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AJ Fong, Department of Pure Mathematics, University of Waterloo

"The mathematics of juggling (and perhaps a geometric application)"

Before videos could be easily transmitted over the internet, mathematical notation for juggling patterns was used by jugglers to share instructions and new patterns with each other. After introducing these, I will show that a mild generalisation of this gives a natural partial order on juggling patterns. If time permits, I will describe a relatively recent result that demonstrated that juggling patterns can be used to index a natural stratification of Grassmannians, which naturally arises in Poisson geometry, total positivity and cluster algebras.

MC 5417

Franklin Tall, University of Toronto

"An undecidable extension of Morley’s theorem on the number of countable models"

We show that Morley’s theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, large cardinals, and Inner Model Theory.

MC 5479

Corey Jones, North Carolina State University

"Constructing actions of fusion categories on C*-algebras"

A fusion category is an algebraic object that simultaneously generalizes finite groups and their representation categories. Fusion categories can ``act" on C*-algebras by bimodules, extending the familiar concept of a group acting by automorphisms to a non-invertible setting. Building actions of specific fusion categories on specific C*-algebras is hard. In this talk, we will discuss a general method that allows for the construction of actions of fusion categories on interesting C*-algebras with minimal algebraic input. As an application, we construct actions of exotic fusion categories on noncommutative tori. Based on joint work with David Evans.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Jakub Krásenský, Czech Technical University in Prague

"Criterion sets for quadratic forms over number fields"

By the celebrated 15 theorem of Conway and Schneeberger, a classical positive definite quadratic form over Z is universal if it represents each element of {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim and B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost. Since then, the question of uniqueness for particular situations has been studied by several authors.

We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. Here again, the existence of criterion sets for universality is known, and Lee determined the set for Q(sqrt5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Tianyi Zheng, UC San Diego

"Random walks on self-similar groups and conformal dimension"

Conformal dimension was introduced in the late 1980s by P. Pansu; it is a natural invariant in the study of the geometry of hyperbolic spaces and their boundaries. In this talk we will discuss how conformal geometry can be used to study random walks on iterated monodromy groups, in particular, random walk entropy bounds when the limit set has Ahlfors-regular conformal dimension strictly less than 2. Based on joint work with N. Matte Bon and V. Nekrashevych.

MC 5501

Rahim Moosa, Department of Pure Mathematics, University of Waterloo

"Binding groups for rational dynamics"

I will report on ongoing work with Moshe Kamensky toward developing a theory of binding groups for quantifier-free types in ACFA, well-suited for applications to rational algebraic dynamics.

MC 5479

Talk #1: Ted Fu, University of Waterloo

"On Waring's problem for large powers"

Let G(k) be the least number s having the property that every sufficiently large natural number is the sum of at most s positive integer k-th powers. In this talk, I will present how Brüdern and Wooley implement smooth numbers technologies in their minor arc analysis and derive G(k) ≤ ⌈k(log k + 4.20032)⌉.

Talk #2: Aidan Boyle, University of Waterloo

"Waring’s problem: Beyond Freiman’s Theorem"

Suppose that we are given a non-decreasing sequence of positive integers (k_i) where each term is at least 2. Given a positive integer j, we seek to understand the circumstances in which there exists a positive integer s := s(j) such that every sufficiently large natural number n can be written as a sum of s positive integers to the respective powers k_j, ..., k_j+s-1. Freĭman asserted that such representation exists if and only if the infinite summation of all 1/k_i diverges. We provide an effective version of this theorem, and in particular, comment on instances in which the exponents form a sequence of consecutive terms of an arithmetic progression.

MC 5417

Faisal Romshoo, Department of Pure Mathematics, University of Waterloo

"A theoretical framework for H-structures"

For an oriented Riemannian manifold $(M^n, g)$, and Lie subgroup $H \subset SO(n)$, a compatible $H$-structure on $(M^n,g)$ is a principal $H$-subbundle of the principal $SO(n)$-bundle of oriented orthonormal coframes.  They can be described in terms of the sections of the homogeneous fibre bundle obtained by $H$-reduction of the oriented frame bundle. Examples of these structures include $U(m)$-structures, $G_2$-structures and $\text{Spin(7)}$-structures. In this talk, we will study a general theory for $H$-structures described in a paper of Daniel Fadel, Eric Loubeau, Andrés J. Moreno and Henrique N. Sá Earp titled "Flows of geometric structures" (https://arxiv.org/abs/2211.05197).

MC 5403

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory IX"

We will discuss effective interpretability of graphs, following Antonio Montalbán's monograph.

MC 5479

Micah Milinovich, University of Mississippi

"Fourier optimization, prime gaps, and the least quadratic non-residue"

There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe two such Fourier optimization frameworks that can be used to study classical problems in number theory: bounding the maximum gap between consecutive primes assuming the Riemann hypothesis and bounding for the size of the least quadratic non-residue modulo a prime assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions. The resulting extremal problems can be stated in accessible terms, but finding the exact answer appears to be rather subtle. Instead, we experimentally find upper and lower bounds for our desired quantity that are numerically close. If time allows, I will discuss how a similar Fourier optimization framework can be used to bound the size of the least prime in an arithmetic progression on GRH. This is based upon joint works with E. Carneiro (ICTP), E. Quesada-Herrera (TU Graz), A. Ramos (SISSA), and K. Soundararajan (Stanford). 

MC 5417