We are pleased to hear of your interest in the Mathematics Department at The University of Georgia. We are proud of the work we've done to enhance the mathematics offerings for excellent mathematics students at The University of Georgia, and the major is a demanding and versatile one. Recent graduates have gone on to do doctoral work at various top-ranked universities (e.g., M.I.T., Stanford, Cornell, Texas, Brown, Rice, Stony Brook, Duke, Indiana, Yale, and Georgia Tech), and some have chosen to do master's or doctoral work here at the University of Georgia. But the majority of our undergraduate mathematics majors get jobs working in more applied areas (and double majors in computer science or business)---actuarial science, software design, computer applications, consulting, ... Some employers who have hired recent mathematics majors are: Arthur Andersen, Bain & Company, C&S Bank, Delta Air Lines, General Electric, Hansen, Inc., Hewatt Associates, MicroStrategy, NASA, National Security Agency, Southern Bell, Turner Broadcasting, the U.S. Navy and Air Force, and William Mercer. You can get some information about some of our alumni, their jobs, and their comments .
We in the mathematics Department are excited to have received a National Science Foundation VIGRE grant that will, among other things, support undergraduate research in the upcoming years and summers, and we hope that you will be able to participate in some of these activities.
In 1984 we officially introduced (what is now numbered) the MATH 2400H-2410H sequence, Calculus with Theory, which is a self-contained calculus course for the best and most-motivated mathematics students, stressing the theoretical foundations of calculus (hence having more of a theorem/proof format than the conventional high school or college calculus course) as well as further applications to physics. Even though most of the students taking the course have received Advanced Placement credit for their high school calculus study, they find the course to be extremely challenging and a different experience; there are usually a few students who've never seen calculus, as well. Students who do well in this course often go right into junior/senior level mathematics courses as sophomores. This course is open to non-Honors students as well. Incoming freshmen who've earned a 5 on the advanced placement Calculus BC should consider taking MATH 3500-3510.
We assume that by now you've completed, or are in the process of completing, one of the single-variable calculus sequences (2200-2210, 2250-2260, 2300H-2310H, or 2400H-2410H). We suggest that all students considering a major in mathematics or a physical science take the new MATH 2250-2260 sequence. Next, students normally take the following classes. The multivariable calculus and differential equations classes are still rather computational in nature. The 3000-level courses are "bridge" courses designed to help you make the transition to the 4000-level courses that are more conceptual and require understanding and writing proofs. We recommend that you take a 3000-level course as early as possible to experience this different "flavor" of mathematics.
N.B. Each of these courses has a prerequisite of MATH 2260, MATH 2210, or MATH 2310H.
As an alternative to MATH 2500 and MATH 3000, we offer the two-semester sequence MATH 3500-3510 (Multivariable Mathematics I and II) starting in the fall semester. This class is at a somewhat higher level, with greater emphasis on concepts and proofs, and highlights the deep connections between linear algebra and multivariable calculus. These courses carry Honors credit, so it's ideal for the students who've completed MATH 2310H or MATH 2410H who wish to continue their Honors study of mathematics. But it's also recommended for the students completing MATH 2260--or for freshmen with a 5 on the Calculus BC Advanced Placement test--who are ready for a more rigorous and conceptual treatment of the material. (To help with this transition, if you haven't taken MATH 2410H, we suggest that you take MATH 3200 concurrently with MATH 3500. As a bit of incentive for you to take this more challenging route, both these classes will count towards the major.)
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To earn a mathematics major, you must complete (with a grade of C or better) 24 hours of MATH courses at the 3000 level or above, excluding MATH 3220. (This includes 8000-level courses for the particularly well-prepared student, but includes only MATH 5200 and MATH 5210 at the 5000/7000-level.) A major program may include no more than 6 credit hours among MATH 4710, 4760, 4780, 4790, 4850, 4900, and 4950, and must include:
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In addition, every mathematics major must complete two courses in Computer Science and/or Physics at or above the level of CSCI 1301 and PHYS 1211. It is strongly recommended that all mathematics majors complete 6 hours of coursework in a related field such as Computer Science, Physics, Statistics, Risk Management, or Management Sciences and Information Technology. You should also be aware that the university will require 15 hours further of 3000- and 4000-level elective courses. If you're not planning to take electives in mathematics or related subjects (see below for some suggestions depending on your interests), you should plan ahead and have some ideas of how you will fulfill this requirement.
Your adviser will help you choose the remaining courses to fit your interests and your career goals. You should check with Julie McEver in room 434C Grad Studies, to be assigned an adviser. If you have already found a particular faculty member who's agreed to be your adviser, be sure to tell Julie.
| The Franklin College of Arts and Sciences requires that you have a graduation check when you have completed 80--85 (semester) hours; ordinarily, this should be done before you register for your last two semesters. You will need to take a checklist of your approved major coursework, signed by your adviser, to the appropriate person at New College. |
It is expected that each
student will complete an exit interview with the Mathematics Department
shortly before graduating.
