The PhD Qualifying Exam in Mathematics took place in the second semester of 2025 at the Federal University of Ceará (UFC), Brazil.
What do you think about the level of difficulty of the questions? The exam must be completed within 4 hours. Here is the link to the questions and solutions: https://pgmat.ufc.br/wp-content/uploads/2025/09/soluCao-prova-exame-prelim-2025-2-analise.pdf
The PhD Qualifying Exam consists of two stages: the Preliminary Exam (written exam) and the Oral Exam. The Preliminary Exam is offered each semester, either from January to March or from August to September, and may include exams in the following areas: Analysis (Functional Analysis / Analysis I), Geometry (Differentiable Manifolds), Singularities (Topology / Singularities), Combinatorics (Advanced Graph Theory), and Dynamical Systems. A student is considered to have passed the Preliminary Exam if, by the end of the third semester (within a period of 18 months counted from the first enrollment), they have passed exams in two areas of the written examination.
The Oral Exam will cover an advanced topic (or a scientific paper) in the student’s area of concentration. The Oral Exam committee shall consist of three researchers, with the student’s advisor serving as the chair of the committee, and at least one of the other members preferably coming from another institution or graduate program. A student will be considered to have passed the PhD Qualifying Exam if, by the end of the fourth semester (within a period of 24 months counted from the first enrollment), they have passed the oral exam. Candidates who fail to meet these requirements will be automatically dismissed from the program.