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We will be talking about Arrow’s Impossibility theorem, and its topological relatives. The span of the talk will be between American political landscape and elementary algebraic topology – feel free to peruse the links and their vicinity!
Follow-up exercise: try to model your decision process next time you are trying to converge with your friends or relatives, where to go out, or what to watch together. What is the topology of your movie-space?…
Consider a Morse function on \(f:\Real^n\to \Real\) with controlled behavior at infinity, – say, \(f=|x|^2\) near infinity. Assume further that all critical values \(a_1<a_2<\ldots<a_k\) are distinct and that all indices of critical points are \(0\) or \(1\). (Condition obviously holds in one variable.)
Clearly, there are many function that satisfy these conditions. In the (again, most transparent) univariate case, the enumeration of topological types of functions with given critical values is the subject of a nice thread of papers by Arnold on “snakes” (see, e.g., here), relating them to enumeration of ramified coverings of Riemannian spheres, up-down sequences and other cute objects.…
What is your math department strength? One of those rather polarizing questions, or an opportunity to gossip without guilt.
Still, important to understand where your department is on the map, if one wants to steer it in some particular direction, or to keep it where it is.
I wrote a short python script, using BeautifulSoup (thank you!) and campus-wide subscription to MathSciNet (thank you too!) – and downloaded the raw numbers: how many items (whichever MathSciNet is indexing: articles, books, theses,…) are authored by folks from ‘1-IL’ (our code) vs.…
“A new wave of a massive D-Dos attack hit our site at 1 o’clock pm last night,” the organizers of the Crimean referendum have just reported.
“Our IT safety experts managed to find out where those attacks came from. It is University of Illinois at Urbana Champaign. The most powerful scanning of servers before the attack was carried out exactly from there.”
It is significant that Urbana – Champaign, with a population of 37,000, has the highest number of subnets with IP addresses.