Math Digests July 2021

Emmy Noether faced sexism and Nazism – 100 years later her contributions to ring theory still influence modern math

The Conversation, July 15, 2021

Emmy Noether is one of the most famous female mathematicians of the twentieth century. In a celebration of the 100th anniversary of some of her greatest work, Tamar Lichter Blanks recounts the difficulties Noether faced as a woman and Jew in a world that discriminated against both groups. Lichter Blanks also describes Noether’s most famous result concerning rings, a set in which one can multiply, add, and subtract.

Classroom Activities: diversity and inclusion, ring theory

  • Read Lichter Blanks’ article in class. Discuss the discrimination Noether faced and how the modern world is similar or different.
  • Introduce rings (see Wolfram MathWorld or Academic Kids for definitions and examples) and assign the following exercises:
    • Prove that the set of integers ($\{ \dots, -2, -1, 0, 1, 2, \dots\}$) forms a ring.
    • Does the set of even integers form a ring? Which axioms does it satisfy?
    • Does the set of odd integers form a ring? Which axioms does it satisfy?

—Leila Sloman

Mathematicians Prove Symmetry of Phase Transitions

Quanta Magazine, July 8, 2021

Scorch a magnetic bar of iron with enough heat, and the solid mass will lose its magnetism. We can observe this type of “phase transition” without a microscope. But zooming into the neighborhood of atoms, we notice why the transition occurs: swaths of atoms that previously aligned magnetically are now in disarray. So-called critical points—where the look and feel of a system teeters between two different states—appear everywhere in nature, from boiling water to crystallizing minerals. “Mathematicians try to bottle this magic in simplified models,” writes Allison Whitten for Quanta Magazine. Whitten’s article describes new mathematics research showing that phase transitions contain rotational invariance—a circular symmetry where the model has the same physical properties regardless of how it’s rotated. A “mathematical elegance” sets in at the critical point, Whitten writes. The finding inches researchers closer to proving that phase transitions have conformal invariance, strong “overall” symmetry.

Classroom Activities: symmetry, fractals, Penrose tiles

  • Discuss the different kinds of symmetry which make up conformal invariance: rotational, translational, and scale symmetry. Which symmetry/symmetries is/are present in a tiling of identical squares? What about a series of concentric circles?
  • Explore fractals, which have scale symmetry, and talk about some examples of fractal-like behavior in nature. (Also see the Wolfram Demonstrations Project on Fractals.)
  • Introduce the Penrose tiles (see this Veritasium video), which do not have translational symmetry and yet do have rare fivefold rotational symmetry like a pentagon. As you zoom out further and further, Penrose tiles have a fractal-like behavior.

—Max Levy

Why Are Gamers So Much Better Than Scientists at Catching Fraud?

The Atlantic, July 2, 2021

How can you tell when good luck is too good? Last year, a popular gamer named “Dream” performed so well in the game Minecraft that his audience took notice—something seemed fishy. “He was the equivalent of a roulette player who gets their color 50 times in a row,” writes Stuart Ritchie in The Atlantic. “You don’t just marvel at the good fortune; you check underneath the table.” Dream’s drama concluded when a diffuse group of gamers published a robust mathematical analysis of his dubious performance. They compared his luck receiving useful items in the game with the probability of getting those items by chance. Then, they used statistical methods to deduce that Dream’s luck was “unfathomably” unlikely. The article closes with a shift to academic science. Mathematical analysis like this one in gaming, Ritchie writes, can maintain the integrity of science by weeding out cheating and fraud there too.

Classroom Activities: probability, p-values, Benford’s Law

  • Tell half of your students to flip a coin 100 times and tally the results. Tell the other half to just make up a sequence of 100 coin flips. Afterward, ask everyone to count their longest streak of heads or tails. You’re likely to find that the real sequences of flips had longer streaks than the fake ones. Making up truly random data is hard! (See this Texas Instruments activity for a more in-depth exploration of such streaks.)
  • Work through this classroom activity on statistical testing from Carleton College: Are Female Mallards Attracted To The Color Green? This worksheet walks students through forming a statistical hypothesis, gathering and analyzing evidence, and interpreting their analysis as a conclusion via a p-value.
  • Discuss Benford’s Law, otherwise known as the law of anomalous numbers, which helps detect fraud in finance. Watch this video from Numberphile about using math to detect fraud with Benford’s Law.

—Max Levy

Fields Medals are Concentrated in Mathematical Families

Scientific American, July 2021

The Fields Medal, the highest honor in mathematics, was initially conceived of in part as an equalizer. In a recent paper in Humanities and Social Sciences Communications, Feng Fu (Dartmouth College) and Ho-Chun Herbert Chang (University of Southern California) write “The award was intentionally given to individuals that would otherwise not receive any recognition, rather than the best young mathematician.” But by analyzing data from the Mathematics Genealogy Project, Fu and Chang find that this plan has fallen by the wayside—44 out of 60 Fields Medalists are academic descendants of either Jean le Rond d’Alembert or Gottfried Liebniz. Moreover, mathematicians of Arabic and African descent are underrepresented among medalists and within the elite community. The in-depth graphic by Clara Moskowitz and Shirley Wu in Scientific American shows the connections between Fields Medalists revealed by Fu and Chang.

Classroom Activities: diversity and inclusion, data visualization

  • Fu and Chang use mathematicians’ names as a proxy for their “lingo-ethnic” identities. Discuss whether this is a good measure or not and what potential errors it might bring up.
  • Have students interpret the graphs shown in Fu and Chang’s paper and reflect on the over/underrepresentation of certain groups in mathematics. Ask questions like:
    • What effect might this have on mathematics research and the community?
    • Do they notice connections to their own lives?
    • Should the Fields Medal Committee return to its original practice of awarding individuals who would otherwise not be recognized, and why or why not?

If (or when!) students have different answers to these questions, have them read and respond to one another’s ideas.

  • Discuss the way the data is visualized in the Scientific American graphic. Is it effective or not? What specific elements of the graphic contribute to that effect?
  • Have students devise alternative ways to visualize the data on Fields Medalists, or have them devise a visualization of another data set. What aspects of the data do they emphasize, and why? (Teach Data Science offers further resources on teaching data visualization.)

—Leila Sloman

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