Below are some remarks intended mostly to help students choose among different upper-division Mathematics courses within particular areas. For more detailed advice about course selection, students are encouraged to consult a faculty advisor to their major.

**Math 109:**Math 109 helps students to develop their skills in writing proofs and is a prerequisite for many upper-division Mathematics courses. Mathematics majors are encouraged to take Math 109 as soon as they have fulfilled the prerequisites. In particular, Mathematics majors admitted as freshmen should aim to complete Math 109 by the end of their sophomore year, while transfer students should take Math 109 during the Fall of their first year at UC San Diego, if possible.**Algebra Courses:**Many students will face a choice between taking Math 100AB or Math 103AB. Math 100AB is the much more challenging sequence. Students interested in pursuing a Ph.D. in Mathematics should choose Math 100AB rather than Math 103AB. However, Math 100AB is recommended only for outstanding students who received a high grade in Math 109, preferably in the A range. Students who do not perform well in Math 109 are usually not successful in Math 100AB and are encouraged to take Math 103AB instead.**Analysis Courses:**Many students will face a choice between taking Math 140AB or Math 142AB. Math 140AB is the much more challenging sequence. Students interested in pursuing a Ph.D. in Mathematics or Statistics should choose Math 140AB rather than Math 142AB. However, Math 140AB is recommended only for outstanding students who received a high grade in Math 109, preferably in the A range. Students who do not perform well in Math 109 are usually not successful in Math 140AB and are encouraged to take Math 142AB instead.**Combinatorics Courses:**The Mathematics department offers four courses in Combinatorics: Math 154, Math 158, Math 184, and Math 188. Math 154 and Math 184 have minimal overlap, and students can benefit from taking either or both of these courses. Math 158 should be viewed as a more advanced and theoretical version of Math 154, while Math 188 should be viewed as a more advanced and theoretical version of Math 184. Math 158 and Math 188 are recommended only for outstanding students who received a high grade in Math 109, preferably in the A range, while other students are encouraged to take Math 154 or Math 184 instead.**Probability and Statistics Courses:**Mathematics majors who plan to take exactly one course in Probability and Statistics are encouraged to take Math 183. Students who plan to study Probability and/or Statistics in greater depth should instead begin with Math 180A. After Math 180A, students can proceed to take further courses in Probability (Math 180BC), the Mathematics of Finance (Math 194), Mathematical Statistics (Math 181AB), or Applied Statistics (Math 185 and Math 189). Math 181A and Math 181B form a sequence, and students who take Math 181A are strongly encouraged also to take Math 181B. However, Math 181CDEF, which are offered less regularly, need not be taken in any particular order.**Differential Equations Courses:**The Mathematics department offers two upper-division courses in Partial Differential Equations (Math 110 and Math 148) and two upper-division courses in Ordinary Differential Equations (Math 130 and Math 146). These four courses are designed to have minimal overlap, and students can take any number of them for credit. Math 110 and Math 130 focus largely on applications, and require only lower-division courses as background. Math 146 and Math 148 are more advanced proof based courses which provide a rigorous treatment of the mathematical theory, as well as advanced applications. Math 146 and Math 148, along with Math 144, a course in Fourier analysis, are intended for students who have already taken Math 140AB or Math 142AB.

**Upper Division Courses as Preparation for Graduate School**

**Ph.D. in Mathematics:**Students planning to pursue a Ph.D. in Mathematics should take Math 100ABC and Math 140ABC to ensure that they have the strongest possible foundation in Algebra and Analysis. It is best for students to complete at least one of these sequences by the end of their third year of undergraduate study, so that later they will have the opportunity to take other advanced undergraduate courses that require Analysis or Algebra as a prerequisite. Complex Analysis and Topology are also important for Ph.D. level work, and students are encouraged to take at least Math 120A and Math 190A. Finally, students are encouraged to acquire a broad background in Mathematics by taking undergraduate courses in a variety of different areas of both Pure and Applied Mathematics.**Ph.D. in Statistics:**Because a strong background in Real Analysis is important for Ph.D. level work in Statistics, students planning to pursue a Ph.D. in Statistics are strongly encouraged to take Math 140ABC. Students planning to pursue a Ph.D. in Statistics should also take Math 180A, Math 181AB, and Math 185, and are encouraged to take additional upper-division courses in Probability and Statistics as well. It is particularly important for students to complete Math 180A and Math 181AB by the end of their third year of undergraduate study. Experience in computer programming is also very important for students planning to pursue a Ph.D. in Statistics.**Masters in Statistics or Data Science:**Most of the advice given above for students planning to pursue a Ph.D. in Statistics also applies to students who plan to pursue a Masters degree in Statistics. However, students planning to pursue a Masters degree in Statistics could take Math 142AB instead of Math 140ABC. Students interested in pursuing a Masters degree in Data Science are also encouraged to take courses in Linear Algebra (Math 102 or Math 170A) and Optimization (Math 171AB or Math 173AB). Experience in computer programming is also very important for students planning to pursue a Masters degree in Statistics or Data Science.