Talk #1 (1:00pm-2:30pm)
Anton Iliashenko, Department of Pure Mathematics, University of Waterloo
"Betti numbers of nearly G_2 and nearly Kähler manifolds with Weyl curvature bounds"
We use the Weitzenböck formulas to get information about the Betti numbers of nearly G_2 and nearly Kähler manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.
Talk #2 (2:30pm-4:00pm)
Amanda Petcu, Department of Pure Mathematics, University of Waterloo
"Some Calculations Regarding G2 and the Isometric Flow (Part 2)"
In the paper the authors consider the following setup for a 7-dimensional manifold M. Given a hypersymplectic structure ω on X4 they consider the manifold M=X4×T3 where T3=S1×S1×S1. With this setup, they consider a closed 3-form φ on M that gives a G2 structure. This is due to the hypersymplectic structure on X4. In the case where X4 is compact the authors deform the hypersymplectic structure on X4 to a hyperkahler triple. Then the G2-structure φ on M=X4×T3 has vanishing torsion forms when ω is a hyperkahler triple implying that φ determines a G2-metric. In this talk, I will loosen the conditions on X4 to pre-hypersymplectic and compute the forms φ and *φ=ψ. We will also compute the four torsion forms for M and determine what conditions are needed in order for the torsion forms to vanish. Finally given the full torsion tensor T of M, we will compute Div(T) in terms of the four torsion forms and determine what conditions we might need in order for Div(T) to vanish.
MC 5403