The Mathematical and Computational Sciences at Western are represented by four separate departments: Applied Mathematics, Computer Science, Mathematics, and Statistical & Actuarial Sciences. The Department of Mathematics has established research groups in several areas of contemporary mathematics including algebra, analysis and analytic geometry, homotopy theory, and noncommutative geometry.
Each week the Department of Mathematics hosts a wide variety of seminars and events. For a comprehensive list of events, please consult our departmental calendar.
This talk is based on the paper `` Dirac operators and Geodesic metric on Harmonic Sierpienski gasket and other fractals" by Lapidus and Sarhad.
First, the Sierpinski gasket will be introduced as the unique fixed point of a certain contraction on the set of compact subsets of the Euclidean plane. Then, by defining the graph approximation of the Sierpinski gasket, I will define the energy form on that space. I will talk about Kusuoka's measurable Riemannian geometry on the Sierpinski gasket and introduce counterparts of the Riemannian volume, the Riemannian metric and the Riemannian energy in that setting. Thereafter harmonic functions on the Sierpinski gasket will be introduced as energy minimizing functions. Using those functions we can define the harmonic gasket. I will also talk about Kigami's geodesic metric on the harmonic gasket. Using a spectral triple on the unit circle, a Dirac operator and a spectral triple for the Sierpinski gasket and the harmonic gasket will be constructed. Next, we will see that Connes' distance formula of noncommutative geometry which provides a natural metric on these fractals, is the same as the geodesic metric on the Sierpinski gasket and the kigami's geodesic metric on the harmonic gasket. It will be shown also that the spectral dimension of the Sierpinski gasket is the same as its Hausdorff dimension. Finally some conjectures about the harmonic gasket will be stated.
The derived category of coherent sheaves on a smooth projective variety is an important object of study in algebraic geometry. One import device relevant for this study is the notion of tilting sheaf. We first show the existence of tilting bundles on some Brauer-Severi schemes, and as an application, we construct tilting bundles on some arithmetic toric varieties.
Our old website has not been maintained for more than two years, and is no longer accessible from this page. If you require access to it, please email Stuart Rankin.