Jon Xu（University of Melbourne）「Chevalley groups and finite geometry」

This work is the result of an effort to create "interdisciplinary" communication and collaboration between the finite geometry community and the representation theory communities in Australia. The idea was that Chevalley groups could be a bridge between the two languages and the different problems of interest in the two communities.

The first result of the thesis is the realisation of key examples of ovoids ('smooth, thin' objects coming from finite geometry) as the points of a flag variety of a suitably chosen (twisted) Chevalley group (coming from representation theory). The precedent in the work of Tits and Steinberg on the Suzuki-Tits ovoid indicated that this was a fruitful research direction.

We then define an incidence structure for each Schubert cell and each pair of maximal parabolic subgroups of a Chevalley group. This provides a way of analyzing the Schubert cell using the viewpoint of finite projective geometry. Then, in pursuit of the question of what causes the "thinness" that distinguishes ovoids, we prove our second result of the thesis, which is a computation of the "thickness" of the incidence structures that come from Schubert cells.

This work is based on my PhD thesis is supervised by Arun Ram (University of Melbourne) and cosupervised by John Bamberg (University of Western Australia).