Comprehensive Exam Presentation

Speaker: Nadia Alluhaibi (Western)

" the geometry of complex hyperbolic and automorphic forms of the unit ball quotients"

Time: 10:00

Room: MC 107

I will describe the ball model and the Siegel space model of the n-dimensional complex hyperbolic space H^n_C. The matrix group SU(n,1) acts on H^n_C. The group of holomorphic isometries of H^n_C is PU(n,1).

Let \Gamma be a discrete subgroup of SU(n,1) which acts freely and properly discontinuously on H^n_C. I will give the definition of an automorphic form for \Gamma. I will talk about constructing automorphic forms associated to certain submanifolds of H^n_C / \Gamma .

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Singular Levi-flat hypersurfaces and holomorphic webs (Part II)"

Time: 15:30

Room: MC 107

Levi flat hypersurfaces are characterized by vanishing of the Levi form on them. Their regular part is foliated by complex hypersurfaces, this is called the Levi foliation. It is an open question whether one can extend this foliation to the ambient space. As example of Brunella shows, near singular points the extension may exist in general only as a singular holomorphic web. A smooth holomorphic web is simply the union of several foliations. A singular web is a more general object which loosely can be thought of as a foliation with branching. In this talk I will give a detailed background concerning holomorphic webs, and will discuss some recent progress on extension of the Levi foliation. This is joint work with A. Sukhov.

Index Theory Seminar

Speaker: Matthias Franz (Western)

"Representations of compact connected Lie groups II"

Time: 11:00

Room: MC 108

This time we discuss highest weights, various constructions of irreducible representations and the Weyl character formula.

Comprehensive Exam Presentation

Speaker: Sajad Sadeghi (Western)

"Dirac Operators and Geodesic Metric on the Sierpinski Gasket and the Harmonic Gasket"

Time: 11:30

Room: MC 107

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.