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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

Katarzyna Wyczesany, Carnegie Mellon University

"Dualities on sets and how they appear in optimal transport"

In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi-involution is of a special form, which arose from the consideration of optimal transport problem with respect to costs that attain infinite values. We will discuss how this unified point of view on order reversing quasi-involutions helps to deepen the understanding of the underlying structures and principles. We will provide many examples and ways to construct new order reversing quasi-involutions from given ones. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

This seminar will be held both online and in person:

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

Christoph Kesting, McMaster University

"The Klein j-Function is not Pfaffian over the Real Exponential Field"

James Freitag showed that the Klein j-function is not pfaffian over the complex numbers. In this talk, I will give a brief introduction to pfaffian functions, their current place in model theory and Freitag's result. Then I will discuss recent work expanding Freitag's result to a restriction of the j-function to the imaginary interval (0, i) not being pfaffian over the real exponential field.

MC 5479

Jiahui Huang, Department of Pure Mathematics, University of Waterloo

"Arc-Floer conjecture"

For a hypersurface singularity, the arc-Floer conjecture states an isomorphism between the compactly supported cohomology of $X_m$, the m-th restricted contact locus (of algebraic nature), and the Floer homology of $\varphi^m$, the m-th iterate of the monodromy on the Milnor fiber (of topological nature). In particular, this gives the Floer homology a mixed Hodge structure.

It was known by a result of Denef and Loeser that the Euler characteristic of $X_m$ agrees with the Lefschetz number of $\varphi^m$, which is given by the Euler characteristic of its Floer homology. The conjecture predicts an equivalence at the level of cohomology. It has been proven for plane curves by de la Bodega and de Lorenzo Poza. We shall look at the case where the singularity is the affine cone of a smooth projective hypersurface.

MC 5417

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Non-reduced schemes"

Last time, we looked at the case where the ground field is not algebraically closed. Now we will drop the hypothesis that the ring of regular functions is not an integral domain and explain what the simplest schemes of this sort look like. We will also introduce the central concepts of limits and flatness and begin to discuss them in detail.

MC 5417