# Postgraduate Seminar

Hello!

Welcome to the Warwick Mathematics Postgraduate Seminar, where graduate students share the outcomes of their research to their peers.

This term, all talks will be held in **B3.02 at 12 noon on Wednesday** (except when stated otherwise). The seminar will take a hybrid format so that students can join us virtually. This is the linkLink opens in a new window to join the seminar virtually.

For those who will join us in person, we will provide at the end of the seminar a light lunch :)

Do you want to give a talk in this seminar? This is what you have to doLink opens in a new window .

If you have any question, do not hesitate to get in contact with us! This seminar is organised by Alvaro Gonzalez HernandezLink opens in a new window and Katerina SanticolaLink opens in a new window.

## Term 1 - Year 2022 - 2023

###### Week 1: Wednesday 5th October

##### Sunny SoodLink opens in a new window - Homological stability for $O_{n,n}$

Motivated by Hermitian K-Theory, we study the homological stability of the split orthogonal group $O_{n,n}$.

Specifically, let $R$ be a commutative local ring with infinite residue field such that $2 \in R^{*}$. We prove that the natural homomorphism $H_{k}(O_{n,n}(R) ; \mathbb{Z}) \rightarrow H_{k}(O_{n+1,n+1}(R); \mathbb{Z})$ is an isomorphism for $k \leq n-1$ and surjective for $k \leq n$.

This will be an excellent opportunity to introduce esoteric concepts such as group homology and hyperhomology spectral sequences at the postgraduate seminar.

This is all joint work with my supervisor Dr Marco Schlichting.

###### Week 2 : Wednesday 12th October

##### Paul PanteaLink opens in a new window - Keeping exotic spheres as pets

TBA

###### Week 3 : Wednesday 19th October

##### Ruzhen Yang - Beilinson spectral sequence and its reverse problems on $\,\mathbb{P}^2$

Derived category is widely accepted as the natural environment to study homological algebra. We will study the structure of the bounded derived category of coherent sheaves on projective space via the semi-orthogonal decomposition (based on the Beilinson's theorem) and comparison (by a theorem by A. Bondal).

As an example we will give explicit free resolutions of some sheaves on $\,\mathbb{P}^2$ using the Beilinson spectral sequence. We will also discuss the reverse problem where we give a condition to when the complex given by the spectral sequence is a resolution of the ideal sheaf of three points.