Here are some informal comments about the subject matter of the Mathematics courses, including prerequisites and the next semester(s) we intend to offer the class. A word about prerequisites: We require that you have taken (and earned at least a C in) the classes that will best prepare you for the course at hand. If you believe that you have the necessary background but lack an official prerequisite, you should consult with your adviser or the Associate Department Head, who can grant you access to the class. (This will often be the case for students who've done well in MATH 2400H-2410H or MATH 3500-3510.)
In addition to the courses on this list is a fun class that does not count towards the math major (but does count toward the required 39 upper division hours). MATH 3220, Advanced Problem Solving, is a pass/fail one-hour course which can be repeated up to three times. Various methods and strategies of problem-solving are covered. During the fall semester, students often work past Putnam exam problems in preparation for the annual competition in December. (Prerequisite or corequisite: MATH 3000 or MATH 3100 or MATH 3200 or MATH 3500.)
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COURSE |
PREREQ |
DESCRIPTION |
NEXT OFFERED |
| MATH 4000-4010: Modern Algebra and Geometry | MATH 3000, MATH 3200 | A first course in modern algebra, emphasizing its interplay with number theory, geometry, and other applications. You will learn about the isometries of the plane, which primes can be written as the sum of two squares, compass and straightedge constructions (e.g., the famous fact that one cannot trisect a general angle using compass and straightedge), group actions and counting arguments, symmetry groups, elements of Galois theory (e.g., the amazing fact that there can be no formula expressing the roots of the general polynomial of degree at least five in terms of radicals), and will be introduced to projective geometry. | 4000: fall, spring, summer; 4010: spring |
| MATH 4100: Real Analysis | MATH 3100, MATH 3200 | The conceptual underpinnings of calculus, studied in metric spaces: limits, numerical sequences and series, continuity, differentiable functions, integrals, and different notions of convergence of sequences and series of functions. | fall semester |
| MATH 4110: The Lebesgue Integral and Applications | MATH 4100 or MATH 4200 |
It was discovered toward the end of the nineteenth century that the Riemann integral we all learn in calculus is woefully ill-equipped to deal with problems that arise in real-life applications. For example, the limit of a uniformly bounded sequence of integrable functions needn't even be integrable. The Lebesgue integral is defined in such a way as to have all the "nice" properties, and we see in this course that it arises naturally in dealing with questions from Fourier analysis and probability. | spring 2006 |
| MATH 4120: Multivariable Analysis | MATH 3510, MATH 4100, or MATH 4200 | A rigorous treatment of multivariable differential and integral calculus, including the inverse and implicit function theorems, an introduction to differentiable manifolds, differential forms, and the n-dimensional Stokes' Theorem, the ultimate version of the Fundamental Theorem of Calculus. | spring 2008 |
| MATH 4150: Complex Variables | either MATH 2500 and MATH 3100 or MATH 3510 | A beautiful blend of theory and computation,
this course explores the magical effect of introducing complex numbers
into calculus. The central point is that when a complex function is
differentiable, then it is represented by a convergent power series;
this, in turn, is understood by representing its values by an integral
(an appropriate "weighted average" of nearby values). Even more
pleasant are the unexpected applications to "real" questions: using
residues, you will find that evaluating definite integrals like  and  is not so
difficult; applications to physics and engineering will be discussed as
time allows. |
spring semester |
| MATH 4200: Point Set Topology | either MATH 3100 and MATH 3200 or MATH 2410 | A study of continuity in the setting of abstract
metric and topological spaces. The notions of connectedness and
compactness (which are key to the intermediate value theorem and
maximum value theorem, respectively, from introductory calculus) are
studied in more general settings. Can there be a space-filling curve?
