American Mathematical Society

11. Wolfgang Haken Born in Berlin in 1928, Wolfgang Haken showed an early taste for mathematics. At the age of 15 he was drafted into a World War II anti-aircraft battery while continuing with his schoolwork which he concluded in early 1946. At this time, most German universities were not accepting students below the age of 23, but the University of Kiel was an exception and Haken began his studies there when just 17, the youngest student in the university.

This month’s topics: Quipus and quantum computing Geometry, a human language? Frank Wilczek’s column in the the Wall Street Journal (April 14, 2022) had the title “A Quantum Leap, With Strings Attached; The Inca system of quipu—tying a series of knots to record information—is providing a surprising model to modern physics and quantum computing.” Left, a quipu from the American Museum of Natural History, image taken from L. Leland Locke’s The Ancient Quipu (AMNH, 1923).

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1. Introduction The concept of quadratic forms can be traced back to ancient civilizations such as the Babylonians and Greeks. The Greeks, particularly Euclid in his famous work Elements, presented geometric methods for solving quadratic equations. The Greeks’ focus on geometry and their methods continued to influence mathematicians for centuries. Rules for quadratic equations were also discussed in The Nine Chapters on the Mathematical Art, composed in China by 200 BCE.

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Conversation Jeremy Avigad Johan Commelin Heather Macbeth Adam Topaz Introduction It has been a long day and you are making your way through a paper related to your work. You suddenly come across the following remark: “…since $x$ and $y$ are eigenvectors of $f$ with distinct eigenvalues, they are linearly independent.” Wait—how does the proof go? You should really know this. Here $x$ and $y$ are nonzero elements of a vector space $V$ and $f : V \to V$ is a linear map.

Computation The use of treewidth in graph algorithms can be illustrated by graph coloring, a problem that is $\mathsf{NP}$-complete for arbitrary graphs but polynomial-time solvable for graphs of bounded treewidth.

Using AI to formalize mathematics may help close gaps in AI's ineffective reasoning capabilities. Math formalization is a recent concept. Around 1666, German polymath Gottfried Wilhelm Leibniz imagined a characteristica universalis, a precise language of symbols to express math and science concepts consistently and unambiguously. With works like Principia Mathematica (1910), mathematicians standardized the logic and symbols of math.

When Po-Shen Loh began coaching the United States Math Olympiad team in 2013, artificial intelligence was hardly on his mind. The professor at Carnegie Mellon University has always focused on human education, but recent developments in machine learning have inspired him to wonder about the future of math-capable AI. In 2022 research startup OpenAI released their ChatGPT application, which could answer math and text prompts.