Number Theory

Vanishing Sums of Roots of Unity: from Integer Tilings to Projections of Fractal Sets

Speaker: 
Caleb Marshall
Date: 
Wed, Oct 8, 2025
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

A vanishing sum of roots of unity (VSRU) is a finite list $z_1,\ldots,z_K$ of $N$-th complex roots of unity whose sum is zero. While there are many simple examples—including the famous "beautiful equation" of Euler, $e^{i \pi} + 1 = 0$—such sums become extremely complex as the parameter $N$ attains more complex prime power divisors (and we will see several classical examples illustrating this idea, as well as new examples from my work).

One fruitful line of inquiry is to seek a quantitative relationship between the prime divisors of $N$, their associated exponents, and the cardinality parameter $K$. A theorem of T.Y. Lam and K.H. Leung from the early '90's states: $K$ must always be (at least) as large as the smallest prime dividing $N$. This generalizes the well known observation that that sum of all $p$-th roots of unity (where $p$ is any prime number) must vanish; and, one notices that Euler's equation is one example of this fact.

In this talk, we will discuss two significant strengthenings of this result (one due to myself and I. Łaba, another due to myself, G. Kiss, I. Łaba and G. Somlai), which are derived from complexity measurements for polynomials with integer coefficients which have many cyclotomic polynomial divisors. As applications, we give connections in two other areas of mathematics. The first is in the study of integer tilings: additive decompositions of the integers $Z = A+B$ as a sum set, where each integer is represented uniquely. The second application is to the Favard length problem in fractal geometry, which asks for bounds upon the average length of the projections of certain dynamically-defined fractals onto lines.

This talk is based upon my individual work, as well as my joint work with I. Łaba, as well as my joint work with G. Kiss, I. Łaba and G. Somlai. All are welcome, and the first 15-20 minutes will include introductory ideas and examples for all results discussed in the latter portion of the talk.

Class: 

Short Proofs For Some Known Cohomological Results

Speaker: 
Abbas Maarefparvar
Date: 
Wed, Sep 24, 2025
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

In this talk, we first introduce the Brumer-Rosen-Zantema exact sequence (BRZ), a four-term sequence related to strongly ambiguous ideal classes in finite Galois extensions of number fields. Then, using BRZ, we obtain some known cohomological results in the literature concerning Hilbert's Theorem 94, the capitulation map, and the Principal Ideal Theorem. This is a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute for Research in Fundamental Sciences).

Class: 

Unimodal Sequences : From Isaac Newton to the Riemann Hypothesis

Speaker: 
M. Ram Murty
Date: 
Thu, Apr 24, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
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.

Class: 

Extensions of Birch–Merriman and Related Finiteness Theorems

Speaker: 
Fatemehzahra Janbazi
Date: 
Thu, Apr 10, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

A classical theorem of Birch and Merriman states that, for fixed 𝑛the set of integral binary 𝑛-ic forms with fixed nonzero discriminant breaks into finitely many GL2(ℤ)-orbits. In this talk, I’ll present several extensions of this finiteness result.

In joint work with Arul Shankar, we study a representation-theoretic generalization to ternary 𝑛-ic forms and prove analogous finiteness theorems for GL3(ℤ)-orbits with fixed nonzero discriminant. We also prove a similar result for a 27-dimensional representation associated with a family of 𝐾3surfaces.

In joint work with Sajadi, we take a geometric perspective and prove a finiteness theorem for Galois-invariant point configurations on arbitrary smooth curves with controlled reduction. This result unifies classical finiteness theorems of Birch–Merriman, Siegel, and Faltings.

Class: 

Diophantine tuples and their generalizations

Speaker: 
Chi Hoi Yip
Date: 
Thu, Mar 27, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

A set {𝑎1,𝑎2,…,𝑎𝑚}of distinct positive integers is a Diophantine 𝑚-tuple if the product of any two distinct elements in the set is one less than a square. In this talk, I will discuss some recent results related to Diophantine tuples and their generalizations. Joint work with Ernie Croot, Seoyoung Kim, and Semin Yoo.

Class: 

Zeros of L-functions in low-lying intervals and de Branges spaces

Speaker: 
Antonio Pedro Ramos
Date: 
Tue, Apr 1, 2025
Location: 
Online
Zoom
Abstract: 

We consider a variant of a problem first introduced by Hughes and Rudnick (2003) and generalized by Bernard (2015) concerning conditional bounds for small first zeros in a family of L-functions. Here we seek to estimate the size of the smallest intervals centered at a low-lying height for which we can guarantee the existence of a zero in a family of L-functions. This leads us to consider an extremal problem in analysis which we address by applying the framework of de Branges spaces, introduced in this context by Carneiro, Chirre, and Milinovich (2022).

Class: 

Elliptic curves, Drinfeld modules, and computations

Speaker: 
Antoine Leudière
Date: 
Thu, Mar 13, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
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.

Class: 

Almost sure bounds for sums of random multiplicative functions

Speaker: 
Besfort Shala
Date: 
Tue, Mar 11, 2025
Location: 
Online
Zoom
Abstract: 

I will start with a survey on sums of random multiplicative functions, focusing on distributional questions and almost sure upper bounds and $\Omega$-results. In this context, I will describe previous work with Jake Chinis on a central limit theorem for correlations of Rademacher multiplicative functions, as well as ongoing work on establishing almost sure sharp bounds for them.

Class: 

Number Theory versus Random Matrix Theory: the joint moments story

Speaker: 
Andrew Pearce-Crump
Date: 
Mon, Mar 10, 2025
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.

Class: 

Fourier optimization and the least quadratic non-residue

Speaker: 
Emily Quesada-Herrera
Date: 
Thu, Mar 6, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
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.

Class: 

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