What happens if you feed ChatGPT some logical puzzles? I took one, from Smullyan’s textbook on basics of mathematical logics. It is easy to solve. Predictably, ChatGPT failed to do so, but in rather instructive ways. Here’s the puzzle: [On an island, the inhabitants are either knights, and always tell the truth, or knaves, and […]
A survey of the remunerations of graduate students in math PhD programs in public universities initiated and conducted by Hans Christianson, the Director of Graduate Studies at UNC at Chapel Hill is in, and the results are not hugely surprising. With a few exceptions, the (adjusted for cost of living) yearly stipends are pretty comparable,
math PhD programs through materialistic lens Read More »
This visual continues the exploration started here. I used the dataset of MathSciNet records of the publications by department members, collected by the IGL participants. Each article reviewed by MSN carries a few MSC codes, roughly classifying them. I used these codes to create a sankey diagram, showing who published in what area. A few
math department and its research Read More »
\(\def\Real{\mathbb{R}}\def\tree{\mathcal{T}}\) Consider a path metric space \(M\) and a continuous function \(f:M\to \Real\). For simplicity, all such pairs here will have sublevel sets of \(f\) compact (so that the function is bounded from below). Then one can form the so-called merge tree (see, e.g. here, for motivation, and some history), a new topological space (let’s
\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\def\xx{\mathbf{x}}\def\ex{\mathbb{E}}\) Consider \(n\) points in Euclidean space, \(\xx={x_1,\ldots, x_n}, x_k\in \Real^d, n\leq d+1\). Generically, there is a unique sphere in the affine space spanned by those points, containing all of them. This centroid (which we will denote as \(o(x_1,\ldots,x_n)\)) lies at the intersection of the bisectors \(H_{kl}, 1\leq k\lt l\leq n\), hyperplanes of points equidistant
Brownian centroids Read More »
\(\def\Real{\mathbb{R}}\def\Z{\mathcal{Z}}\def\sg{\mathfrak{S}}\def\B{\mathbf{B}}\) Consider a collection of vectors \(e_1,\ldots,e_n\) in the upper half-plane, such that \( e_k=(x_k,1)\) and \( x_1>x_2\gt \ldots \gt x_n\). Minkowski sum of the segments ( s_k:=[0,e_k]) is a zonotope ( \Z). Rhombus in this context is the Minkowski sums \( \Z(k,l)=s_k\oplus s_l, 1\leq k\lt l\leq n\) of a pair of the segments, perhaps
domain restrictions and topology Read More »
There is no excerpt because this is a protected post.
Protected: kod*fest Read More »
\(\def\Comp{\mathbb{C}}\def\Proj{\mathbb{P}}\def\Nat{\mathbb{N}}\) This workshop is about complex analytic techniques usable in applications from classical combinatorial problems to asymptotic representation theory and cluster algebras. The scope is approximately what is covered by Pemantle-Wilson(-Melczer) book. I plan to report on what is going on here (reporting is sporadic and idiosyncratic). Lectures are streamed. Day One (all times PDT):
analytic combinatorics in several variables (aimath, san jose 4-9.4.2022) Read More »
In connection with our IGL project, I remembered a result that might be useful, and is not widely enough known. Let \(z_1\leq \ldots\leq z_N\) be a sequence of real numbers forming the spectrum of a Hermitian operator \(A\) acting in \(N\)-dimensional Hermitian space \(U\):\[P_A(z):=\det(zE-A)=\prod_k (z-z_k).\] Proposition: Let \(V\) be a codimension \(1\) hyperplane in \(U\),
derivatives and random matrices Read More »
