The ELLIS PhD program is a European initiative that supports excellent young researchers by connecting them to leading researchers in AI. In particular, PhD students are supervised by two advisors from different countries: an advisor and a co-advisor. In this work we summarize the procedure that, in its final step, matches students to advisors in the ELLIS 2020 PhD program. The steps of the procedure are based on the extensive literature of two-sided matching markets and the college admissions problem [Knuth and De Bruijn, 1997, Gale and Shapley, 1962, Rothand Sotomayor, 1992]. We introduce PolyGS, an algorithm for the case of two-sided markets with quotas on both sides (also known as many-to-many markets) which we use throughout the selection procedure of pre-screening, interview matching and final matching with advisors. The algorithm returns a stable matching in the sense that no unmatched persons prefer to be matched together rather than with their current partners (given their indicated preferences). Roth [1984] gives evidence that only stable matchings are likely to be adhered to over time. Additionally, the matching is student-optimal. Preferences are constructed based on the rankings each side gives to the other side and the overlaps of research fields. We present and discuss the matchings that the algorithm produces in the ELLIS 2020 PhD program.