CMU Mathematics Department (HyFlex/Virtual) Colloquia

2021 -- 2022

Organizer

• Department of Mathematics Colloquium Coordinator C-Y. Jean Chan

• Because there might be a time delay in updating the webpage, please always check with the Coordinator for the available dates.

Meeting Times and Platforms

Typical Colloquium Talks are Thursday, 4:00–4:50pm, in virtual or HyFlex format. Media is to be announced.
The following table gives the information for each Thursday event, including Department-wise Meetings (3:30pm), and courtesy book keeping for Tuesday department-wise meetings (3:30pm).

Schedule (in reversed chronological order)

Abstracts

Speaker: Chase Bender (October 21, 2021)
Title: Rigid Motions and Rotations
Abstract: We classify the isometries of Euclidean spaces, the so-called rigid motions, using elementary methods of linear algebra. We briefly discuss the group structure of the rigid motions, namely that this group is the semidirect product $\mathbb{R}^n \rtimes O(n)$. We then consider rotations about the origin in two and three dimensions as subgroups of rigid motions. In particular, we show that a 2-dimensional rotation corresponds to a matrix of the form \[ \begin{pmatrix} \cos\theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \] and we prove Euler's rotation theorem that a 3-dimensional rotation is a 2-dimensional rotation in a plane about its normal line called the axis of rotation. Finally, we prove a version of Rodrigues' rotation formula which gives each rotation as a composition of two reflections in planes.

Speaker: Rich Fleming (November 4, 2021)
Title: The Quotient Lifting Property
Abstract: A variation on the standard lifting property leads to some interesting consequences. We say that for a closed subspace $J$ of a Banach space $X$, the pair $(X,J)$ has the quotient lifting property (QLP) if for any bounded linear operator $S$ from a Banach space $Y$ to $X/J$, there is a bounded operator $T$ on $X/J$ to $X$ such that $\pi \circ T = S$ and $\|T\| = \|S\|$, where $\pi$ is the usual quotient map. This property implies that $J$ is proximinal and the metric projection $P_J$ on $X$ to $J$ has a linear selection. Just as the standard lifting property characterizes $\ell^1$ spaces, the QLP for every pair $(X,J)$ is shown to characterize Hilbert spaces of dimension greater than two.

Speaker: Rasha Almughrabi (November 9, 2021)
Title: Introduction to Bergman Spaces
Abstract: The modern subject of Bergman spaces is a blend of complex function theory with functional analysis and operator theory. The main goal of this talk is to introduce the basic definitions and notations for Bergman space as a subspace of all square integrable functions defined on some domain Ω⊂ℂ𝑛. We prove some properties of Bergman spaces, such as the Bergman inequality, and being a Hilbert space. The latter property guarantees the existence of the Bergman kernel. Bergman Kernel is a function on Ω×Ω which has interesting characteristics, but the most important characteristic of the Bergman kernel function is its reproducing property.

Speaker: Chase Bender (November 23, 2021)
Title: Quaternions and Rotations
Abstract: We begin with a discussion of the basic properties of the \emph{quaternions} $\mathbb{H}$; the algebra of formal combintations $a + bi+cj+dk$ where \[i^2 = j^2 = k^2 = ijk = -1 \] for all real numbers $a, b, c, d$ in $\mathbb R$. These properties include the geometric interpretation of the quaternionic product, the fact that $H$ is a real division algebra, and that the Euclidean norm is multiplicative (as in the real and complex cases). We then realize the unit 3-sphere in $\mathbb{H}$ about the origin as a compact multiplicative group $\operatorname{Sp}(1)$ known as the symplectic group of degree 1. This group turns out to be the double cover of the rotation group $\operatorname{SO}(3)$. We then classify all finite subgroups of $\operatorname{SO(3)}$ by adopting a classical argument of Hermann Weyl, and use this to determine all finite multiplicative groups of quaternions.

Speaker: Rasha Almughrabi (December 7, 2021)
Title: Bergman spaces and Bergman Kernels
Abstract: This talk is a continuation for what we covered in the first part. We start by characterizing Bergman kernels and show that a function belonging to a Bergman space and satisfying the reproducing property and being conjugate symmetric must be the Bergman Kernel for that space. Then we will prove that the Bregman kernel of the product of two domains equals the product of the kernels of each domain. This property is essential to calculate Bergman kernel of higher dimensional domains. We will also give a representation any kernel in terms of a given orthonormal basis and using this formula to compute the Bergman kernel of the unit disc. Then, we will introducing one of the most interesting facts called “transformation property” which connects Bergman kernels of biholomorphic domains.

Speaker: Rabeya Basu (February 22, 2022)
Title: On General Quadratic Groups
Abstract: In 1966/67 A. Bak introduced the concept of form rings and form parameters to give a uniform definition of classical type groups. In this talk we will discuss the definition, examples and a few applications in view of classical algebraic K-theory.

Speaker: Jordan Gill (March 3, 2022)
Title: Transitionary Periods, Attributions, and Their Effect on Mathematical Beliefs
Abstract: In this presentation we will discuss how periods of transition can affect student’s mathematical beliefs, particularly their mathematical mindset and their mathematics self-efficacy. Students undergoing two different types of transitions (from high school to college and from less to more advanced mathematical topics) were interviewed on how their beliefs had been affected by the transitions. Results showed that the attributions students made for their results (either helpless or effort-based) coincided with the shifts in MSE and mindset that were observed. Specifically, for students who experienced misaligned performance results (e.g. an exam score that did not match MSE), effort-based attributions resulted in shifts to higher MSE (for those who experienced success) or maintaining high MSE (for those who experienced failure). Conversely, students who made helpless attributions experienced lowered levels of MSE but preserved their initial mindset, whether it was growth or fixed.

Past Department Colloquia: Spring 2020/Fall 2019 Spring 2019 Fall 2018 Spring 2018 Fall 2017 Spring 2017 Fall 2016

Look for Colloquia Archives for past activities