Mathematics in Pisa

Perfezionamento

In November 2012, I started the Perfezionamento (a PhD) at Scuola Normale Superiore Pisa in Mathematics. I was very fortunate to have Prof. Umberto Zannier as my supervisor, and I submitted my thesis in Mai 2016.

I defended my thesis successfully in September 2016.

The title of my thesis is Specialization and Reduction of hyperelliptic continued fractions, it investigates the appearance of prime numbers in the denominators in continued fractions of square roots of polynomials.

Such phenomena are closely tied to reduction questions, which are investigated in detail here. The first parts deal with general aspects of hyperelliptic continued fractions, and could serve as a detailed introduction to the topic.

Tutoring at SNS

From 2013 to 2015 I tutored, together with Laura Capuano and Fabrizio Barroero the second year course of Prof. Zannier at SNS. This lecture treats miscellaneous topics in algebra, geometry and number theory and was held two years in sequence.

Computations

During my PhD, I spent quite some time doing calculations, or at least writing code to do calculations. In some cases, this has actually helped my research, but certainly it was a lot of fun.

I even attempted to write my own computer algebra system. Despite its limitations, it has still the nicest implementation for continued fractions of general Laurent series I wrote so far. And I have written a few of them, some in Mathematica.

Nowadays I mostly use SageMath from org-babel for my mathematical computions, and I collected a few useful functions for that.

Hyperelliptic continued fractions computation

For my research, I also built a library for computing and analysing hyperelliptic continued fractions (essentially continued fractions of the square root of a polynomial).The download includes all the necessary Sage/Python code and some example instruction on how to use the library. Please note an installation and basic knowledge of SageMath is required.

The main features of the library include:

  • Computation of complete and partial quotients and convergents of square roots and other quadratic functions.
  • Reduction/Specialization of continued fractions over number/function fields.
  • Computation of period lengths.
  • An effective test of periodicity over number fields (modulo two primes).
  • Semiautomatic analysis and presentation of hyperelliptic continued fractions (generates LaTeX code).
  • Advanced reduction analysis: listing of valuations, degrees and more of partial quotients and convergents.
  • Listing of bad reduction primes.

Talks

  • February 2013: A survey of the polynomial Pell equation, Scuola Normale Superiore Pisa
  • October 2015: Gaussnormen für polynomiale Kettenbrüche(Gauss norms for polynomial continued fractions), Doctoral students seminar at University of Basel
  • February 2016: Prime denominators in hyperelliptic continued fractions, Scuola Normale Superiore Pisa

Conferences

  • June 2013: Arithmetics & Geometry: 25 Years Number Theory Seminar, ETH Zürich
  • July 2014: Diophantine Geometry, Unlikely Intersections and Algebraic Dynamics, Cetraro
  • Mai 2015: Final ERC Meeting in Diophantine Geometry, Rome
  • June 2015: Summary on ERC scientific activities in Diophantine problems at SNS, Cetraro
    Talk: Calculating Gauss norms in polynomial continued fractions
  • September 2015: Third Italian Number Theory Meeting, Pisa
  • June 2016: Leuca2016, Celebrating Michel Waldschmidt's 70th birthday, Patù
  • August 2016: Workshop on Arithmetic and Geometry, Cetraro

Defence