In this report the uniform deconvolution problem will be discussed. It is a statistical problem where the observations we would like to make are distorted by an independent additive noise sampled from a standard uniform distribution. The sampling density is therefore the convolution of the distribution function of interest and the standard uniform density. However, we would like to know the distribution of the data without the uniform convolution, or in other words, we want to infer the deconvolution of the distribution of the observables by the uniform density, based on the observable noisy data. We will first define this problem mathematically and find a connection between the distribution of the observed data and the distribution that we would like to know. Having constructed this relation, we aim to estimate the unknown distribution based the noisy observations. It turns out that this problem is related to the current status problem. This is a problem in which patients are tested for a disease to find out the time of onset of the disease. However, the observations only tell us whether patients are infected or not at the time of the test, but not the time of onset of the disease. We would like to know the times of onset as we can use these to estimate the distribution of the time of onset. To estimate this distribution, we will use the connection to the to the uniform deconvolution problem. Thus, when the relation between the distribution of the observations and the unknown distribution of the variables of interest is created, we would like to estimate the distribution function only using finitely many noisy observations. To estimate this, estimators like the MLE or MoM cannot be used as we would need to assume that the family of possible distributions can be parameterized smoothly by a Euclidean parameter. In our case, the family of possible distributions is much larger and does not satisfy this assumption, so we have to use nonparametric estimation methods. We will show two different methods of this nonparametric estimation; the nonparametric maximum likelihood estimator and the kernel density estimator. For the nonparametric maximum likelihood estimator, we first constructs the (log-)likelihood function of the problem and then we maximizes it over all possible distribution functions in the allowed family. This maximizing function is defined as the NPMLE. There is not a general method to obtain this distribution so we will have to use specific properties of our problem. Then we will use the method of isotonic regression to obtain the maximum likelihood estimator for the unknown distribution function. The other method that will be used to estimate the unknown distribution in a nonparametric way is kernel density estimation. This method constructs a distribution function more directly than the NPMLE. The intuition behind this estimator is simple, if there are a lot of observations near a value, the probability density in that value should be high, and the other way around. Finally, we will simulate the uniform deconvolution problem and solve it using the two discussed methods. The methods and their respective strengths and limitations will be compared. The kernel density estimator turns out to give a more smooth and accurate estimate but depends on an optimally chosen parameter which, in practice, cannot be calculated exactly and differs from case to case. The nonparametric maximum likelihood estimator does not depend on any parameter and the way of calculating it is the same in every case, but is only able to produce step functions.

This report introduces spherical harmonics, functions defined on the surface of a sphere that play a central role in mathematical analysis, especially in problems with spherical symmetry. They appear in many fields, such as 3D representation within computer graphics, simulation light behaviour and angular momentum within quantum mechanics.

We begin by developing the theory from first principles. We look at what homogeneous harmonics polynomials are and explain how spherical harmonics arise by restricting these polynomials to the unit sphere. Using this we discuss properties such as orthogonality and dimension. We also discuss zonal harmonics, which are symmetric around a chosen axis.

In three dimensions, we solve Laplace’s equation in spherical coordinates to derive explicit formulas for spherical harmonics. Associated Legendre polynomials will play a key role here. This directly connects with angular momentum, which will also be looked at in this report. This report aims to give students an introduction on spherical harmonics and how they can be used.

The Union-Closed Sets Conjecture (UCSC), posed by Peter Frankl in 1979, asserts that every finite union-closed family of sets contains an element that appears in at least half of its member sets. Despite this seemingly simple formulation, the conjecture has remained unresolved for decades. Recently in November 2022 Gilmer developed a novel approach based on the information theoretic concept: \emph{entropy}, which has offered fresh insights and promising partial results. In this thesis, we provide a self-contained introduction to entropy and explore its utility in combinatorics, specifically its application to the UCSC. We thoroughly examine the groundbreaking entropy-based proof that establishes a constant lower bound of \(\frac{3 - \sqrt{5}}{2}\approx0.382\). Additionally, we shortly discuss the small further improvements made to this bound. We then consider a related conjecture proposed by Nagel, which states: \textit{the \(k\)th most frequent element in a union-closed family appears in at least a fraction \(\frac{1}{2^{k-1} +1}\) of the sets.} We shall discuss how the new entropy approach could also be used there to improve the bounds on the sizes of the family.
All these explorations have been focused on understanding recent developments and to create a unified narrative. Due to time constraints, the thesis did not pursue new results. However, the comprehensive understanding gained has given some promising directions for future research.

Sophus Lie (1842-1899) known as the founder of the theory of transformation groups, originally aimed to study solutions of differential equations via their symmetries. Over the decades this theory has evolved into the theory of Lie groups. These Lie groups are of an analytic and geometric nature, but Sophus Lie's principal discovery was that these groups can be studied by their "infinitessimal generators" leading to a linearization of the group. The group structure endows this linearized space with a special bracket operation, [x,y]=xy-yx, which gives rise to Lie algebras.

The main applications for Lie algebras stem from physics, notably in quantum mechanics and particle physics. It turns out that representations of Lie algebras are the way to describe symmetries of physical systems. So, it becomes an important task to figure out what all the possible representations are. Thus, our main goal for this thesis is to classify all finite-dimensional semisimple Lie algebra representations.