Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2023 For more information, contact Félix Baril Boudreau . Talks are usually at noon on Monday in room M1040 (Markin Hall). All times are Mountain Time. The live access zoom link is available here and on researchseminars.org, and is also sent to our mailing list.
 The next talk: Feb 6 at noon via zoom Cristhian Garay (CIMAT Guanajuato, Mexico) Generalized valuations and idempotization of schemes 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.
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 Date Speaker Title Jan 9 Youness Lamzouri A walk on Legendre paths at noon in M1040 (Institut Élie Cartan de Lorraine, France) PIMS Distinguished Speaker Series 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. Jan 16 Neelam Kandhil On linear independence of Dirichlet L-values at 9:30am via zoom (Institute of Mathematical Sciences, Chennai, India) 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. Jan 23 Antonella Perucca Recent advances in Kummer theory at 9:30am via zoom (University of Luxembourg) 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. Jan 30 Oussama Hamza Filtrations, arithmetic and explicit examples in an equivariant context at noon via zoom (University of Western Ontario) 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. Feb 6 Cristhian Garay Generalized valuations and idempotization of schemes at noon in M1040 (CIMAT Guanajuato, Mexico) 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. Feb 6 Cristhian Garay An invitation to the algebraic geometry over idempotent semirings 3:10–4:45pm in B716 (CIMAT Guanajuato, Mexico) Minicourse, Lecture 1 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. The minicourse will also be streamed online. To attend (one or both sessions) via zoom, register at: https://uleth.zoom.us/meeting/register/tJYuc-uqrz4pE93dDYPcP3Ek7CbrJCFISazV Feb 9 Cristhian Garay An invitation to the algebraic geometry over idempotent semirings 3:10–4:45pm in B716 (CIMAT Guanajuato, Mexico) Minicourse, Lecture 2 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. The minicourse will also be streamed online. To attend (one or both sessions) via zoom, register at: https://uleth.zoom.us/meeting/register/tJYuc-uqrz4pE93dDYPcP3Ek7CbrJCFISazV Feb 13 Speaker TBA Title TBA at noon in M1040 (University of Lethbridge) Abstract TBA Feb 27 Florent Jouve Title TBA at 9:30am via zoom (Université de Bordeaux, France) Abstract TBA Mar 6 John Voight Title TBA at noon via zoom (Dartmouth College, USA) Abstract TBA Mar 13 Renate Scheidler Title TBA at noon in M1040 (University of Calgary) Abstract TBA Mar 20 Joshua Males Title TBA at noon in M1040 (University of Manitoba) Abstract TBA Mar 27 Douglas Ulmer Title TBA at noon in M1040 (University of Arizona) Abstract TBA Apr 3 Harald Andrés Helfgott Title TBA at time tba via zoom (University of Göttingen, Germany, and Institut de Mathématiques de Jussieu, France) Abstract TBA
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