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Here Algebra and Number Theory (ANT) are interpreted broadly. If you work in related aspects of ANT and like to give a talk here, feel free to contact any of the organizers below. For talks in previous years, click here.
All times are in Mountain Time.
Organizers
(Faculty) Thomas Bitoun (thomas.bitoun@ucalgary.ca),
Dang-Khoa Nguyen (dangkhoa.nguyen@ucalgary.ca),
(Postdoc) Samprit Ghosh (samprit.ghosh@ucalgary.ca).
To register click : Zoom Registration Link
Research Seminars Listing: https://researchseminars.org/seminar/ANTS
Calgary ANTS is partially supported by PIMS for the Academic Year 2024-2025.
PIMS Site : https://pims.math.ca/sites/university-calgary
MathTube : UCalgary Algebra and Number Theory Seminar
WINTER 2025
Date : April 24, 1 - 2 pm
Speaker : M. Ram Murty, Queen's University
Title : Unimodal Sequences : From Isaac Newton to the Riemann Hypothesis
Abstract :
We will give an exposition on the recent progress in the study of unimodal sequences, beginning with the work of Isaac Newton and then to the contemporary papers of June Huh. We will also relate this topic to the Riemann hypothesis. In the process, we will connect many areas of mathematics ranging from number theory, commutative algebra, algebraic geometry and combinatorics.
Date : March 13, 1 - 2 pm.
Speaker : Antoine Leudière
Title : Elliptic curves, Drinfeld modules, and computations
Abstract : We will talk about Drinfeld modules, and how they compare to elliptic curves for algorithms and computations.
Drinfeld modules can be seen as function field analogues of elliptic curves. They were introduced in the 1970's by Vladimir Drinfeld, to create an explicit class field theory of function fields. They were instrumental to prove the Langlands program for GL2 of a function field, or the function field analogue of the Riemann hypothesis.
Elliptic curves, to the surprise of many theoretical number theorists, became a fundamental computational tool, especially in the context of cryptography (elliptic curve Diffie-Hellman, isogeny-based post-quantum cryptography) and computer algebra (ECM method).
Despite a rather abstract definition, Drinfeld modules offer a lot of computational advantages over elliptic curves: one can benefit from function field arithmetics, and from objects called Ore polynomials and Anderson motives.
We will use two examples to highlight the practicality of Drinfeld modules computations, and mention some applications.
Date : March 6, 1 - 2 pm
Speaker : Emily Quesada-Herrera, University of Lethbridge
Title : Fourier optimization and the least quadratic non-residue
Abstract : We will explore how a Fourier optimization framework may be used to study two classical problems in number theory involving Dirichlet characters: The problem of estimating the least character non-residue; and the problem of estimating the least prime in an arithmetic progression. In particular, we show how this Fourier framework leads to subtle, but conceptually interesting, improvements on the best current asymptotic bounds under the Generalized Riemann Hypothesis, given by Lamzouri, Li, and Soundararajan. Based on joint work with Emanuel Carneiro, Micah Milinovich, and Antonio Ramos.
FALL 2024
Date : November 21, 2 -3 pm
Speaker : Clifton Cunningham, University of Calgary
Title : Packets and the fine structure of L-functions
Abstract : Automorphic representations, which provide a vast generalization of modular forms, are are grouped together into so-called L-packets according to the L-functions they produce. From this point of view, automorphic representations give a kind of fine-structure to L-functions themselves. While L-packets of automorphic representations are natural from this point of view, they have some deficiencies when one looks for how L-functions transfer between different groups that are related by what we call “functoriality” in the Langlands program. To address these deficiencies, Arthur introduced A-packets of automorphic representations; each A-packet is an enlarged L-packet. However, A-packets have not been defined in the same generality as L-packets. In this talk I will review this story and sketch work by my research group on a generalization of A-packets. The talk includes comments on applications of A-packets to number theory, some of which are highly speculative.
Date: October 24, 2 - 3 pm
Speaker : Renate Scheidler, University of Calgary
Title : Orienteering with One Endomorphism
Abstract: Given two elliptic curves, the path finding problem asks to find an isogeny (i.e. a group homomorphism) between them, subject to certain degree restrictions. Path finding has uses in number theory as well as applications to cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.
Date : October 17, 2 - 3 pm
Speaker : Gaurav Patil, Postdoctoral Fellow, University of Toronto.
Title : Parametrization of rings of finite rank - a geometric approach and their use in counting number fields.
Abstract
We describe parametrizations of rings that generalize the notions of monogenic rings and binary rings. We use these parametrizations to give better lower bounds on the number of number fields of degree n and bounded discriminant.
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