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The Department of Mathematics and Statistics invites you to attend the Ph.D. Oral Defense ofÌýMr. Omid Makhmali

THESIS TITLE:
Differential geometric aspects of causal structures

ABSTRACT
This thesis is concerned with causal structures, which are defined as a field of tangentially nondegenerate
projective hypersurfaces in the projectivized tangent bundle of a manifold. The localÌýequivalence problem
of causal structures on manifolds of dimension at least four is solved using Cartan’s method of equivalence,
leading to an {e}-structure over some principal bundle. It is shown that these structures correspond to
parabolic geometries of type (Dn,P1,2) or (Bn,P1,2) when n ≥ 4 or (D3,P1,2,3). The essential local invariants
are determined and interpreted geometrically. Several special classes of causal structures are considered
including those that are a lift of pseudo-conformal structures and those referred to as conformally flat causal
structures. A rigidity theorem for conformally flat causal structures over compact manifolds is obtained. A
notion of half-flatness is defined for four dimensional causal structures whose null cones are ruled projective
surfaces, which generalizes the notion of half-flat conformal structures of split signature. A twistorial
construction for conformally flat causal structures and half-flat causal structures is given.

COMMITTEE MEMBERS

DEPUTY CHAIR:
Jacques Hurtubise,ÌýProfessor,ÌýMathematics and Statistics, Ã山ǿ¼éÌý

SUPERVISOR:
Niky Kamran,ÌýProfessor,ÌýMathematics and Statistics, Ã山ǿ¼é

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John Toth, Professor,ÌýMathematics and Statistics, Ã山ǿ¼é

INTERNAL MEMBER:
Gantumur Tsogtgerel, Associate Professor,ÌýMathematics and Statistics, Ã山ǿ¼é

·¡³Ý°Õ·¡¸é±·´¡³¢Ìý²Ñ·¡²Ñµþ·¡¸é:
Keshav Dasgupta, AssociateÌýProfessor,ÌýPhysics, Ã山ǿ¼é

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