Pure Honours talks (T2-2024)

Wednesday, 31-July-2024 

1. Sobolev Space and Existence Theorems of Weak Solution of Elliptic PDE

Let $A$ be an $n\times n$ matrix and $x$ is an n-dimensional vector, $c$ be an n-dimensional vector. In the first-year linear algebra course, we already know the equation $Ax=c$ admits a unique solution if and only if the homogeneous equation $Ax=0$ only has a trivial solution. Do we have a similar result for Second Order Elliptic Partial Differential Equations?  Yes, we have, which is the Second Existence Theorem of Weak Solutions.

Weak solutions are defined as elements in the Sobolev space, in our talk, we will 

1. Introduce the Sobolev space and define the weak solutions of Elliptic PDEs. 

2. Then use the Lax-Milgram Theorem to prove the First Existence Theorem of Weak Solutions. 

3. Finally, we will use the Rellich-Kondrachov Compactness Theorem and Theory of Compact Operators to show the Second Existence Theorem of Weak Solutions.

2. Constructing the Hilbert scheme of points as a Nakajima quiver variety.

Hilbert schemes are fundamental to the study of moduli spaces and classification theory in algebraic geometry. However, it is not clear from the definition what geometric structure Hilbert schemes have. In this talk, the Hilbert scheme of points is constructed as a Nakajima quiver variety, relying on both geometric invariant theory and the moment map to do so. This results in a comparatively simple proof that the Hilbert scheme of points is a smooth variety. Thus, the general construction of quiver varieties from representations of double quivers is illustrated, as well as how they can be used in resolving singularities.