at noon
in M1040

The Legendre symbol is one of the most basic, mysterious and extensively studied objects in number theory. It is a multiplicative function that encodes information about whether an integer is a square modulo an odd prime $p$. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798, and has since found countless applications in various areas of mathematics as well as in other fields including cryptography. In this talk, we shall explore what we call "Legendre paths", which encode information about the values of the Legendre symbol. The Legendre path modulo $p$ is defined as the polygonal path in the plane formed by joining the partial sums of the Legendre symbol modulo $p$. In particular, we will attempt to answer the following questions as we vary over the primes $p$: how are these paths distributed? how do their maximums behave? and what proportion of the path is above the real axis? Among our results, we prove that these paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademakher random multiplicative functions. Part of this work is joint with Ayesha Hussain.

It is an open question of Baker whether the Dirichlet L-values at 1 with fixed modulus are linearly independent over the rational numbers. The best-known result is due to Baker, Birch and Wirsing, which affirms this when the modulus of the associated Dirichlet character is co-prime to its Euler's phi value. In this talk, we will discuss an extension of this result to any arbitrary family of moduli. The interplay between the resulting ambient number fields brings new technical issues and complications hitherto absent in the context of a fixed modulus. We will also investigate the linear independence of such values at integers greater than 1.

Kummer theory is a classical theory about radical extensions of fields in the case where suitable roots of unity are present in the base field. Motivated by problems close to Artin's primitive root conjecture, we have investigated the degree of families of general Kummer extensions of number fields, providing parametric closed formulas. We present a series of papers that are in part joint work with Christophe Debry, Fritz Hörmann, Pietro Sgobba, and Sebastiano Tronto.

at noon

Pro-$p$ groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences $(a_n)$ and $(c_n)$, closely related to a special filtration of a finitely generated pro-$p$ group $G$, called the Zassenhaus filtration. These sequences give the cardinality of $G$, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer $n$ such that $a_n=0$ (or $c_n$ has a polynomial growth) if and only if $G$ is a Lie group over $p$-adic fields.

In 2016, Minac, Rogelstad and Tan inferred an explicit relation between $a_n$ and $c_n$. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an equivariant context: when the automorphism group of $G$ admits a subgroup of order a prime $q$ dividing $p-1$.

In this talk, we present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.

at noon
in M1040

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. For example, one can enrich spectra of rings with new points coming from valuations defined on them and taking values in totally ordered abelian groups.

Totally ordered groups are examples of idempotent semirings, and generalized valuations appear when we replace totally ordered abelian groups with more general idempotent semirings. An important example of idempotent semiring is the tropical semifield.

As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of a scheme, which behaves particularly well with respect to idempotization of closed subschemes.

This is a joint work with Félix Baril Boudreau.

Idempotent semirings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization. They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry. However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semirings. In this mini course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semirings and modules over them.

Idempotent semirings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization. They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry. However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semirings. In this mini course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semirings and modules over them.

at noon
in M1040

(University of Lethbridge)

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at noon
in M1040

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at noon
in M1040

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at noon
in M1040

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