Mohammad Alattar

• Doctor of Philosophy
• PhD Student at Durham University

We obtain the Lipschitz analogues of the results Perelman used from Siebenmann's deformation of homeomorphism theory in his proof of the stability theorem. Consequently, we obtain the Lipschitz analogue of Perelman's gluing theorem. Moreover, we obtain the analogous deformation theory but with tracking of the Lipschitz constants.

Under Gromov–Hausdorff convergence, and equivariant Gromov–Hausdorff convergence, we prove stability results of Wasserstein spaces over certain classes of singular and non-singular spaces. For example, we obtain an analogue of Perelman’s stability theorem on Wasserstein spaces.

Under Gromov--Hausdorff convergence, and equivariant Gromov--Hausdorff convergence, we prove stability results of Wasserstein spaces over certain classes of singular and non-singular spaces. For example, we obtain an analogue of Perelman's stability theorem on Wasserstein spaces.

We give applications of equivariant Gromov-Hausdorff convergence in various contexts. Namely, using equivariant Gromov-Hausdorff convergence, we prove a stability result in the setting of compact finite dimensional Alexandrov spaces. Moreover, we introduce the notion of an almost commutative diagram and show that its use simplifies both exposition...

We give applications of equivariant Gromov-Hausdorff convergence in various contexts. Namely, using equivariant Gromov-Hausdorff convergence, we prove a stability result in the setting of compact finite dimensional Alexandrov spaces. Moreover, we introduce the notion of an almost commutative diagram and show that its use simplifies both exposition...