My social media memory from 11 years ago: a cautionary tale about why we don't use mixed fractions.

In topology, we were talking about the Euler characteristic. There’s a theorem that if a sphere is tiled by hexagons and pentagons with three meeting at every corner, then there must be exactly 12 pentagons (like a soccer ball). I brought in this golf ball as an example. Do you see a pentagon?

Ha ha! I love it!

I took a topology exam once. It was both open-book and closed-book.

This is so cool! Given only the "exponential minus log" function, elm(x,y)=exp(x)-ln(y), and the constant 1, you can perform +, -, x, ÷, exp, ln, trig, powers, roots, etc. You can also obtain e and π! See arxiv.org/pdf/2603.21852 and arxiv.org/src/2603.218...

Fantastic page on Wikipedia: Signs of AI Writing. en.wikipedia.org/wiki/Wikiped...
I shared this link with ChatGPT and Claude and asked them to "Write a paragraph about the mathematics program at Dickinson College that incorporates as many of these 'bad' AI traits as possible." Here are the results.

In my office hours this morning, I was talking about triangulation of surfaces. The sphere was hard to draw on the whiteboard, but I had an orange and a Sharpie handy

The question is whether there can be a local max/min that is the only critical point, that is not a global max/min.

max and min that are the only critical points for the functions, but are clearly not absolute extrema.

You can see and play with the actual functions here: www.geogebra.org/m/zat7hak7

See also, www.jstor.org/stable/2689910

A 3D print showing a local maximum that is not a global maximum

A 3D print showing a local minimum that is not a global minimum

New 3D prints: There's a theorem in single variable calculus called "The only critical point in town":

If a continuous function f: ℝ→ℝ has only one critical point and it is a local max/min, then it is a global max/min.

This isn't true for functions of two variables. These prints illustrate a local

I was showing my topology students this claymation video I made 18 years ago (wow!). Can you get from one configuration to the other without breaking a loop? Yes, you can! www.youtube.com/watch?v=S5fP...

I would have felt so much better.

aberration.

Fast-forward 20-years. I reconnected with my grad school friend. We were talking about the old days. He said, "Remember that first algebra exam that we all failed?" I had no idea!! At the time, I was embarrassed, frustrated, and doubting myself. Had I known that we all struggled,

a straight-A math student in college. Then I got to grad school and failed my first exam. I'd already found grad school challenging, and this exam grade really made me doubt my ability and my choice to go on to grad school. Fortunately, I stuck with it, and that grade was an

created perfect exams and the grades always followed a predictable distribution, that would be great. In reality, sometimes exams are too hard, sometimes they are too easy. Knowing how the class did can help a student understand how they are doing with the material.

Here's a personal story. I was

As I said to @tienchihmath.bsky.social, I've gone back and forth about this. I often feel the way you do. However, I think that for some students, getting a grade in a vacuum can be really stressful and can make it hard to calibrate how well or poorly they are doing in the class. If every professor

I don’t spend a lot of time on it either way. But I focus on problems that trip up multiple students

I’ve gone back and forth in my thinking on this very issue

For me: I usually do. So, I was surprised when a colleague mentioned that their class did poorly on an exam and said, "I may have to go through the exam solutions in class."

Professors: When you return an exam, do you typically talk about the results at the question level? Like, project the exam solutions and talk about them? Or talk about specific problems that tripped up a lot of students?

Or do you just return the exam, state the mean/median, etc., and move on?

Fun hands-on maths project to try at home in celebration of the beginning of spring term. Even if you've played with #MobiusStrip before, you can always find a new "What if?" Q to explore.

johnkerl.org/doc/ortho/or...

My son, the computer science major: "My question is, how do Grandma and Grandpa send such small, low-resolution photos [from their iPhone/iPad]? I wouldn't know how to do that if I tried!" 😂😂😂

The Magnificent Möbius Band YouTube video by David Richeson

In topology today, I had the students cut Möbius bands in various ways (an activity fun for kindergarteners to college seniors). I was teaching topology when COVID hit in spring 2020. I made this Möbius band-cutting video for students who were at home.

Here are links to the Franks and Bangert articles:
link.springer.com/article/10.1...
www.worldscientific.com/doi/10.1142/...
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result was to show that a certain map on the annulus had infinitely many periodic points.

This isn't my area of expertise, but Franks was my PhD advisor, and he proved this result not long before I entered grad school. I thought it was really cool.
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all other spheres (Bangert). Together, their work proved the remarkable theorem:

Every Riemannian manifold that is topologically a sphere has infinitely many closed geodesics!

I was telling my topology class about this theorem because we were discussing the annulus, and the key to Franks's
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cross your path multiple times, but you end up back where you started from, heading in the same direction. What can we say about those?

In the early 1990s, Franks and Bangert each published an article covering a special case: the sphere has positive Gaussian curvature (Franks) and
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In 1929, Lyusternik and Schnirelmann proved the conjecture. Although the idea of the proof was correct, it contained a flaw that was fixed in the 1980s by Grayson.

Their result was about simple closed geodesics. What about closed geodesics in general? You head off in some direction, you may
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In 1905, Poincaré conjectured that any such sphere must have at least three simple closed geodesics. That is, they are closed curves without self-intersections. For example, for an ellipsoid x²/a²+y²/b²+z²/c²=1, think of the ellipses where the ellipsoid intersects the coordinate planes.
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