Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's meth ....Continued Fractions and Torsion on Hyperelliptic Curves. Scientific advance should not blindly add to our knowledge; a true advance brings insights that collapse different issues into one. Understanding more is to need to remember less. For an important class of examples, this project identifies the study of a fundamental invariant of a quadratic number field, its regulator and hence its class number, with maximum torsion on the Jacobian variety of an hyperelliptic curve. The investigator's methods will surprise some longstanding problems into submission and in particular will lead them to reveal full data on torsion on hyperelliptic curves of low genus.
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Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, ....Asymptotic Geometric Analysis and Machine Learning. Phenomena in large dimensions appear in a number of domains of Mathematics and adjacent domains of science (e.g. Computer Science), dealing with functions of infinitely growing number of parameters. Here, we focus on several questions naturally linked to Asymptotic Geometric Analysis which have natural applications to Statistical Learning Theory. We intend to use geometric, probabilistic and combinatorial methods to investigate these problems, with an emphasis on modern tools in Empirical Processes Theory and the theory of Random Matrices.Read moreRead less