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 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
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.
Part of the requirements for Stage 3 involves taking two mathematics courses. Below are the courses that I took since Fall 2024 that did not satisfy any of the Stage 1 or 2 requirements:
Stats 526: Discrete State Stochastic Processes (Fall 2024)
Math 700: Directed Reading in Elliptic PDE (Fall 2024)
Math 650: Fourier Analysis (Winter 2025)
Math 636: Topics in The Topology and Dynamics of Rational Maps (Fall 2025)
Pat 505: Video Game Music (Winter 2026)
The remaining requirements are related to research and the thesis defense. I have given several talks related to the topics that I studied (see Presentations for more details):
Domino Tilings of Black-and-White Temperleyan Cylinders. Student Analysis Seminar, University of Michigan, February 24th 2026.
Discrete Complex Analysis: T-holomorphic Functions. Student Combinatorics Seminar, University of Michigan, October 20th 2025.
WUSTs, LERWs, and SLEs. Student Analysis Seminar, University of Michigan, January 28th 2025.
Introduction to Random Lozenge Tilings. Student Analysis Seminar, University of Michigan, December 3rd 2024.
T-embeddings and Origami Maps. Student Combinatorics Seminar, University of Michigan, November 11th 2024.
I also contributed to a paper about my research work (see Research for more details):
D. Chelkak and Z. Deiman. Domino tilings of black-and-white Temperleyan cylinders. Submitted.