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.

DEVELOPMENT OF AN ADJUSTABLE SUTURE TECHNIQUE FOR TRABECULECTOMY

This research presents a new design of an adjustable suture that could provide a better intraocular pressure (IOP) control in the post treatment of trabeculectomy surgery and limit associated complication with the current suturing techniques. A better control in tension suture brings a great deal of advantages to this surgical technique compared with the traditional adjustable suture. A length adjustment can be added in advance to a 10-0 nylon suture which enables suture tension to be released during the postoperative period of trabeculectomy surgery. This adjustment has a D-ring geometry made of 10-0 nylon suture adhered to a 10-0 nylon surgical suture which is used to close the scalar flap. The D ring was adhered with about 180 microdroplet of Loctite 4311that was found to form a strong joint to connect the D ring to the main 10-0 nylon suture and strong enough to carry the added tension instead after cutting the central suture between the two joints of the D ring. The geometry of adjustment is the key factor of maintaining the IOP at the normal range and keeping the scleral flap tight enough and secure so that aqueous humor continues to percolate under the subconjunctiva. It has been found that a 365, and 450 µm length extensions can release suture tension postoperatively and relieve the intraocular pressure within the eye by 33, and 66% respectively. The fabrication process of the new adjustable suture was divided into two steps: fabrication of micro jig and forming microdroplets. A micro jig was fabricated in order to form and bond a precise length extension to the new design of the adjustable suture. In addition, a new liquid separation technique has been followed in this study in order to generate micro adhesive droplets as small as 50µm for bonding the new adjustable suture structure.