TV personality focusing on home and hospitality.

Martha Graham was an American modern dancer, teacher, and choreographer (she died in 1991).

Martha Grimes is an American writer of detective fiction.

Frankly, the only one I had heard of was Martha Stewart.

But later they decided Jerry was right and gave him the points (Ken kept his points).  Here was Jerry's argument:

James Stewart was a well known  actor (died in 1997).

James Graham- the show said he was a British General in the 17th Century. I (Bill, not Jerry) looked up James Graham on Wikipedia- there are a lot of them. Four were under the category Military.  Of those, two were British. Neither was a General. One lived 1649-1730 so it may be him.

James Grimes-the show said he was a Senator involved with the impeachment of Andrew Johnson.  I (Bill) looked it up- Grimes was a Republican and it was his party that was trying to impeach, but he (and six other Republicans) broke rank and voted for acquittal. He and the others were bothered that the process had been manipulated. There were rumors they were bribed with patronage jobs or cash though I could not tell if this was true.

The category is problematic.

It is implicit that the three people have to be famous.

This raises two questions and a challenge

Two Questions: What is fame?  How do you measure it? 

Challenges: Let x,y \(\ge\) 2. 

 Find x  last names L1,...Lx  such that there are y first names F1,...,Fy

with F1-L1, F1-L2,...,F1-Lx famous AND

        F2-L1, F2-L2,..., F2-Lx famous AND

DOT DOT DOT

        Fy-L1, Fy-L2,...,Fy-Lx famous.

For the example above x=3 and y=2. You either try the easier case x=2, y=2, or a harder case like x=3, y=3.