Finite groups and geometries: a view on the present state and on the future

The model: groups of Lie-Chevalley type and buildingsThis paper is not the presentation of a completed theory but rather a report on a search progressing as in the natural sciences in order to better understand the relationship between groups and incidence geometry, in some future sought-after theory Τ. The search is based on assumptions and on wishes some of which are time-dependent, variations being forced, in particular, by the search itself.A major historical reference for this subject is, needless to say, Klein's Erlangen Programme. Klein's views were raised to a powerful theory thanks to the geometric interpretation of the simple Lie groups due to Tits (see for instance), particularly his theory of buildings and of groups with a BN-pair (or Tits systems). Let us briefly recall some striking features of this.Let G be a group of Lie-Chevalley type of rank r, denned over GF(q), q = pn, p prime. Let Xr denote the Dynkin diagram of G. To these data corresponds a unique thick building B(G) of rank r over the Coxeter diagram Xr (assuming we forget arrows provided by the Dynkin diagram). It turns out that B(G) can be constructed in a uniform way for all G, from a fixed p-Sylow subgroup U of G, its normalizer NG(U) and the r maximal subgroups of G containing NG(U).

Impact of tumor board recommendations on treatment outcome for locally advanced head and neck cancer

Background/Aims: To identify physician selection factors in the treatment of locally advanced head and neck cancer and how treatment outcome is affected by Tumor Board recommendations. Methods: A retrospective analysis of 213 patients treated for locally advanced head and neck cancer in a single institution was performed. All treatments followed Tumor Board recommendations: 115 patients had chemotherapy and radiation, and 98 patients received postoperative radiation. Patient characteristics, treatment toxicity, locoregional control and survival between these two treat- ment groups were compared. Patient survival was compared with survival data reported in randomized studies of locally advanced head and neck cancer. Results: There were no differences in comorbidity factors, and T or N stages between the two groups. A statistically significant number of patients with oropharyngeal and oral cavity tumors had chemoradiation and postoperative radiation, respectively (p < 0.0001). Grade 3-4 toxicities during treatment were 48 and 87% for the postoperative radiation and chemoradiation groups, respectively (p = 0.0001). There were no differences in survival, locoregional recurrences and distant metastases between the two groups. Patient survival was comparable to survival rates reported by randomized studies of locally advanced head and neck cancer. Conclusion: Disease sites remained the key determining factor for treatment selection. Multidisciplinary approaches provided optimal treatment outcome for locally advanced head and neck cancer, with overall survival in these patients being comparable to that reported in randomized clinical trials. Copyright © 2008 S. Karger AG.