Kevin Berkopes, PhD

CEO & Co-Founder of @MathTrackU & Intangled Learning. Aspiring craftsman, supporter of mistakes, believer in apprenticeship through intanglement.

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: The Words Came Last https://t.co/5WmoOvH074 via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

@dlopezIND @IBJnews We can’t compete on venture investment with the coasts & we shouldn’t try. We can win through partnerships between startups, growth stage, and mature companies. Indiana can become the partnership state. That is our future if we want it.

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: Growth is Growth Volume 1
A little about how to collaborate and learn through mutual respect.
@MathTrack_MTI
https://t.co/4XWG59DcAr via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: You Never Get Out https://t.co/t7OzcdP71Y via @LinkedIn

[Steven Strogatz @stevenstrogatz]

Why should we care whether humans still understand mathematics if AI can solve the problems for us? Fair question. But we can say this at least: Historically, understanding has been a route to predictive power — with enormous benefits for our lives.

https://t.co/44cUXi8sce

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: The Books Became My Mentors https://t.co/19m28RJBsX via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: Nobody Knew That https://t.co/lHjRrQfGzV via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: Math Don't Like Me https://t.co/9dTZuXgH7h via @LinkedIn @MathTrack_MTI

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: The Utopia Was on the Test https://t.co/tfeuzfDcxj via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

The Word for What Is Happening to Teachers Is Not Burnout https://t.co/yNQPAE85sj

[Jared Duker Lichtman @jdlichtman]

This is quite a remarkable result:

Posed in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry, but was just disproved by Alpoge, Matthew, and Claude Fable 5.

The Jacobian conjecture roughly says that a multivariable polynomial F has an inverse function (made out of polynomials) provided the Jacobian is non-singular (i.e. matrix of partial derivatives has non-zero determinant). This condition is neccessary by the Inverse Function Theorem from multivariable calculus. The hard question is whether it is also sufficient.

Evidently, Fable found that

F(x,y,z) = ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z)

has det(J_F) = -2 non-zero. However F is not invertible, since F sends three different pts (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to the same image (-1/4, 0, 0).

Beyond the disproof itself, it would be value to know if a suitably refined conjecture is recoverable. Per @Acer, GPT5.6 has proposed:
"A constant-Jacobian polynomial local biholomorphism with no loss of sheets at infinity—e.g. a proper Keller map—is an automorphism."
I would be interested to know if any algebraists (e.g. @levent @littmath) have a reaction to this...

A further twist to the story:
Not only was the Jacobian conjecture one of the central open problems in algebraic geometry, it was (a special case of) Yitang Zhang's PhD problem!

The catch was that Zhang's advisor had him solve it, assuming a lemma of his advisor. But that lemma turned out to be false! As a result, Zhang's thesis crumbled and he then struggled to get recommendation letters and a permanent academic position. Despite all this, Zhang went on to prove bounded gaps between primes!

This is one of the most inspiring stories in modern mathematics, and was a motivation for me to work in the same area for my doctorate.

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in the series, The Uncomfortable Truth about #math teaching: The Teacher They Loved https://t.co/qZAMeywHSR via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

The Only Apprenticeship Everyone Serves https://t.co/hm3GO3zgZR

[Kevin Berkopes, PhD @berkopes]

The Asset That Isn’t on Your Books https://t.co/xTPFMHvDu3

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: Why We Became Intangled Learning https://t.co/5FGeXxEiOp via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: The University Isn't the Source of Expertise. That's the Whole Problem. https://t.co/6wuUoRYFGJ via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

Check out the latest article in my newsletter: The University Can't Give You What It Doesn't Have https://t.co/b7LcpJbQuu via @LinkedIn

[Kevin Berkopes, PhD @berkopes]

We are an amalgam of the people around us, not just an addition. Our environments shape us, but individual experiences and even parents evolving over time make us unique. Siblings from the same house can be vastly different. #PersonalGrowth #SelfReflection

[Kevin Berkopes, PhD @berkopes]

The Teacher's Twinge: Why Faculty Confidence Is the Hidden Lever of Math Education https://t.co/OSu6p1F91v

[Kevin Berkopes, PhD @berkopes]

What the Math Wars Are Hiding https://t.co/G4JkWenuAn