Events

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

William Verreault, University of Toronto

On the minimal length of addition chains

An addition chain is a sequence of increasing numbers, starting with 1 and ending with n, such that each number is the sum of two previous ones in the sequence. A challenging problem is, given a positive integer n, to find the minimal length of an addition chain leading to n. I will present bounds on the distribution function of this minimal length, which are sharp up to a small constant. This is joint work with Jean-Marie De Koninck and Nicolas Doyon.

MC 2034

Elisabeth Werner, Case Western Reserve University

Affine invariants in convex geometry

In analogy to the classical surface area, a notion of affine surface area (invariant under affine transformations) has been defined. The isoperimetric inequality states that the usual surface area is minimized for a ball. Affine isoperimetric inequality states that affine surface area is maximized for ellipsoids. Due to this inequality and its many other remarkable properties, the affine surface area finds applications in many areas of mathematics and applied mathematics. This has led to intense research in recent years and numerous new directions have been developed. We will discuss some of them and we will show how affine surface area is related to a geometric object, that is interesting in its own right, the floating body.

MC 5501

Robert Cornea, University of Waterloo

Stable Pairs on P2 via Spectral Correspondence

In this talk we will consider stable wild Vafa-Witten-Higgs bundles (or stable pairs for short) (E, ϕ) on P^2 where E is a rank two holomorphic vector bundle and ϕ : E -> E(d) is a holomorphic bundle map with d > 0. There is a way to construct stable pairs on called the spectral correspondence. This states that given a stable pair (E,ϕ) on P^2, there exists a surface Y and a 2:1 covering map pi: Y -> P^2 such that E is the push forward of a line bundle on Y and ϕ comes from the multiplication of a section on Y. So studying stable pairs (E,ϕ) on P^2 boils down to finding 2:1 covering maps Y -> P^2 and line bundles on Y. The study of constructing rank two vector bundles on P^2 via 2:1 coverings was studied by Schwarzenberger in 1960. We will demonstrate examples of stable pairs when d=1 and explain the cases briefly for d=2 and 3.

MC 5479

Noah Slavitch, University of Waterloo

Measurable Cardinals and Non-Constructible Sets

In this talk we will explain how the existence of a measurable cardinal implies that V≠L, that is, that there exist nonconstructible sets.

MC 5479

Liam Orovec, University of Waterloo

Greedy beta-expansions for families of Salem numbers

We give criteria for finding the greedy beta-expansion for 1 under families of Salem numbers that approach a given Pisot number. We show these expansions are related to the greedy expansion under the Pisot base. This expands the work of Hare and Tweedle to include more Pisot numbers and more families of Salem numbers.

MC 5403

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

Erik Seguin, University of Waterloo

Selected Topics on Fourier Algebras of Locally Compact Hausdorff Groups

We discuss some selected topics on Fourier algebras of locally compact Hausdorff groups.

MC 5403

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

Miho Mukohara, University of Tokyo

On a Galois correspondence for minimal actions of compact groups on C*-algebras

Inclusions arising from compact quantum group actions on factors have been studied by Izumi-Longo-Popa and Tomatsu. For a minimal action of a compact group on a factor, there is an isomorphism from the lattice of closed subgroups onto that of intermediate subfactors between the factor and the fixed point subfactor. The correspondence between intermediate subfactors and subgroups is called a Galois correspondence. As a duality result, a Galois correspondence for discrete group actions is also known. Analogues for actions on C*-algebras were also studied by Izumi, Cameron-Smith, and others. In this talk, I will discuss a Galois correspondence for compact group actions on C*-algebras. A crucial result for our main theorem is the proper outerness of finite index endomorphisms of purely infinite simple C*-algebras. This was shown by Izumi recently. If time permits, I will also explain an extension of our main result to actions of compact quantum groups of Kac type and a relationship between our main result and the C*-discrete inclusion introduced by Hernández Palomares and Nelson.

MC 5417 or Join on Zoom