## General advice for passing preliminary examinations and coursework

The primary goals of the first two years of graduate school are two-fold: the first is to complete the prelim requirements; the second and possibly more important is to find a dissertation supervisor. Students should not lose sight of the second objective while working on the first.

Students entering with strong backgrounds may be able to complete the prelim requirements within one year. All students should be able to complete the prelim requirements within two years, ordinarily without needing to take more than two prelim courses per semester. Because students are expected to earn credit in three courses during each long semester, this leaves time for a student to take the department’s Introduction to Teaching course (M398T), which is required no later than the first semester in which the student is employed as a TA, and to take various topics or conference (reading) courses, which are useful for sharpening a research specialization and identifying an appropriate dissertation supervisor.

What follows are some guidelines for what would ordinarily be adequate or excellent preparations for our preliminary examinations and coursework. Note that “adequate” preparation is what would generally be expected for a student to succeed in the prelim courses themselves; passing the exams typically requires an “excellent” level of preparation, including graduate courses and extensive individual study. By nature of their generality, these guidelines are neither mandatory nor comprehensive. Indeed, there are many other advanced undergraduate and beginning graduate courses which help one to develop the necessary mathematical sophistication needed to do well in prelim courses and exams. An individual student should consult with the Graduate Adviser or a faculty mentor to determine what specific preparation is most suitable for him or her.

(Note: The course examples below are those offered by UT Austin. For courses offered by other universities, please make sure that the material covered is comparable to that covered in UT courses.)

### Algebra

- Adequate: two semesters of undergraduate algebra (M373K/L).
- Excellent: several advanced undergraduate courses in group theory, number theory, and coding theory, or experience in a graduate algebra course at another university.

### Analysis

- Adequate: two semesters of undergraduate real analysis (M365C/D).
- Excellent: one or more undergraduate courses beyond real analysis, a course in complex analysis (M361), or experience in graduate analysis courses at another university. (Note that our complex analysis prelim course is essentially independent of the real analysis prelim material.)

### Applied Mathematics

- Adequate: experience with ordinary and partial differential equations and Fourier series.
- Excellent: an undergraduate course on partial differential equations (M372), a graduate course in real analysis and measure theory (M381C), or a beginning graduate course in functional analysis.

### Numerical Analysis

- Adequate: undergraduate courses on ordinary differential equations and numerical methods (M427K and M348, respectively).
- Excellent: undergraduate real analysis (M365C), complex analysis (M361), linear algebra (M341), and PDE (M372K). Basic programming skills (in any language) are also useful.

### Probability

- Adequate: undergraduate real analysis (M365C), linear algebra (M341), and probability (M362K).
- Excellent: graduate courses in measure theory and probability.

### Topology

- Adequate: an undergraduate topology course (M367K), and good knowledge of linear algebra (M341). For differential topology, advanced calculus and some knowledge of ordinary differential equations (M427K) are useful.
- Excellent: an undergraduate course on curves and surfaces (M365C) or experience in a graduate topology course at another university.