Since obtaining his PhD in 1990 Kay Magaard has worked on various aspects of group and representation theory. The questions he has worked on relate to various aspects of the maximal subgroup problem and can be phrased in terms of representation theory.
Another theme of Magaard's research relates to group actions on curves. The central theme here is how the braid-, and more generally the mapping class, groups act on Nielsen classes. Results concerning these actions have applications in number theory and geometry.
Related to the work on braid group actions is the question of constructive recognition of simple groups which has lead him to develop algorithms for the constructive recognition of exceptional groups of Lie type.
Himstedt, F.; Le, T.; Magaard, K.; Characters of the Sylow p-subgroups of the Chevalley groups $D_4(p^n)$, J. Algebra, 332 (2011), 414 -- 427
Magaard, K.; Shaska, T.; V\"olklein, H.; Genus 2 Curves that Admit a Degree 5 Map to an Elliptic Curve, Forum Math. 21 (2009), no. 3, 547--566.
Lübeck, F.; Magaard, K.; O'Brien, E.; Constructive recognition of $SL_3(q)$, J. Algebra (2007)
Magaard, K.; Völklein, H.; On Weierstrass points of Hurwitz curves. J. Algebra 300 (2006), no. 2, 647--654.
Magaard, K.; Shpectorov, S.; Völklein, H.; A GAP Package for Braid Orbit Computation and Applications, Experimental Mathematics 12:4, (2004) 385--394.
Gluck, D.; Magaard, K.; Riese, U.; Schimd, P.; The solution of the $k(GV)$ problem, J. Algebra, 279 (2004), 694--719.
Guralnick, R. M.; Magaard, K.; Saxl, J.; Tiep, P. H.; Cross characteristic representations of symplectic and unitary groups. J. Algebra 257 (2002), no. 2, 291--347.
Frohardt, D.; Magaard, K.; Composition Factors of Monodromy Groups, Annals of Mathematics, 154 (2001).