What is a Manifold? (4/6)

After our luxurious treatment of 1-d manifolds, we turn to 2-d manifolds.

My story of surfaces starts in a beautifully weird morning when I got up to realize that life in the usual Euclidean plane had changed dramatically. Vectors had shortened, areas had shrunk, and infinity was just a few feet away!

Riddle of the Month (November)

Welcome back to this month’s mathematical riddle (and can you believe it’s almost December)! Today we have a neat logic puzzle with an amusing twist on the traditional knight-or-knave problems that are popular in the literature.

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Mathematical Democracy: Mission Impossible? Maybe not…

In 1950, a 29-year-old PhD candidate at Columbia published a stunning theorem that later won him a Nobel Prize: “There is no such thing as a fair voting system.”  Or so the legend goes.  Let’s dive into this claim and see to what extent Arrow’s Impossibility Theorem does and does not quash our hopes for a fair representative democracy.  For a background of voting systems, see Rina Friedberg’s post last month or Stephanie Blanda’s 2014 post on how Olympic host cities are chosen.

What is a Manifold? (3/6)

Intrinsic descriptions

One immediate benefit of considering coordinate-free descriptions of geometric objects is that we may talk about “curves” that are not a priori embedded in $\mathbb{R}^3$. In other words, we don’t have to start with a subset of $\mathbb{R}^3$ to be able to study 1-dimensional objects. There is already quite a nontrivial question we can ask: what curves can be embedded in a plane? The answer will be provided as a condition on $k$ and $\tau$, and this description has the advantage of having nothing to do with $\mathbb{R}^3$ or any other non-intrinsic data. Later we will talk about surfaces (2-dimensional manifolds) that do not live in 3-space, but rather in 4-space. Having an intrinsic way of seeing objects is liberating and opens up new possibilities. Continue reading “What is a Manifold? (3/6)” »

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Conquering Students’ Mid-Semester Complacency

About a week after their first test, during the first round of midterms of the semester for undergraduate students, mid-semester complacency came swooping into my classroom. My once invigorated students suddenly started going through the motions of class rather than actively engaging with me and the material during class. Worried that I was going to lose them to this complacency for the rest of the semester, I frantically wracked my brain for ways in which I could grab their attention and help them refocus on the material at hand.

After a particularly grueling example involving a word problem and a piecewise function, I asked my students, “What questions do you have regarding the process that we just went through?” and gave them a couple of minutes to review their notes and formulate questions. Gripped by the mid-semester complacency, my students lethargically looked through their notes and shook their heads indicating to me that there were no questions. Instead of taking their word and moving on, I was inspired to not move on and instead ask a question that would lead to discussion and had the potential to pull them out of the complacency that they were experiencing. Continue reading “Conquering Students’ Mid-Semester Complacency” »

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