The first edition

The talks:

See below for a more detailed description.

• 17/05/23.
  Elliptic curves: 1-Introduction to Elliptic curves. Speaker: Francesco Battistoni.
  Video: https://youtu.be/VEwZxRcSpeY

• 24/05/23.
  The harmonic measure of a square:  1-Introduction. Speaker: Francesco Monzani.
  Video: https://youtu.be/eAJsNQ-Am4E

• 31/05/23.
  Elliptic curves: 2-Elliptic curves and Cryptography. Speaker: Matteo Doni.
  Video: https://youtu.be/a58kB1GKR_o

• 14/06/23.
  Elliptic curves: 3-Elliptic curves over C. Speaker: Martina Monti.
  Notes: link to Google Drive
  Video: https://youtu.be/S_bzRWvkAh0

• 21/06/23.
  The harmonic measure of a square: 2-Harmonic measure: a PDEs perspective. Speaker: Mattia Martini.
  Notes: link to Google Drive
  Video: https://youtu.be/cbRFEvrXRpM

• 23/06/23.
  Elliptic curves: 4-Moduli spaces of (framed) elliptic curves. Speaker: Benedetta Piroddi.
  Video: https://youtu.be/B2n5Yqx-1Zs

• 28/06/23.
  Elliptic curves: 5-Curves over arithmetic fields. Speaker: Krishna Kumar Madhavan Vijayalakshmi.
  Notes: link to Google Drive
  Video: https://youtu.be/7cL3uCZHQcY

• 11/07/23.
  The harmonic measure of a square: 3-Hitting the ends: various techniques for computing the harmonic measure of a rectangle’s ends. Speaker: Giovanni Bocchi.
  Notes: link to Google Drive
  Video: https://youtu.be/cdMJgsZJ3OY

• 12/07/23.
  Elliptic curves: 6-The L-function of an elliptic curve and modular forms. Speaker: Simone Maletto.
  Video: https://youtu.be/aV196LCWCas

• 18/07/23.
  The harmonic measure of a square: 4-Brownian motion over manifolds. Speaker: Filippo Mastropietro.
  Video: https://youtu.be/kFogvvYuxKw 

• 19/07/23.
  Elliptic curves: 7-Weil conjetures. Speaker: Federico Mocchetti.
  Video: https://youtu.be/DTB6Zh-65wU

We normally meet in the Aula dottorato of the Mathematics department in Via Cesare Saldini 50, Milano, at 14:30. A streaming link will be made available under request to the organizers. Subject to change.

Elliptic Curves

1. Introductive lesson: what are elliptic curves, why are they called like this; exposition of the several reasons for their importance, including the group law, Fermat's Last Theorem and Cryptography

2. Elliptic curves and cryptography: a real world application of the theory of elliptic curves

3. Elliptic curves over C: biholomorphism with complex tori, morphisms between elliptic curves, Jacobian

4. Moduli spaces of (framed) elliptic curves: framed elliptic curves, construction of their moduli space. Comparison with the non-framed setting and reasons to choose one or the other.
   Reference for this talk: Hain, R. (2014). Lectures on Moduli Spaces of Elliptic Curves. arXiv [Math.AG]. Retrieved from http://arxiv.org/abs/0812.1803

5. Curves over arithmetic fields: elliptic curves over finite fields, Hasse's bound, Weil conjectures,  elliptic curves over number fields,  rational points and Mordell-Weil group, Falting's theorem, primes of good and bad reduction, Birch and Swynnerton-Dyer conjecture

6. The L-function of an elliptic curve and modular forms: definition of the L-function of an elliptic curve and link with modular forms; statement of the modularity theorem.

7. Weil conjectures: proof of Weil's conjectures for elliptic curves. Introduction to étale cohomology and first steps in Deligne's proof of the Riemann hypothesis for a general variety.
   References for this talk are:

   1. Joseph H. Silverman. The Arithmetic of Elliptic Curves. https://doi.org/10.1007/978-0-387-09494-6. Section V.2

   2. Pierre Deligne. La conjecture de Weil. I. https://doi.org/10.1007/BF02684373

   3. Nicholas M. Katz. An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields. https://doi.org/10.1090/pspum/028.1

   4. Emmanuel Kowalski. Deligne's proof of the Weil conjectures for varieties over finite fields. http://www.math.ethz.ch/~kowalski/deligne.pdf

Our main references will be:

• Washington, L.C. (2008). Elliptic Curves: Number Theory and Cryptography, Second Edition (2nd ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9781420071474

• Joseph H. Silverman. The Arithmetic of Elliptic Curves. https://doi.org/10.1007/978-0-387-09494-6

The harmonic measure of a square

1. Introduction: In this introductory talk, we will introduce the concept of harmonic measure of a domain as the first hitting point of a Brownian Motion and we will discuss the connection with harmonic functions. Then, we will pose our problem: find the more explicit possible description of the harmonic measure of the unit square. As a possible first approach, we discuss the conformal invariance of the BM. The main reference is [MY10].

2. Harmonic measure: a PDEs perspective: in this talk, we will delve into the connection between Brownian motion and the Dirichlet problem, giving an introduction to the so-called probabilistic potential theory. Specifically, we will show how it is possible to construct solutions for partial differential equations (PDEs) using probabilistic methods. Lastly, we will make explicit the link between these problems and the harmonic measure, reformulating our problem, at least heuristically, in terms of PDEs with non-regular boundary conditions.
   References for this talk are:

   • Schilling, René L. and Partzsch, Lothar. Brownian Motion: An Introduction to Stochastic Processes, Berlin, Boston: De Gruyter, 2012. https://doi.org/10.1515/9783110278989

   • Mörters, P., & Peres, Y. (2010). Brownian Motion (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge: Cambridge University Press. doi: 10.1017/CBO9780511750489

   • Baxter, J. (1985). Classical Potential Theory and Its Probabilistic Counterpart (J. L. Doob). SIAM Review, 27(3), 460–462. doi: 10.1137/1027124

3. Hitting the ends: various techniques for computing the harmonic measure of a rectangle’s ends: A particle at the center of a 10x1 rectangle undergoes 2D Brownian motion till it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides (i.e. the harmonic measure of the ends)?
   Main reference for this seminar is: Bornemann F, Laurie D, Wagon S, Waldvogel J. The SIAM 100-Digit Challenge. Society for Industrial and Applied Mathematics; 2004. doi:10.1137/1.9780898717969

4. Brownian motion over manifolds: we show how the properties of brownian motion on a riemannian manifold influence the analytic and geometric properties of the manifold itself.

Our main references will be:

• [MMO22] Marchione, Manfred Marvin, and Enzo Orsingher. 2022. "Hitting Distribution of a Correlated Planar Brownian Motion in a Disk" Mathematics 10, no. 4: 536. https://doi.org/10.3390/math10040536

• [BCY16] Bishop, Christopher J., and Yuval Peres. Fractals in Probability and Analysis. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2016.  https://doi.org/10.1017/9781316460238

• [MY10] Mörters, Peter, and Yuval Peres. Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press, 2010.  https://doi.org/10.1017/CBO9780511750489

Here you can find a list of all the topics that were suggested in the beginning.