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Tuesday, September 27, 2016

**Abstract:** I will begin by briefly reviewing the foundations of stable motivic homotopy theory. In the non-equivariant stable motivic homotopy category, I will construct a k-spectrum MGLO whose topological realization over the field k=C is MO. I will give a complete description of the coefficient ring of MGLO up to knowledge of the coefficients of motivic HZ/2. Next I will discuss how MGLO is related to the Z/2-equivariant k-spectrum MGLR. MGLR is a motivic analogue of Landweber's real oriented cobordism MR. Just as taking fixed points of MR at the pre-spectrum level gives MO, taking fixed points MGLR at the pre-spectrum level gives MGLO. Restricting attention to the field k=C, I will discuss new research relating to MGLR. Finally, I will construct a k-spectrum MGLSO whose topological realization over the field k=C is MSO. Restricting to k=C, and completing at p an odd prime, MGLSO splits as a wedge sum of suspensions of motivic Brown-Peterson spectra BPGL. Restricting to k=C, and completing at p=2, MGLSO splits as a wedge sum of suspensions of motivic HZ and HZ/2.