Can there be a function 
that is continuous at precisely the irrational points? the
rational points? |
fall semester |
| MATH 4220: Differential Topology | either MATH 4120 or MATH 3510 and either MATH 4100 or MATH 4200 | The course focuses on differentiable manifolds, a generalization of the curves and surfaces studied in multivariable calculus). Among the results proved are the Brouwer fixed point theorem (generalizing the intermediate value theorem from calculus), the fundamental theorem of algebra ("every nonconstant complex polynomial has a root"), the hairy ball theorem ("you cannot comb the hair on a hairy billiard ball") and the surprising meteorological fact that at any instant there is a pair of antipodal (opposite) points on the surface of the earth with identical barometric pressure and identical temperature. And intersection theory allows you to see immediately that a sphere and a torus are not "equivalent." | fall 2007 |
| MATH 4250: Differential Geometry | either MATH 2500 and MATH 3000 or MATH 3510 | The notion of curvature is quite intuitive for
curves (e.g., straight lines have none) and less so for surfaces (why
can we use a sheet of paper to make a cylinder but not the surface of a
doughnut?). You will find your multivariable calculus and linear
algebra to be powerful tools in discussing such issues. Other topics of
note are the Fary-Milnor Theorem (which tells us that total curvature
of at least  is required to
make a knot), the Gauss-Bonnet Theorem (which relates the total
curvature of a surface to its topology), and an introduction to
non-Euclidean geometry. |
spring semester |
| MATH 4400: Number Theory | MATH 4000 | A careful study of the simplest of all mathematical structures, the integers. Topics include: congruences, distribution of primes, factorization of integers, "Pythagorean triples" (e.g., 3,4,5), and Fermat's Last Theorem (which states that there are no positive integral solutions of xn + yn = zn when n>2, proved in a tour de force by Andrew Wiles in 1995). In addition, various real-world applications, such as cryptology, will be discussed. | spring 2008 |
| MATH 4450: Cryptology and Computational Number Theory | MATH 4000 | For centuries, cryptology, the science of secret codes, has been almost solely in th realm of government and the military. With the advent of electronic communication and commerce it has become vitally important to have secure cryptographic systems available to ordinary people. Many of the modern methods that are in use or have been suggested are based on computational problems in number theory and algebra, especially factoring large numbers and the "discrete logarithm" problem (given two elements in a group with one a power of the other, find the exponent). In this course, we take a tour of various cryptographic systems, learning their strengths and weaknesses. Along the way, we study modern algorithms for recognizing prime numbers, factoring composite numbers, and computing discrete logarithms. The mathematical structures introduced will include finite fields and elliptic curves. | fall 2007 |
| MATH 4500-4510: Numerical Analysis | suggested corequisite: MATH 3100; for 4500, MATH 3000 and some experience with computer programming; for 4510, MATH 4500, MATH 2700, and MATH 2500 or MATH 3510 | How does one use a computer to solve a large system of linear equations? Will the computer always produce an accurate solution? How might one use a computer to design a new alphabetic font orthe shape of a new car? This course will address such questions, along with numerical methods for solving differential equations, numerical methods for determining the eigenvalues of large matrices, interpolation and approximation of functions, and splines. Some hands-on computer programming is involved to experiment with those numerical schemes, although no specific programming language is a prerequisite. | 4500: fall; 4510: spring |
| MATH 4600: Probability Theory | either MATH 3510 or MATH 2500 and MATH 3100 or MATH 2260 and MATH 2500 | Consider such questions as: What is the likelihood that a poker hand will be a straight flush? What is the likelihood that in a room of 23 people there are two people with the same birthday? Topics in discrete probability include: conditional probability, independence, expectation value, Bernoulli trials, binomial and Poisson distributions. Then one deals with continuous random variables, the law of large numbers, the central limit theorem, and applications (e.g., to random walks, diffusion, Markov processes). This course is particularly recommended for those students intending to study statistics and actuarial science. | fall semester |
| MATH 4670: Combinatorics | MATH 3000 and MATH 3200 | A study of properties of discrete sets. Problems involve enumeration of combinatorial objects (e.g., the number of permutations with forbidden positions, the number of distinct circular necklaces made from beads with some different colors). Techniques include the pigeonhole principle and its generalizations, the inclusion/exclusion principle, generating functions and recurrence relations, and Polya's counting principle. | fall semester |
| MATH 4690: Graph Theory | MATH 3000 and MATH 3200 | A graph is just a collection of vertices with certain pairs of vertices connected by edges. Graphs can be used to solve problems like finding the shortest (or least expensive) airline route between two cities, matching job candidates to available positions, efficient routing of letter carriers, minimizing the number of computer registers needed to store all variables used simultaneously in a computer program, designing sports tournaments, or even classifying the regular polyhedra. The famous Four Color Problem ("Can you color the countries on any map using just four colors in such a way that any two contiguous countries are different colors?") is readily translated into a question about graphs, and graph theory ultimately provided a solution; sometimes the easier Five Color Theorem is proved in the course. | spring semester |
