# Honours Seminar

**Talk 1 (Benjamin Leedom, 4pm): The space of endomorphisms of a vector space.**

Abstract: Using the example of endomorphisms of a vector space, we will explore the mathematics of moduli spaces, the difference between algebraic and topological notions of what these moduli spaces are - and how to work on reconciling them.

**Talk 2 (Hongzhi Liao, 4:30pm): Semismooth Netwon method and the relevant application in optimization problems.**

Abstract: Optimization is a significant part of opsearch. The task of optimization is to find which points can let the objective functions achieve their minimum of maximum value, if these points do exist. If an objective function is differentiable, we can use Newton method to achieve its minimizer. Meanwhile, Newton method converge to minimizer very fast, it has quadratic convergence rate. Thus, Newton method is quite useful in real problems. However, we cannot use Newton method in non-differentiable function directly. To solve the problem, we need to generalize the definition of differential, include B-subdifferential and C-subdifferential. With these new defined concepts, we can generalize Newton method to semismooth Newton method. The semismooth Newton method can be used to find minimizers of semismooth functions (semismooth functions also needs to be defined). Although semismooth functions do not have the same good property as differentiable functions, semismooth Newton method can also find minimizers of semismooth functions with superlinear convergence rate. It is also very fast and practical. Besides, not only in finite dimensions, semismooth Newton method can even be used to infinite-dimensional functions. This may can solve some non-discrete problems. When semismooth Newton method is established, we can use it to solve many real problems, such as picture denoise, inverse problems and determining parameters in some PDE equations and show its advantages comparing with other optimization methods.

**Talk 3 (Jake McCall, 5pm): Equivariant topology and discrete geometry **

Abstract: A brief overview of how to model (and solve) discrete geometry problems using equivariant topology.

**Talk 4 (Roger Murray, 5:30pm): An Introduction to Elliptic Curves**

Abstract: An elliptic curve is a defined (naively) as the set of points satisfying a simple polynomial equation. Mathematicians have been fascinated by elliptic curves for nearly 2000 years, and in the past two centuries the theory of these curves has been expanded immensely. In this talk I aim to give an introduction to elliptic curves.