Honours presentations, in order, of Elie Sikh, Abdellah Islam, Daniel Czapski, and Godfrey Wong.
Note that we changed the order of the talks from the previously announced program.
Each talk is 20 minutes long followed by 5 minutes of questions and by 5 minutes break.
We will start at 2:05.
Speaker: Elie Sikh
Title: Intersection Theory from a Differential Viewpoint
Abstract: Intersection theory is one of the main branches of algebraic geometry and refers to the study of the intersections of subvarieties. We choose to showcase intersection theory through the lens of differential topology, allowing us to develop a geometric intuition for the underlying mathematics. Through this approach we see how local differential properties are ultimately linked to the global topological properties of the object being studied. We also explore the interesting case of analytic subvarieties, where the complex structure restricts the possible types of intersections. We focus on presenting results of Hirsch, and Griffith and Harris in a manner approachable for undergraduate students seeking to get a feel for the topic.
Speaker: Abdellah Islam
Title: Toric Varieties
Abstract: The study of algebraic geometry investigates properties of varieties such as their singularities and their (co)homologies. The family of varieties which contain an embedded torus as an open dense subset, named toric varieties, give explicit examples of varieties to test hypotheses and theorems. We outline the construction of toric varieties and give geometric examples.
Speaker: Daniel Czapski
Title: Asymptotic eigenvalue distributions of Gaussian Hermitian random matrices
Abstract: Early elements of random matrix theory can be seen in the works of Wishart on random covariance matrices, dating to the early 1920s. The study of random matrices, however, began in earnest with the works of Eugene Wigner from the 1950s on the application of random Hermitian matrices to problems in atomic and nuclear physics. The statistical behaviour of the eigenvalues of randomly chosen matrices – spectral statistics – is a key problem that has seen significant development since that time and remains an active area of research. The first three major results, proved primarily by Dyson and Wigner in the 1950s and 1960s, relate to the “average” distribution of eigenvalues of N-dimensional Hermitian matrices, their asymptotic distribution as N increases without bound, and the distribution of eigenvalues of Hermitian matrices of very large dimension. We review these three major results, briefly discuss a classical method of proof and touch on both generalisations and current research directions.
Speaker: Godfrey Wong
Title: Permutation polynomials over finite fields: Construction and statistics
Abstract: Permutation polynomials are polynomials over the finite field F_q of q elements that permute F_q. Every permutation polynomial of degree at most 6 has been classified. However, there is no coherent theory on the construction of permutation polynomials of any degree yet. For example, finding the exact and even an asymptotic number of permutation polynomials of a given degree n < q-1 still an open problem.
We will study the construction of permutation polynomials and then give a new lower bound on the number of permutation polynomials of large degree. We will then extend the result to permutation rational functions, which are rational functions over F_q that permutes F_q union infinity.