# Daily Archives: January 15, 2020

## “Have FUN!” – Joe Gallian

Didn’t do well in high school. Didn’t think he would go to college. After getting his PhD, he applied to 145 jobs and got one, just one… University of Minnesota Duluth. Then, he became the father of undergraduate research. I am talking about Joe Gallian.

During JMM 2020, Joe talked about his REU program. Then, Shah Roshan Zamir (a former student of Joe), premiered a documentary about Joe and the program, which has ran continuously since 1987. It was fun, informative, and it conveyed how much fun students have in the program. It will be shown again at 2020 MAA MathFest and will eventually be available in YouTube.

In the documentary, after singing a Beatles son, Joe says “When you go to a new place, make yourself valuable.” For him, it was believing in and conducting undergraduate research. In his program, arguably the most successful math REU, he has some guidelines. The first one: participants should have FUN. Joe organizes weekly field trips, students play Math Charades and other games, and throughout the year, the program has developed all sorts of funny traditions: Duluth mile, erase the whiteboard, halftime show, crossing the river, and many others. You have to ask him about them…

Crossing the river. Ask Joe…

His alumni, which include Manjul Bhargava and Melanie Wood, paint a picture of a humble, fun, caring, and supportive advisor. In the Documentary, he takes things lightly, has fun singing Beatles songs, and shares great pieces of advice. “Find something to do, something to love, and something to look forward to.” And “Have FUN!”

## MAA Invited Address by Scott Adamson: Mazes, riddles, zombies and unicorns

Scott Adamson, of Chandler-Gilbert Community College, kicked off his talk by asking the audience this: “How can we create a more joyful mathematical learning experience where the beauty of mathematics can be shared and be engaged in by our students?”

I’m reminded of the need to answer this question every time I introduce myself as a graduate student in math and am met with the classic response—“I was so bad at math in school.” But it can be difficult to try out new methods of teaching when instructors are pressed for time and resources. Adamson articulated the specific way our education system is failing students—by asking them to memorize formulas without any understanding of the underlying mathematics—through examples he’s seen in his work. Students, for instance, learn how to calculate the average of a set of numbers, but often they don’t understand what it means conceptually. Teachers, he explained, frequently try to make math “fun” by introducing gimmicks—following mathematical steps to reach the end of a maze or attacking zombies with a line of best fit—but in the process, send the message that math is too painful to endure without turning it into a game. At the college level, there is rarely an attempt to make math fun at all.

The result is students who implement formulas mechanically and cannot solve problems that differ from what they’ve seen in class. Adamson suggested an inexpensive way to getting students engaged in problem-solving: vertical non-permanent surfaces, or VNPSs (that is, whiteboards, or chalkboards, or some approximation). Research by Peter Liljedahl has shown that students who work on permanent surfaces—those where you can’t erase your work—participate less, persist less in their problem-solving, and take significantly longer to start writing—two minutes as opposed to 20 seconds. Any non-permanent surface is good, but vertical surfaces are slightly better than horizontal surfaces. They let students work together more easily, and also look around the classroom at one another’s work for ideas.

Although students may struggle at first with a teaching method that requires them to solve new kinds of problems, Adamson suggested students are empowered when they are able to “discover” math for themselves.

## A conversation with Samuel Hansen

Samuel Hansen discusses their math podcast, Relatively Prime, and JMM 2020.

## Finding solutions to x^3+y^3+z^3=n?

“Tony gave the best lecture I ever saw at Rice.” When Brendan Hassett introduces a speaker with such words, you know you are in for a treat. Tony Várilly-Alvarado gave an inviting AMS Invited Address Wednesday morning.

“Can you find three integers (you can use negative numbers) $x$,$y$, and $z$ such that $x^3+y^3+z^3=35$?” he playfully asked. What about $x^3+y^3+z^3=34$?

While $(3,2,0)$ and $(3,2,-1)$ provide solutions for these questions, if you consider $x^3+y^3+z^3=33$ the first solution appeared in 2019 and involves integers with 16 digits. And if you consider $x^3+y^3+z^3=32$, there are no solutions! What is going on? To explain this, Tony took us through the wonderful world of projective space and elliptic curves.

Let’s first look at an easier question by “only” considering tuples $(x,y,z)$ such that $x^2+y^2=z^2$, i.e., Pythagorean triples. Since once you have a solution, any multiple of it is also a solution, then we only need to consider $x,y,$ and $z$ to be co-primes. In other words, we can look at it with a projective lense.

To find solutions, rescale $z$ to be 1 and consider rational solutions to $(x/z)^2+(y/z)^2-1=0$.  Setting $X=x/z$ and $Y=y/z$, we reduce the problem to finding rational solutions to $X^2+Y^2=1$ in projective space aka where you are allowed to rescale $X$ and $Y$. Using a bit of projective geometry, it turns out that the map

$$(X:Y)\to (2XY:X^2-Y^2:X^2+Y^2)$$

describe all such solutions, hence all Pythagorean triples are of the form $(2XY,X^2-Y^2,X^2+Y^2)$ or multiples of it.

By converting algebraic equations to geometric objects in projective geometry, Tony explained that the equation $x^3+y^3=z^3$ has actually no non-zero solutions (this is the $n=3$ version of Fermat’s Last Theorem) and that, going back to the initial question, $x^3+y^3+z^3=n$ seems to only have solutions when $n\neq \pm4 \mod 9$. To this date, this remains an open problem.

To be fair to Tony, he also talked about several other interesting objects: perfect cuboids, magic squares and how using similar methods, one can provide evidence for the veracity of conjectures involving these objects.

I left the room impressed by the talk. It was engaging, interesting, and quite understandable despite it touching technical mathematical tools. Well done Tony!