Earning a Doctor of Philosophy (Ph.D.) in mathematics involves taking some of the highest-level mathematics courses a college has to offer. These courses serve to prepare students for the school's Ph.D. qualifying exams. They cover many advanced topics such as number theory and applied mathematics.
Typical Courses for a Mathematics Ph.D.
A Doctor of Philosophy (Ph.D.) in Mathematics might appeal to students who want to teach mathematics at the university level, conduct research, or work in consultation. Students might be given the opportunity to focus on one area of mathematics, such as statistics, or will take several advanced-level maths courses. Common subjects include advanced algebra, analysis and topology. Mastery of these courses allows students to move forward and begin preparing to research and write a thesis.
<h3 id="section---ImportantFactsAboutMathematicsPh.DPrograms">Important Facts About Mathematics Ph.D Programs</h3><p></p><table border="1"><tr><td> Prerequisites </td><td> Graduate degree or enrollment, GRE scores</td></tr>
<tr><td> Concentrations </td><td> Mathematics, applied mathematics, teaching</td></tr>
<tr><td> Degree Requirements </td><td> Dissertation, qualifying exam</td></tr>
<tr><td> Possible Careers </td><td> Actuarial mathematics, finance, software design and testing, statistical analysis, weather and climate forecasting</td></tr>
<tr><td> Median Salary (2021)</td><td> $108,100 (<i>mathematicians</i>)</td></tr>
<tr><td> Job Outlook (2021-2021)</td><td> 0% (<i>mathematicians</i>)</td></tr>
</table><p><i>Source: U.S. Bureau of Labor Statistics</i>
</p><h3 id="section---AbstractAlgebra">Abstract Algebra</h3><p>Graduate-level abstract algebra concerns the study of algebraic structures such as modules, fields, groups and rings. Students learn about the properties of these structures and the theorems that apply to them. Classes may also cover how these ideas apply to related branches in mathematics, such as homological algebra and algebraic number theory. As a result of these courses, students will learn about:
</p><ul><li>Abelian groups
</li><li>Galois theory
</li><li>Integral domains
</li></ul><h3 id="section---MathematicalAnalysis">Mathematical Analysis</h3><p>At least one analysis course will likely be required as part of a mathematics Ph.D. degree program. These courses may cover real analysis, complex analysis or numerical analysis. Often students will have the opportunity to take more than one class, if desired. Mathematical analysis closely examines the properties and limits of functions and sequences. The particular topics of study will depend on the chosen course:
</p><ul><li>Real analysis
<ul><li>Analytic functions and sequences of real numbers
</li><li>Measure theory
</li></ul></li><li>Complex analysis
<ul><li>Analytic functions and sequences of complex numbers
</li><li>Holomorphic functions
</li></ul></li><li>Numerical analysis
<ul><li>Algorithms for problems in continuous mathematics
</li><li>Approximation theory
</li></ul></li></ul><h3 id="section---Topology">Topology</h3><p>Topology is a field of study with connections to concepts in geometry and concerns space and the surfaces of objects. It is also based off developments in set theory. This branch of mathematics has applications in areas such as physics, chemistry and <a href="https://learn.org/articles/What_Is_the_Curriculum_of_a_Bachelors_Degree_in_Computer_Science.html">computer</a> graphics. As part of the mathematics Ph.D. program, topology classes will cover:
</p><ul><li>Metric, covering and function spaces
</li><li>The fundamental group
</li><li>Topological equivalence</li></ul>
