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The French mathematician Nicolas de Condorcet was a passionate believer in democracy, but he also discovered a frustrating paradox within it: A majority can sometimes vote itself into a logical loop. Mathematicians have now found that this “logical loop” maps onto a strange topological shape: the Klein bottle, a surface with no distinct inside or outside. On this episode of The Quanta Podcast, host Samir Patel speaks with writer Alex Stone about this new connection between this centuries-old voting paradox and the Klein bottle — and how it illustrates just how difficult it is to translate a group of individual preferences into a single choice. This topic was covered in a recent story for Quanta Magazine. Each week on The Quanta Podcast, Quanta Magazine editor-in-chief Samir Patel and Senior Editor Hannah Waters speak with the people behind the award-winning publication to navigate through some of the most important and mind-expanding questions in science and math.

In the UK, many waders nest on farmed grassland, with species such as Lapwing (alongside) and Oystercatcher enjoying the security of an all-round view, often on a flat field. By the time that chicks are a few days old, the grass will have grown and it’s probably time to either cut the field for silage … Continue reading Giving waders time to nest

Leaving aside AI for research at the moment, as anyone with a pulse knows at this point, we are in a period of rapid change in terms of AI tools and specifically their role in higher education.   There is an arms race among universities about what tools should be made available to students.  There are equity of access issues here - wealthy students are likely to pay out of pocket for more expensive subscriptions, regardless of what universities provide.  Princeton announced that it's providing everyone with chatGPT subscriptions (the equivalent of the $20/mo GPT 6 Astra).  Cornell is doing something similar, including Claude.   This is likely just the beginning, as the major AI firms want to hook customers for life provide a great opportunity to many users, and prestigious universities don't want to risk being perceived as inadequate relative to their peers. In the meantime, it's honestly not clear if undergraduate students even want this.  This WSJ article talks about how demand is through the roof at Harvard for a reading-intensive humanities course.  The reaction from some students at Princeton to being given GPT subscriptions is not exactly a ringing endorsement. There is no question that these tools, when used correctly, can be powerful tools to develop educational materials.  A nice example that has gone a bit viral in the last week is this site, by Nathan Liow, an undergraduate CS major at UT Austin.  He's been using Claude Opus 5.5 to create interactive visuals to go with the notes he took from his AP courses, leading to some very clean demonstrations for calculus, statistics, and physics.  Without these tools, it would be a huge investment of effort to create these kinds of visualizations.  We must figure out the ways that AI capabilities can help learning, rather than removing all of the "intellectual friction" needed to build up analytical skills, critical thinking, and knowledge. Separately:  Tomorrow is Election Day in the US.  If you are eligible, please vote!  The only way representative government can work remotely effectively is if the electorate stands up and votes.  (Corollary: Anyone who pins their political strategy on suppressing voter turnout does not believe in representative government.)