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    • Counting, Recounting and Matching

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    • Chat Time Sam!

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    • Igor Rostislavovich Shafarevich: Ш (may we say "shah"?) of Number Theory

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      A magician from Moscow did stitcha maze of Galois groups and abelian varieties whichled math to progress by leaps and bounds,scale new heights and break new grounds.It behoves us to thank Igor Shafarevich!

      Shafarevich was an algebraist and geometer of the highest orderwho pioneered several topics in both. The fundamental resultshe proved, as well as the conjectures he made, were beacons formathematical progress and universal landmarks. The first part Ш(pronounced “Sha”) of his surname was borrowed to denote an intriguing,elusive object called the ‘Tate–Shafarevich group’. Shafarevichheld strong views on the philosophy of mathematics andwrote at length on it. Here, we pick out some of the mathematicalgems resulting from Shafarevich-craft, so to say. In particular,we discuss his theorems and conjectures on Galois groups over Qand the role of his conjecture on curves and abelian varieties inthe proof of Mordell’s conjecture.

    • Unearthing the Banach–Tarski Paradox

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      Greeks used the method of cutting a geometric region intopieces and recombining them cleverly to obtain areas of figureslike parallelograms. In such problems, the boundaryis ignored. However, in our discussion, we will take everypoint of space into consideration. The human endeavour tocompute lengths, areas, and volumes of irregular complicatedshapes and solids created the subject of ‘measure theory’.The paradox of the title can be informally described as follows.Consider the earth including the inside stuff. It is possibleto decompose this solid sphere into finitely many piecesand apply three-dimensional rotations to these pieces suchthat the transformed pieces can be put together to form twosolid earths! The whole magic lies in the word ‘pieces’. Thepieces turn out to be so strange that they cannot be ‘measured’.

    • Robert Langlands Wins the 2018 Abel Prize

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    • Editorial

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