Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.

ROLE OF STAT1 AND CXCL10 IN THE TUMOUR IMMUNE MICROENVIRONMENT OF HIGH-GRADE SEROUS OVARIAN CANCER

High-grade serous ovarian cancer (HGSC) is the most prevalent epithelial ovarian cancer characterized by late detection, metastasis and resistance to chemotherapy. Previous studies on the tumour immune microenvironment in HGSC identified STAT1 and CXCL10 as the most differentially expressed genes between treatment naïve chemotherapy resistant and sensitive tumours. Interferon-induced STAT1 is a transcription factor, which induces many genes including tumour suppressor genes and those involved in recruitment of immune cells to the tumour immune microenvironment (TME), including CXCL10. CXCL10 is a chemokine that recruits tumour infiltrating lymphocytes (TILs) and exhibits angiostatic function. The current study was performed to determine the effects of differential STAT1 and CXCL10 expression on HGSC disease progression and TME. STAT1 expression and intratumoural CD8+ T cells were evaluated as prognostic and predictive biomarkers via immunohistochemistry on 734 HGSC tumours accrued from the Terry Fox Research Institute-Canadian Ovarian Experimental Unified Resource. The combined effect of STAT1 expression and CD8+ TIL density was confirmed as prognostic and predictive companion biomarkers in the second independent biomarker validation study. Significant positive correlation between STAT1 expression and intratumoral CD8+ TIL density was observed. The effects of enforced CXCL10 expression on HGSC tumour growth, vasculature and immune tumour microenvironment were studied in the ID8 mouse ovarian cancer cell engraftment in immunocompetent C57BL/6 mice. Significant decrease in tumour progression in mice injected with ID8 CXCL10 overexpressing cells compared to mice injected with ID8 vector control cells was observed. Multiplexed cytokine analysis of ascites showed differential expression of IL-6, VEGF and CXCL9 between the two groups. Endothelial cell marker staining showed differences in tumour vasculature between the two groups. Immune transcriptomic profiling identified distinct expression profiles in genes associated with cytokines, chemokines, interferons, T cell function and apoptosis between the two groups. These findings provide evidence that STAT1 is an independent biomarker and in combination with CD8+ TIL density could be applied as novel immune-based biomarkers in HGSC. These results provide the basis for future studies aimed at understanding mechanisms underlying differential tumour STAT1 and CXCL10 expression and its role in pre-existing tumour immunologic diversity, thus potentially contributing to biomarker guided immune modulatory therapies.

An impulsive differential equation model for Marek's disease

Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are referred to as impulsive differential equations, systems of differential equations coupled with discrete mappings in state space. In this thesis we employ impulsive differential equations to model disease transmission within an industrial livestock barn. In particular we focus on the poultry industry and a viral disease of poultry called Marek's disease. This system lends itself well to impulsive differential equations. Entire cohorts of poultry are introduced and removed from a barn concurrently. Additionally, Marek's disease is transmitted indirectly and the viral particles can survive outside the host for weeks. Therefore, depopulating, cleaning, and restocking of the barn are integral factors in modelling disease transmission and can be completely captured by the impulsive component of the model. Our model allows us to investigate how modern broiler farm practices can make disease elimination difficult or impossible to achieve. It also enables us to investigate factors that may contribute to virulence evolution. Our model suggests that by decrease the cohort duration or by decreasing the flock density, Marek's disease can be eliminated from a barn with no increase in cleaning effort. Unfortunately our model also suggests that these practices will lead to disease evolution towards greater virulence. Additionally, our model suggests that if intensive cleaning between cohorts does not rid the barn of disease, it may drive evolution and cause the disease to become more virulent.