PhD Oral Exam - Almaz Butaev, Mathematics
When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.
Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.
Van Schaftinen  showed that the inequalities of Bourgain and Brezis ,  give rise to new function spaces that refine the classical embedding W1,n (Rn) ⊂ BMO(Rn). It was suggested by Van Schaftingen  that similar results should hold in the setting of bounded domains Ω ⊂ Rn for bmor (Ω) and bmoz (Ω) classes.
The first part of this thesis contains the proofs of these conjectures as well as the development of a non-homogeneous theory of Van Schaftingen spaces on Rn. Based on the results in the non-homogeneous setting, we are able to show that the refined embeddings can also be established for bmo spaces on Riemannian manifolds with bounded geometry, introduced by Taylor .
The stability of parabolic equations with time delay plays important role in the study of non-linear reaction-diffusion equations with time delay. While the stability regions for such equations without convection on bounded time intervals were described by Travis and Webb , the problem remained unaddressed for the equations with convection. The need to determine exact regions of stability for such equations appeared in the context of the works Mei on the Nicholson equation with delay .
In the second part of this thesis, we study the parabolic equations with and without convection on R. It has been shown that the presence of convection terms can change the regions of stability. The implications for the stability problems for non-linear equations are also discussed.