Here I list my progress in the Mathematics Ph.D. program at the University of Michigan Ann Arbor, which I started in Fall 2021. This can give you a sense of how far along I am in the program, in addition to giving me a clear picture of what I have done and what I still have to do. Detailed requirements are given on the math department's website: Program Requirements.
I took the following courses at the University of Michigan:
Fall 2021
Math 591: General and Differential Topology
Math 593: Algebra I (Rings and Modules)
Winter 2022
Math 594: Algebra II (Groups and Galois Theory)
Math 597: Analysis II (Real Analysis)
Fall 2022
Math 556: Applied Functional Analysis
Math 650: Fourier Analysis
Math 656: Introduction to Partial Differential Equations
Winter 2023
Math 592: Introduction to Algebraic Topology
Math 657: Nonlinear Partial Differential Equations
Math 681: Mathematical Logic
Fall 2023
Math 602: Real Analysis II (Functional Analysis)
Stats 625: Probability and Random Processes
Winter 2024
Stats 525: Probability Theory
Math 604: Complex Analysis II
Fall 2024
Stats 526: Discrete State Stochastic Processes
Math 700: Directed Reading (Elliptic PDE)
Winter 2025
Math 650: Fourier Analysis
Stage 1 of the program is the Qualifying Review (QR). The main goal is to pass QR exams and/or take courses in various mathematical fields to demonstrate mathematical proficiency. I needed to pass two QR exams by January 2023, and I needed to complete Stage 1 by January 2024.
The mathematical subjects, with corresponding "alpha courses," that can be used to satisfy Stage 1 requirements are listed below.
Algebra
Math 593: Algebra I (Rings and Modules)
Math 594: Algebra II (Groups and Galois Theory)
Topology
Math 591: General and Differential Topology
Math 592: Introduction to Algebraic Topology
Analysis
Math 596: Analysis I (Complex Analysis)
Math 597: Analysis II (Real Analysis)
Applied Analysis
Math 556: Applied Functional Analysis
Math 572: Numerical Methods for Differential Equations
Out of the 8 alpha courses listed above, a total of 6 must be completed by passing either the QR exam or the corresponding alpha course for each subject; at least 3 QR exams must be passed. Moreover, an additional graduate mathematics course outside of those 6 subjects must be passed.
As of Winter 2023, I have completed all the Stage 1 requirements.
Here are the QR exams I passed:
Algebra I (January 2022)
Complex Analysis (January 2022)
Real Analysis (August 2022)
Algebraic Topology (May 2023)
Here are the alpha courses I passed:
Math 594: Algebra II (Winter 2022)
Math 556: Applied Functional Analysis (Fall 2022)
Here is the additional graduate mathematics course I passed:
Math 681: Mathematical Logic (Winter 2023)
The Requirements
The University of Michigan has a General Mathematics Master's Program and an Applied Mathematics Master's Program, one of which may be obtained while working towards a Ph.D. Although this is optional, the Master's degrees are generally satisfied through coursework in the Ph.D. program.
As of Winter 2023, I have completed the requirements for the General Mathematics Master's Program. Here are the courses that I used for the program:
Math 556: Applied Functional Analysis (Fall 2022)
Math 592: Introduction to Algebraic Topology (Winter 2023)
Math 593: Algebra I (Fall 2021)
Math 594: Algebra II (Winter 2022)
Math 597: Analysis II (Winter 2022)
Math 650: Fourier Analysis (Fall 2022)
Math 656: Introduction to Partial Differential Equations (Fall 2022)
Math 681: Mathematical Logic (Winter 2023)
Stage 2 of the program is the "path to candidacy." The main goals are to complete coursework via "distribution" and "cognate" requirements and to pass the preliminary exam after finding an advisor. I needed to complete Stage 2 by September 2024, but I asked for an extension on this deadline to January 2025.
The mathematical subjects that can be used to satisfy the Stage 2 distribution requirement are grouped together below. Specific advanced courses which satisfy the distribution requirement can be found on the math department's website: Math Ph.D. Course Registration.
Subject 1: Algebra, Algebraic Geometry, Algebraic Number Theory
Subject 2: Analysis, Analytic Number Theory, Probability
Subject 3: Topology, Differential Geometry
Subject 4: Applied Analysis, Numerical Analysis
Subject 5: Applied Discrete Mathematics, Combinatorics, Logic
A total of 6 graduate mathematics courses must be completed to satisfy the distribution requirement. Of the 5 subjects listed above, at least 1 course from at least 3 of the subjects must be completed. In general, the courses chosen cannot be the same ones used to satisfy Stage 1 requirements.
A total of 4 credit hours of cognate courses in areas other than mathematics must be completed to satisfy the cognate requirement. This includes cross-listed courses listed on the math department's website: Cross-Listed Courses.
The preliminary exam, taken in front of the advisor and a second examiner, is an oral exam intended to prepare for future research. The amount of material is approximately equal to two semesters' worth of advanced courses.
As of Winter 2025, I have completed all the Stage 2 requirements.
Here are the advanced courses I have passed:
Math 592: Introduction to Algebraic Topology (Winter 2023, Subject 3)
Math 602: Real Analysis II (Fall 2023, Subject 2)
Math 604: Complex Analysis II (Winter 2024, Subject 2)
Math 650: Fourier Analysis (Fall 2022, Subject 2)
Math 656: Introduction to Partial Differential Equations (Fall 2022, Subject 2)
Math 657: Nonlinear Partial Differential Equations (Winter 2023, Subject 4)
Math 681: Mathematical Logic (Winter 2023, Subject 5)
Here are the cognate courses I have passed:
UC 415: Methods in Research for the Natural Sciences (Fall 2021)
Stats 525: Probability Theory (Winter 2024)
I passed the preliminary exam on January 7th 2025. Here was the full syllabus for my exam. Advanced topics included:
Random lozenge tilings
Domino tilings in Temperleyan domains
Connections to imaginary geometry
Recent papers on t-embeddings
The Requirements
Stage 3 of the program is the thesis research and defense. The main goal is to conduct mathematical research and write up the results in a thesis, which is then presented through a dissertation defense to several faculty members. Stage 3 will take at least 2 years to complete.
More details about my progress will be written here eventually...