We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is an application of the Goerss--Hopkins--Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E}_{n$ with its $C}_{2$-action by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C}_{2$-fixed points. Then, I will talk about the slice spectral sequence of a $C}_{4$-equivariant spectrum. This spectrum is a variant of the detection spectrum of Hill--Hopkins--Ravenel and is very closely related to the height 4 Lubin--Tate theory.
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Quantization is known as a way of constructing (noncommutative) quantum mechanical system from (commutative) classical mechanical system, mathematically forming operators from ordinary functions. The word `` quantize'' means ``discretize'' and we discuss where it came from by reviewing the motivation and background of quantum mechanics. Then we discuss mathematical formulation of (deformation) quantizations and introduce two $C}^{*$-algebras, noncommutative two tori and quantum Heisenberg manifolds, which are main examples of Rieffel's strict deformation quantizations. We also introduce noncommutative (or quantum) Yang-Mills theory developed by Connes and Rieffel on noncommutative tori, and then review some of the results of Yang-Mills connections on noncommutative two tori and quantum Heisenberg manifolds if we have time.