Starts 5 Feb 2025 11:00
Ends 5 Feb 2025 17:30
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Number Theory Day 2025
ICTP
Leonardo Building - Luigi Stasi Seminar Room
Organizers: Emanuel Carneiro (ICTP), Pietro Corvaja (University of Udine) and Umberto Zannier (SNS - Pisa)
Program:
11:00 - 11:50: David Masser (University of Basel, Switzerland)
Title: Equidistribution and heights
Abstract: Since Yuri Bilu (1997) we know that the conjugates of an algebraic number α of small height tend to be equidistributed near the unit circle. With Roger Baker (2023) we refined some aspects of weak convergence, for example with the test function log |1 − z|. We gave several applications, for example to the number of conjugates in the upper half plane: this number differs from half the degree d by at most 8000 h^{1/3} d for the absolute logarithmic height h (now assumed non-zero) of α. We will describe some of this and also more recent applications to heights themselves, such as those of rational functions evaluated at roots of unity.
13:30 - 14:20: Mithun Das (ICTP, Italy)
Title: Effective equidistribution of Galois orbits for mildly regular test functions
Abstract: Abstract: We provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the N-dimensional algebraic torus, identifying the qualitative dependence of the convergence on the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005) and D'Andrea, Narváez-Clauss, and Sombra (2017). This is also related to previous works by Pritsker (2011), Baker and Masser (2023), and Amoroso and Pleissis (2024). This is a joint work with Emanuel Carneiro.
15:00 - 15:50: Carlo Pagano (Concordia University, Canada)
Title: Hilbert 10 via additive combinatorics
Abstract: In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for Z. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields L/K, an elliptic curve E/K with rk(E(L))=rk(E(K))>0. This was done under BSD by Mazur and Rubin. In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves unconditionally, thus settling Hilbert 10 for every finitely generated infinite ring.
16:30 - 17:20: Gregory Debruyne (Ghent University, Belgium)
Title: Recent advances surrounding the Ingham-Karamata Tauberian theorem
Abstract: The Ingham-Karamata theorem is a cornerstone in Tauberian theory: it retrieves asymptotic information of a function from a regularity hypothesis on the function and from information about its Laplace transform. I will discuss some of my recent work surrounding this theorem and explain some applications in number theory.
Program:
11:00 - 11:50: David Masser (University of Basel, Switzerland)
Title: Equidistribution and heights
Abstract: Since Yuri Bilu (1997) we know that the conjugates of an algebraic number α of small height tend to be equidistributed near the unit circle. With Roger Baker (2023) we refined some aspects of weak convergence, for example with the test function log |1 − z|. We gave several applications, for example to the number of conjugates in the upper half plane: this number differs from half the degree d by at most 8000 h^{1/3} d for the absolute logarithmic height h (now assumed non-zero) of α. We will describe some of this and also more recent applications to heights themselves, such as those of rational functions evaluated at roots of unity.
13:30 - 14:20: Mithun Das (ICTP, Italy)
Title: Effective equidistribution of Galois orbits for mildly regular test functions
Abstract: Abstract: We provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the N-dimensional algebraic torus, identifying the qualitative dependence of the convergence on the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005) and D'Andrea, Narváez-Clauss, and Sombra (2017). This is also related to previous works by Pritsker (2011), Baker and Masser (2023), and Amoroso and Pleissis (2024). This is a joint work with Emanuel Carneiro.
15:00 - 15:50: Carlo Pagano (Concordia University, Canada)
Title: Hilbert 10 via additive combinatorics
Abstract: In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for Z. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields L/K, an elliptic curve E/K with rk(E(L))=rk(E(K))>0. This was done under BSD by Mazur and Rubin. In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves unconditionally, thus settling Hilbert 10 for every finitely generated infinite ring.
16:30 - 17:20: Gregory Debruyne (Ghent University, Belgium)
Title: Recent advances surrounding the Ingham-Karamata Tauberian theorem
Abstract: The Ingham-Karamata theorem is a cornerstone in Tauberian theory: it retrieves asymptotic information of a function from a regularity hypothesis on the function and from information about its Laplace transform. I will discuss some of my recent work surrounding this theorem and explain some applications in number theory.