Sheaves as G intended
18/03/24, University of Manchester (Student Algebra Seminar)
In 1949, Andre Weil published the Weil conjectures, a series of problems which Weil suggested could be resolved with an appropriate notion of "Cohomology" of Algebraic Varieties. Sheaf Cohomology seemed the obvious way to proceed, apart from one problem; algebraic varieties don't have enough open sets to make sheaves well behaved! So what if you "added" more open sets?...
Advanced Hole Counting
09/02/24, University of Manchester (Pure Postgraduate Seminar)
The art of hole counting is an ancient and profound one... a circle? One hole. The plane? No holes. It's nearly too easy... But how far can we push hole counting? What even is a hole in \(\overline{\mathbb{F}}_5[X,Y] / \langle X^2 + Y^2 -1 \rangle \) ?
Hochschild Cohomology
15/12/23, University of Manchester
What is Hochschild Cohomology? And why should we care? This talk is an hour long introduction to Hochschild Cohomology done for a reading group; the transcript can be found here.
Spectral Sequences
03/11/23, University of Manchester
This talk is an attempt to motivate how you could come about spectral sequences yourself, through trying to filter a the cycles and boundaries in a double complex.
A brief introduction to Hecke Algebras
06/10/23, University of Manchester (Pure Postgraduate Seminar)
A 5 minute summary of the talk below.
Hecke Algebras of \(\operatorname{GL}_n(\mathbb{Q}_p) \)
10/03/23, University of Oxford
Representations of p-adic groups are the bread (butter lacking) of the Local Langlands Correpondence; in understanding them, we are naturally lead to beautiful objects called Hecke Algebras. Here we give a brief overview of the theory of Hecke Algebras in relation to representation theory, building up to the Iwahori Decomposition of \( \operatorname{GL}_n(\mathbb{Q}_p)\)
