Jelena Grbić

Senior Lecturer in Pure Mathematics School of Mathematics University of Manchester Alan Turing Building Oxford Road Manchester M13 9PL office: 2.114 tel: +44 (0)161 2755849 fax: +44 (0)161 2755819 e-mail: jelena.grbic@manchester.ac.uk

Research

Research Interests: Algebraic Topology and Homotopy Theory.

My current research interests lie in Algebraic Topology with emphasis on unstable Homotopy Theory.

Toric Topology: Toric Topology is a new topological discipline concerned with a class of problems on the borders between the topology of torus actions, commutative and homological algebra, and combinatorial geometry, which investigates the combinatorial structures associated with spaces acted upon by the torus. My main goal in this area is to associate to objects introduced earlier in combinatorics and algebra different "topological models" and then study these from the point of view of homotopy theory. I studied the unstable homotopy type of the complement of a complex coordinate subspace arrangement by fathoming out the connection between its topological and combinatorial structures. One consequence of that result is an application in commutative algebra: certain local rings are proved to be Golod.

Loop space decompositions and representations of groups: Motivated by the Barratt conjecture and homotopy exponent problems, I have studied the natural maps from loop suspensions to loop spaces. The methods for analysing properties of certain unstable maps depart from classical unstable homotopy theoretical constructions. I apply the homology functor to natural maps from loop suspensions, and obtain certain functorial coalgebra transformations. This approach leads to the study of the algebra of natural linear transformations of tensor algebras and related groups of natural coalgebra transformations. These algebras and groups are studied by means of combinatorial group theory. A geometrical realisation of the algebraic results gives a description of the groups of homotopy classes of maps from the James construction to loop spaces. Further applications to homotopy theory involve the study of the composite of the loop map on an n-fold Whitehead product and the k-th James-Hopf invariant, and the establishment of a rate of exponent growth for the Barratt conjecture.

Research Papers

Curriculum Vitae (pdf)

Teaching

Algebraic Topology
(MATH3/4/61072)