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Abstract: In 1900, in the second ICM, Hilbert proposed a list of 23 problems that shaped much of mathematics throughout the past century. The tenth problem asked for an algorithm capable of deciding the solvability over the integer of any polynomial equation, in any number…
We study the renormalization group method and its applications in probability theory.
For an elliptic curve E defined over Q, the Mordell-Weil group E(Q) is a finitely generated abelian group. We prove that there are infinitely many elliptic curves E over Q for which E(Q) has rank 2. Our elliptic curves will be given by explicit models and their ranks will be found using a 2-…
The probability community has obtained fruitful results about the connectivity of random graphs in the last 50 years. Random topology is an emerging field that studies higher-order connectivity of random simplicial complexes, which are higher-order generalizations of graphs. Many classical…