Understanding materials based on their underlying topological order has caused a paradigm shift in condensed matter physics in how we classify and describe emergent phenomena in systems. Controlling and engineering these exotic orders is key for next-generation electronics and topological quantum computers. In this talk, I will discuss how the interplay of real and reciprocal space topological order can provide tunable topological phases in a class of multiferroics called the hexagonal manganites[1,2]. Using first-principles calculations and Landau theory, I will describe how the multiferroic orders in these compounds give rise to tunable topological defects and nodal-line semimetals[3,4]. Finally I will discuss how the ultrafast control of topological phases can be achieved by the targeted probing of symmetry-breaking phonons.
 S.M. Griffin, M. Lilienblum, K.T. Delaney, Y. Kumagai, M. Fiebig and N.A. Spaldin.Phys. Rev. X 2 (4), 041022 (2012)
 F.T. Huang, X. Wang, S.M. Griffin, Y. Kumagai, O. Gindele, M.-W. Chu, Y. Horibe, N.A. Spaldin, S.-W. Cheong.Phys. Rev. Lett., 113 (26), 267602 (2014)
 Q.N. Meier, M. Lilienblum, S.M. Griffin, K. Conder, E. Pomjakushina, Z. Yan, E. Bourret, D. Meier, F. Lichtenberg, E.K.H. Salje, N.A. Spaldin, M. Fiebig and A. Cano. Phys. Rev. X, 7, 041014 (2017)
 S.F. Weber, S.M. Griffin and J.B. Neaton. arXiv:1902.10085 (2019)
 S.M. Griffin. To appear in J. Cond. Mat. Phys. (2019)