Below are the solutions to the Fall 2019 prelims offered at UMN. The solutions bear striking resemblance to those of the previous prelims written by Montie Avery, Cora Brown, Bruno Poggi, Harini Chandramouli, Sunita Chepuri, Adrienne Sands, Michelle Mastrianni and Jasper Weinburd, among others whose names I have unknowingly and apologetically omitted.

Besides thanking the aforementioned and all who helped them, I thank John O'Brien for his contribution to the solutions.

To err is human, but 'too err' is me (bad pun, I know). Consequently, there may be errors in my solutions. Should you notice one, please let me know so that I can rectify it.

• Algebra

• Complex Analysis

• Manifolds and Topology

• Real Analysis