Title: A mathematical model to compare mass ivermectin treatment and vector control policies for river blindness (Onchocerciasis)
Abstract: In this talk, we investigate the dynamics of river blindness (Onchocerciasis) with an SEIR epidemiology model. Onchocerciasis is a parasitic disease caused by a filarial worm (Onchocerca volvulus) and transmitted through blackflies (Simulium Latreille). The transmission mechanism of the disease is mathematically modelled by a system of ODEs. We take the effects of mass drug policies (ivermectin), vector control strategies, and different treatment rates for exposed and infected individuals into account to assess the effectiveness of these methods. Mass drug policies, although they are safer, take considerably longer to wipe out the parasite. On the other hand, vector control strategies eliminate the transmission of the parasite rather quickly but result in mass human casualties. In light of our simulation results, we propose a strategy that uses a combination of both of these methods.
Title: Illumination number of cap bodies in small dimensions
Abstract: Despite recent progress on the Illumination conjecture, it remains open in general, as well as for specific classes of bodies. Bezdek, Ivanov, and Strachan showed that the conjecture holds for symmetric cap bodies (a specific class of convex bodies) in sufficiently high dimensions. Further, Ivanov and Strachan calculated the illumination number for the class of 3-dimensional centrally symmetric cap bodies to be 6.
In this talk/presentation, I will show that even the broader class of all 3-dimensional cap bodies has the same illumination number 6. In addition, I will show that the conjecture also holds for any n-dimensional cap bodies, where 3 < n < 20. The proof is based on probabilistic arguments and integer linear programming.
This talk is based on joint work with A. Arman and A. Prymak.
Title: Building the foundations for a new Ramsey-type problem
Abstract: Ramsey numbers have gained notoriety over the last century for the difficulty of computing exact values or even improving known bounds, and have given rise to a large family of Ramsey-type problems. The classical theorem of Ramsey can be rephrased in terms of independent sets: in large enough graphs, either the graph or its complement will have a large independent set.
In this talk, I present a natural analogous problem to that of the classical Ramsey numbers. Observing that an independent set is a special case of an acyclic set in a graph — that is, a set of vertices which induces no cycles — we instead study how in large enough graphs, either the graph or its complement will have a large acyclic set. In addition to presenting key theorems and constructions, I will discuss some small non-trivial exact values and best-known bounds.
Title: Development and Application of t-Distributed Stochastic Neighbourhood Embeddings
Abstract: t-Distributed Stochastic Neighbourhood Embedding (tSNE) is a technique developed in the past couple decades which has become a commonly employed data visualization tool for statisticians. The main problem of focus is on how to map high-dimensional data into two or three dimensions such that the distribution of how the transformed data is clustered is as close as possible to its original high-dimensional distribution. tSNE is a technique aimed at addressing this problem. This talk will focus on building up the relevant theory for the derivation of the technique complete with some applications of it.
Title: A single patch and Climate-Based Metapopulation Malaria Model with Human Travel and Treatment
Abstract: In this talk, we present single-patch and metapopulation models for malaria transmission that incorporate human movement between regions with varying climatic conditions, the use of effective and counterfeit treatments, and time-periodic parameters for mosquito dynamics. We develop an algorithm to compute the threshold condition $R_0$ for epidemiological systems derived from compartmental deterministic models. The analysis shows that when $R_0<1$, the disease-free equilibrium (DFE) is locally asymptotically stable, while $R_0>1$ leads to instability of the DFE. Under additional reasonable assumptions, we establish conditions under which the DFE is globally asymptotically stable, showing that, in many cases, this occurs precisely when $R_0≤1$.
Title: Computing Fourier Coefficients of Non-Holomorphic Eisenstein Series
Abstract: The classical (holomorphic) Eisenstein series with the form
\begin{equation*}
E_k(z) = \frac{1}{2} \displaystyle\sum_{\substack{m,n \in \mathbb{Z} \\ (m,n) \neq (0,0)}} (mz + n)^{-k}
\end{equation*}
for $z \in \mathbb{H}$ has the Fourier coefficients
\begin{equation*}
E_k(z) = \zeta(k) + \frac{(2\pi)^k (-1)^{k/2}}{(k - 1)!} \sum_{N=1}^\infty \sigma_{k-1}(N) q^N,
\quad \text{where } q = e^{2\pi i z}.
\end{equation*}
We see that the sum-of-divisors function $\sigma$ pops up in the non-constant coefficient part, which can be obtained via an elementary examination.
When we change our focus to the non-holomorphic Eisenstein series with the form
\begin{equation*}
E_s(z) = \displaystyle\sum_{\substack{c,d \in \mathbb{Z} \\ \gcd(c,d) = 1}} \frac{y^s}{|cz+d|^{2s}},
\quad \text{for } z \in \mathbb{H} \text{ and } \Re(s) > 1,
\end{equation*}
the coefficients inevitably change. More explicitly, their Fourier expansion is
\begin{equation*}
E_s(z) = y^s + \frac{\xi(2s-1)}{\xi(2s)} y^{1-s} + \frac{2y^{\frac{1}{2}}}{\xi(2s)} |m|^{s-\frac{1}{2}}
\sigma_{1-2s}(m) K_{s-\frac{1}{2}}(2\pi |m|y) e^{2\pi i mx}.
\end{equation*}
However, we keep seeing $\sigma$ in the non-constant coefficient part, and it comes from a more complicated but rigorous series of computations over local fields. In this talk, we will attempt to explain the background setup of these computations. Finally, if time permits, we will talk about the Whittaker functions that appear throughout our computations for a moment.