 Complete the following phrase:  Vahlen had studied under Frobenius and written his dissertation on additive number theory. He had been published in mathematical journals like Crelles Journal and Acta Mathematica. Vahlens mathematical interests included partition theory, non-Euclidean geometry and differential equations. His published works on applied mathematics included papers on ballistics, aerodynamics, the magnetic compass and celestial mechanics. By 1911 Vahlen was teaching in Greifswald. Between 1914 and 1915 he served on the Western Front of World War I. He was wounded during his service on the Eastern Front 1916-1919 and left the army a decorated, high-ranking officer. He joined the Nazi Party after the failed 