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Emily Quesada-Herrera, University of Lethbridge

Fourier optimization and the least quadratic non-residue

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.

MC 5479

Faisal Romshoo, University of Waterloo

A canonical form theorem for elements of spin(7)

We will first demonstrate the maximal torus theorem at the Lie algebra level for the exceptional Lie algebra g_2 by proving a canonical form theorem for the elements of g_2 following arXiv:2209.10613. Then, we will proceed to prove a canonical form theorem for the elements of the Lie algebra spin(7).

MC 5479

Veronika Shelestunova, RBC Capital Markets

Teaching Stream

The Career Talks seminar series invites professionals from various fields to share their personal career journeys and insights on how they achieved success. Each session offers valuable advice and guidance for current graduate students. By hearing firsthand experiences, attendees gain a deeper understanding of the challenges and opportunities that lie ahead in their professional lives.

MC 5501

Refreshments will be available during the talk

Blake Madill & Zack Cramer, University of Waterloo

Teaching Stream

The Career Talks seminar series invites professionals from various fields to share their personal career journeys and insights on how they achieved success. Each session offers valuable advice and guidance for current graduate students. By hearing firsthand experiences, attendees gain a deeper understanding of the challenges and opportunities that lie ahead in their professional lives.

MC 5501

Habiba Kadiri, University of Lethbridge

An explicit version of Chebotarev’s Density Theorem.

This talk will first provide a (non-exhaustive) survey of explicit results on zero-free regions and zero densities of the Riemann zeta function and their relationship to error terms in the prime number theorem. This will be extended to Dirichlet L functions and Dedekind zeta functions, where new challenges arise with potential exceptional zeros. We will explore estimates for the error terms for prime counting functions across various contexts, with a specific attention to number fields. Chebotarev’s density theorem states that prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias and Odlyzko in 1977. In this article, we present an explicit refinement of their result. Key aspects of our approach include using the following: smoothing functions, recently established zero-free regions and zero-counting formula for zeros of the Dedekind zeta function, and sharp bounds for Bessel-type integrals. This is joint wok with Sourabh Das and Nathan Ng.

MC 2034

Christine Eagles, University of Waterloo

The Zilber dichotomy in DCF_m II

We continue to read Omar Le\'on S\'anchez' paper on the Zilber dichotomy in partial differentially closed fields

MC 5403

Gerrik Wong, University of Waterloo

Tidy Subgroups and Ergodicity

We will continue talking about applications of tidy subgroups to ergodic automorphisms on totally disconnected locally compact groups.

MC 5403

Jérémy Champagne, University of Waterloo

Equidistribution and the probability of coprimality of some integer tuples

" What is the probability of two random integers being coprime? "

This question, sometimes called " Chebyshev’s Problem », is very natural and happens to have a very straightforward answer. Using only elementary methods, one can easily show that the natural density of pairs (m,n) with gcd(m,n)=1 is exactly 1/zeta(2)=6/pi^2=60.8..%.

Knowing this, one might seek certain g:N->N for which the density of n’s with gcd(n, g(n))=1 is also 1/zeta(2), which give a certain sense of randomness to the function g. Many functions with that property can be found in the literature, and we have a special interest for those of the form g(n)=[f(n)] where f is a real valued function with some equidistributive properties modulo one; for example, Watson showed in 1953 that g(n)=[αn] has this property whenever α is irrational. In this talk, we use a method of Spilker to obtain a more general framework on what properties f(n) must have, and also what conditions can replace coprimality of integer pairs.

MC 5403

Jesse Huang, University of Waterloo

Enumerative Mirror Symmetry

Continuing on with the introduction to mirror map and Yukawa coupling, we will discuss Gromov-Witten invariants and quantum cohomology which give rise to the statement of enumerative mirror symmetry. The statement extends to certain non-Calabi-Yau toric varieties, whose mirror information can be extracted from compactificatification of SYZ discussed on Monday.

MC 5479

Andrew Hanlon, Dartmouth College

Mirror Symmetry Seminar: Compactifying 2D mirror symmetry for the algebraic torus

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