## Projects

Below you will find a potential list of projects and the instructions for the paper to write. The paper is due by May, 9th.

### Description of the project

You have to write a paper of 5-6 pages (excluding the bibliography) on one of the subjects in the list below. To write your paper, you will have to use the amsart document class to write your paper. You will set the margins at 2 centimeters and the font at 12pt. You will also need to find the right AMS Mathematics Subjects Classification for your paper. For the content, please check the evaluation grid.

If you want some tips on writing a good paper, then I recommend the following ressources:

### Spectrum of an Element in a $$\mathbf{C^\ast}$$-Algebra (Lynette and Aleksander)

The aim of this project is to generalize the concept of the ''spectrum'' to an element in a $$C^\ast$$-Algebra. Then, the second goal is to prove that if $$a$$ is an element of a $$C^\ast$$-Algebra, then its spectrum $$\sigma (a)$$ is non-empty. This proof involves, as we would suspect due to the use of the Fundamental Theorem of Algebra in the matrix case, complex analysis methods.

Main references:

• W. Rudin, Real and Complex Analysis.

### Norlund Summability Methods

In this project, you will study the Norlund methods of summability. The goal will be to prove the inclusion theorem between two Norlund summability methods.

Main references:

• J. Boos, Classical and Modern Methods in Summability, section 3.3.

### Power Series Summability Methods

In this project, you will study a type of summability methods: the power series summability methods. The goal will be to give a prove of the inclusion theorem between two power series summability methods $$P$$ and $$Q$$.

Main references:

• J. Boos, Classical and Modern Methods in Summability, section 3.6.

### Methods of Sonnenschein and Karamata Matrices

The goal of this project is to describe the Sonnenschein method of summability and the Karamata Matrix. You will present the conditions for the summability method of Karamata to be regular.

Main references:

• J. Boos, Classical and Modern Methods in Summability, section 3.5.
• W. T. Sledd, Regularity conditions for Karamata matrices.

### The Hardy Space $$H^2$$ of The Unit Disk (Dennis)

For an analytic function $$f$$ on the unit disk, define $$\sup_{0 < r < 1} \int_0^{2\pi} |f(r e^{i\theta})|^2 \, d\theta < \infty .$$ Analytic functions in the unit disk satisfying the above constraint are said to be in the Hardy space $$H^2$$. In this project, you will present this space and show some of its important properties, such as the existence of radial limits almost everywhere. [Comment: Some knowledge in Measure Theory is required to fully appreciate this project.]

Main references:

• K. Hoffman, Banach Spaces of Analytic Functions.
• P. L. Duren, Theory of $$H^p$$ Spaces.

### Abel's Limit Theorem (taken)

The goal of this project is to prove the Abel's limit Theorem.

Main Reference:

• Section 3.4 of Complex Analysis by Donald Marshall.

### Pseudohyperbolic and Hyperbolic (Poincaré) Metrics (Kawika and Sam Miller)

The goal of this project is to introduce the pseudohyperbolic and hyperbolic metrics on the unit disk $$\mathbb{D}$$. You will describe its properties

Main references:

• Lecture notes (Mainly Chapter 3).
• Exercise 3.9 (D. Marshall, Complex Analysis).
• S. R. Garcia, J. Mashreghi, W. T. Ross, Finite Blaschke Products and Their Connections, Section 2.1 and Section 2.3.

### Solving the Dirichlet Problem (Udani)

The goal of this project is to present the solution of the Dirichlet problem for the unit disk. More precisely, you will solve the following problem: given $$f$$ continuous on the unit circle, find a harmonic function $$u$$ on the closure of the unit disk such that $$\left\{ \begin{matrix} \Delta u = 0 & \text{ in } \mathbb{D} \\ u (e^{i\theta}) = f(e^{i\theta}) & \text{ for } \theta \in [0, 2\pi ] \end{matrix} \right.$$

Main reference(s):

• J. B. Conway, Functions of One Complex Variable I, (section X.2).
• D. Marshall, Complex Analysis (section 13.1)
• J. B. Garnett, D. Marshall, Harmonic Measure, (section I.1).
• J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering (section 12.2).

### Julia's Lemma (Taken)

In this project, you will prove Julia's Lemma and show some of its consequences.

Main reference(s):

• S. R. Garcia, J. Mashreghi, W. T. Ross, Finite Blaschke Products and Their Connections, Section 1.6.

### Newton's Method for solving $$f(z) = 0$$ (Sam Glickman)

You will describe the Newton's method for analytic functions $$f$$ in a neighborhood of some point $$a$$.

Main references:

• Exercise 2.15 (D. Marshall, Complex Analysis).
• K. Falconner, Fractal Geometry Mathematical Fundations and Applications, section 14.5.

### Iterations of Rational Functions (Christine)

In this project, you will study the famous sets in complex dynamics: the Julia, Fatou, and Mandelbrot sets. The goal of the project will be to prove that, for a polynomial $$p$$, the Julia set $$J (p)$$ is the closure of the repelling periodic points of $$p$$.

Main reference(s):

• K. Falconner, Fractal Geometry Mathematical Fundations and Applications, section 14.1.

### The $$Z$$-Transform (Alan)

In this project, you will explore the $$Z$$-transform of a sequence $$x_n$$. You will prove some of its properties and, if you want, show how it is used to solve difference equations.

Main reference(s):

• J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering.

### Analytic functions in two complex variables (Taken)

In this project, you will explore the extension of complex analysis to regions in $$\mathbb{C}^2$$.

### Bicomplex Analytic Functions

In this project, you will introduce a new number system, the bicomplex numbers. Then, you goal will be to extend the definitions of analytic functions to bicomplex numbers.

Main reference(s):

• B. Price, Multicomplex Spaces and Functions.

### Quaternionic Analytic Functions

The objective of this project is to show that the definition of the derivative with the difference quotient doesn't give rise to a rich theory of analytic functions.

The second objective is to explore another point of view, for example using the Cauchy-Riemann equations.

A starting point: