Aug 7, 2021 04:19 PM
https://www.quantamagazine.org/mathemati...-20210805/
INTRO: Mathematicians, in their attempt to make sense of the sprawling landscape of mathematical objects, can face similar challenges. That’s especially true for practitioners in the field of descriptive set theory, who try to rate the difficulty of classification problems — sometimes concluding that a given classification task is relatively easy to carry out, and sometimes (as with the Amazon) discovering that it’s too hard. The discipline is just one branch of set theory, the study of collections of objects — they can be numbers, graphs, points in space, vectors, anything — called sets. The real numbers, rational numbers, imaginary numbers and so on are all sets in their own right, leaving no shortage of objects for mathematicians to study.
For decades, one classification problem — involving a particular set of infinitely large objects called torsion-free abelian groups (or TFABs) — stymied researchers. This problem was first raised in 1989 by the mathematicians Harvey Friedman and Lee Stanley in a paper that, according to Paolini, “introduced a new way of comparing the difficulties of classification problems for countable structures, indicating that some things are more complicated than others.”
Now, in a paper posted online earlier this year, Paolini and his former postdoctoral adviser, Saharon Shelah of the Hebrew University of Jerusalem, have finally settled the issue regarding TFABs.
“It is certainly an important paper, which solves an old problem from more than 30 years ago,” said Alexander Kechris of the California Institute of Technology.
“[Their strategy displays] an incredible amount of cleverness in transforming a complicated problem into something easier,” added Chris Laskowski of the University of Maryland, who has collaborated with Shelah on roughly a dozen papers (though not this one). “Many had tried and not succeeded. It’s great to have this settled.” (MORE)
INTRO: Mathematicians, in their attempt to make sense of the sprawling landscape of mathematical objects, can face similar challenges. That’s especially true for practitioners in the field of descriptive set theory, who try to rate the difficulty of classification problems — sometimes concluding that a given classification task is relatively easy to carry out, and sometimes (as with the Amazon) discovering that it’s too hard. The discipline is just one branch of set theory, the study of collections of objects — they can be numbers, graphs, points in space, vectors, anything — called sets. The real numbers, rational numbers, imaginary numbers and so on are all sets in their own right, leaving no shortage of objects for mathematicians to study.
For decades, one classification problem — involving a particular set of infinitely large objects called torsion-free abelian groups (or TFABs) — stymied researchers. This problem was first raised in 1989 by the mathematicians Harvey Friedman and Lee Stanley in a paper that, according to Paolini, “introduced a new way of comparing the difficulties of classification problems for countable structures, indicating that some things are more complicated than others.”
Now, in a paper posted online earlier this year, Paolini and his former postdoctoral adviser, Saharon Shelah of the Hebrew University of Jerusalem, have finally settled the issue regarding TFABs.
“It is certainly an important paper, which solves an old problem from more than 30 years ago,” said Alexander Kechris of the California Institute of Technology.
“[Their strategy displays] an incredible amount of cleverness in transforming a complicated problem into something easier,” added Chris Laskowski of the University of Maryland, who has collaborated with Shelah on roughly a dozen papers (though not this one). “Many had tried and not succeeded. It’s great to have this settled.” (MORE)