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Jules Ribolzi, University of Waterloo 

Definable groups in ACF_0 and stable groups.

The vague goal of this term’s working seminar is the structure of groups definable in the theory of compact complex manifolds. Anyone interested in is welcome.

For the first session will review some notions and tools from definable groups in ACF_0 and stable groups.

MC 5479

Jacques Gideon van Wyk, McMaster University

A Lucky Game of Yahtzee Over Christmas

Yahtzee is a game where you win points by rolling five dice into particular patterns, sort of like hands in Poker. The roll which wins you the most points, and which is the hardest to get, is the Yahtzee, the game's namesake, where all five die land on the same digit.

Over the Christmas holidays, I played Yahtzee with some of my family, and we had a game where we rolled five Yahtzees among five teams in one game. In our experience, this felt like quite the lucky game, as many games of Yahtzee end with no one rolling any Yahtzees at all.

In this talk, we're going to get to the bottom of how lucky we actually were: I'll explain how Yahtzee is played, we'll discuss some strategies you can employ to increase your chances at winning, and, assuming an optimal strategy, we'll figure out how likely it is to get at least five Yahtzees in one game.

MC 5479

(Refreshments will start at 16:30)

Michael Gregory, University of Waterloo

Computability Relative to Random Sets

This presentation explores the interaction between algorithmic randomness and Turing degrees. We focus on 1-random sets and how randomness interacts with computable reducibility. Several fundamental results are discussed that illuminate the placement of random sets within the Turing degrees and the constraints that randomness imposes on computable reductions. In particular, the Kucera-Gacs Theorem is presented, which establishes that every set is weak truth-table reducible to a 1-random set.

MC 5403

Zhihao Zhang, University of Waterloo

Translation-Invariant Function Algebras of Compact Groups

Let G be a compact group and let Trig(G) denote the algebra of trigonometric polynomials of G. For a translation-invariant subalgebra A of Trig(G), one can consider the completions of A under the uniform norm and the Fourier norm. We show in Chapter 2 using techniques developed by Gichev that both completions have the same Gelfand spectrum, answering a question posed in a paper of Spronk and Stokke. In the same paper, a theorem describing of the Gelfand spectrum of the Fourier completion of finitely-generated such algebras A was given. In Chapter 3, we extend this theorem to the case of countably-generated, translation-invariant subalgebras, A. In Chapter 4, we give a brief overview of the Beurling--Fourier algebra, a weighted variant of the classical Fourier algebra studied by Ludwig, Spronk and Turowska. The addition of a weight for these particular algebras invites new spectral data in contrast to its classical counterpart. In Chapter 5, we show for Beurling--Fourier algebras of compact abelian groups, G, that its weight can be used to construct a seminorm on a real vector space generated by the dual of G that remembers the spectral data of the algebra.

MC 2009

Siyuan Lu, McMaster University

Interior C^2 estimate for Hessian quotient equation

In this talk, we will first review the history of interior C^2 estimates for fully nonlinear equations. As a matter of fact, very few equations admit this property, not even the Monge-Ampère equation in dimension three or above. We will then present our recent work on interior C^2 estimate for Hessian quotient equation. We will discuss the main idea behind the proof. If time permits, we will also discuss the Pogorelov-type interior C^2 estimate for Hessian quotient equation and its applications.

MC 5417

Joaquin G. Prandi, University of Waterloo

Iterated Function Systems and the Local Dimension of Measures

Given an iterated function system S in R^d, with full support and such that the rotation in it fixed the hypercube [-1/2,1/2]^d , then S satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. With this in mind, we extend the notion of net intervals from R to R^d. We also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support.

We study the local dimension of the convolution of two measures. We give conditions for bounding the local dimension of the convolution on the basis of the local dimension of one of them. Moreover, we give a formula for the local dimension of some special points in the support of the convolution.

We study the local dimension of the addition of two measures. We give an exact formula for the lower local dimension of the addition based on the local dimension of the two added measures. We give an upper bound to the upper local dimension of the addition of two measures. We explore the special case where the two measures satisfy the convex additive finite-type condition that we introduce.

We introduce the notion of graph iterated function system. We show that we can always associate a self similar to the graph iterated function system.

MC 5417

Spiro Karigiannis, University of Waterloo

Organizational Meeting

MC 5403

Pavel Zatitskii, University of Cincinnati

Extremal problems and monotone rearrangement on averaging classes

We will discuss integral extremal problems on the so-called averaging classes of functions, meaning classes defined in terms of averages of their elements, such as BMO, VMO, and Muckenhoupt weights. A typical extremal problem we consider involves an integral inequality, such as the John--Nirenberg inequality for BMO. One common way to formulate such questions is using Bellman functions. It turns out that such Bellman functions are solutions to specific boundary value problems, formulated in terms of convex geometry. We will also discuss the monotone rearrangement operator acting on the averaging classes, which arises naturally in this context and is useful when solving extremal problems.

MC 5417

Alex Pawelko, University of Waterloo

Riemannian Geometry of Knot Spaces

We will review the construction of knot spaces of manifolds, specifically over G2 and Spin(7) manifolds. We will then see an explicit construction of the Levi-Civita connection of the knot space, and see what this can tell us about the torsion of the induced special geometric structures on knot spaces of G2 and Spin(7) manifolds.

MC 5403

Sergey Grigorian, University of Texas Rio Grande Valley

Geometric structures determined by the 7-sphere

The 7-sphere is remarkable not only for its rich topological and algebraic properties but also for the special geometric structures it encodes. In this talk, we explore how the symmetries and stabilizer subgroups of Spin(7) acting on the 7-sphere, regarded as the set of unit octonions, give rise to G2​-structures on 7-manifolds, SU(3)-structures on 6-manifolds, and SU(2)-structures on 5-manifolds. We will trace how these structures arise naturally via the inclusions of Lie groups and are reflected in the geometry of sphere fibrations. This perspective highlights the role of the 7-sphere as a unifying object in special geometry in dimensions from 5 to 8.

MC 5417