| MATH 4700: Qualitative Ordinary Differential Equations | MATH 2700 and either MATH 3000 or MATH 3500 | A more conceptual approach to the study of ODE's, both single and systems; long-term behavior, stability (which involves eigenvalues), and chaos. Applications to physics, engineering, and biology are included. | fall semester |
| MATH 4720: Introduction to Partial Differential Equations | MATH 2700 and either MATH 2500 or MATH 3510 | This course focuses on the three basic partial differential equations that arise in elementary physics: Laplace's equation (which characterizes harmonic functions), the heat equation, and the wave equation. What is it in the mathematics that explains why a sudden noise is heard only once (in 3D) but a sudden burst of heat has an effect forever? | spring 2008 |
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COURSES OFFERED ON A LESS REGULAR BASIS |
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| MATH 4050: Advanced Linear Algebra | MATH 4000 | This course expands on the culminating topic of MATH 3000: diagonalizability and the spectral theorem. Linear transformations and matrices in the complex setting are emphasized: hermitian and unitary matrices provide the appropriate setting of the spectral theorem. And, in general, when diagonalizability fails, Jordan and rational canonical forms are a powerful tool, in both theoretical and applied mathematics. | spring 2007 |
| MATH 4130: Introduction to Functional Analysis | MATH 4100 and MATH 3000 or 3500 | Introduction to linear algebra and analysis in the infinite-dimensional setting. Topics include complete metric spaces, Banach spaces, Hilbert spaces, and applications to physics and other disciplines. | TBA |
| MATH 4300: Introduction to Algebraic Curves | MATH 4000 | By studying the ring of polynomials in several variables, one sees a rich interplay between algebra and geometry. This course gives a deeper treatment of the algebraic curves, from both a local and a global approach. What does "rationality" of conics have to do with parametrizing Pythagorean triples; why is the cubic curve y2=x3+ax+b not "rational" and why does this lead to "elliptic integrals"? | TBA |
| MATH 4630: Mathematical Analysis of Computer Algorithms | MATH 3000 and MATH 3200 | Computing takes time. Some problems take a very long time, others can be done quickly. Some problems seem to take a long time, and then someone discovers a faster way to accomplish the same task. The study of the amount of computational effort required to perform certain kinds of tasks is the study of complexity, a relatively new branch of mathematics. This course is involved with answering, via complexity, the question: "how quickly can one compute the answer to a question?" | TBA |
| MATH 4780: Mathematical Biology | MATH 2500 and MATH 4700 | This class will cover various discrete, continuous, and probabilistic models for biological systems, reaction kinetics, epidemics, and tumors. It should be of particular interest to students considering further work in biology or medical school. | spring 2009 |
| MATH 4790: Mathematics of Option Pricing |
either MATH 2500 and MATH 3000 or MATH 3510 |
Consider a financial portfolio
that contains some options (European, American, or Exotic). This course
will explain the process of computing derivative prices in terms of
underlying equity prices, emphasizing the mathematics of investments
that replicate equities under arbitrage-free trading, and eventually
leading to the Nobel-prize winning Black-Scholes formula. Spreadsheets
are used to compute option trees; the necessary probability theory and
linear algebra concepts will be developed as needed. |
spring 2008 |
| MATH 4850: History of Mathematics | senior standing in mathematics | A study of (periods of) the history of mathematics by means of mathematical problems. Either a term paper or project will be required. | TBA |
| MATH 4900: Topics in Mathematics | Covering different material not included in the usual curriculum. (May be taken at most twice for credit towards the major.) | spring 2008 | |
| MATH 4950: Research in Mathematics | Students involved in VIGRE undergraduate research may earn from 1 to 3 credit hours. (May be taken for at most 6 credit hours.) | fall, spring, summer | |
Students who are thinking of working in a computer-oriented job are encouraged to take courses in Computer Science. Students who may be interested in the applications of mathematics to science are encouraged to study more physics, chemistry, biology, etc.
Many students from the sciences (and even humanities) find that a double major with mathematics is a particularly rewarding endeavor. An increasing number of students who are interested in a medical or biotechnical career are majoring in mathematics. The mathematics program meshes particularly well with Computer Science (especially if you're interested in either computer graphics or graduate work) and Physics.
Students who may be interested in pursuing doctoral work in mathematics are encouraged to complete MATH 4000-4010, either MATH 4110 or MATH 4120, MATH 4200 (and MATH 4220 or MATH 4250), and some 8000-level (graduate) courses.
Students who are thinking of being high school mathematics teachers should take MATH 4000-4010, MATH 4600, and MATH 5200-5210. We also now offer a dual degree in mathematics and mathematics education (B.S./B.S.Ed.); for more information, read this.
These days, many students are interested in actuarial science. The Society of Actuaries and the Casualty Actuarial Society are currently revamping their curriculum. See the Be An Actuary website for general information on what actuaries do. For information on the new curriculum and exams, check out the article by Jim Daniel at the MAA website. It is recommended that a student interested in pursuing a career in actuarial science take the following classes in mathematics and statistics (along with MATH 2700, 3000, 3100, and 3200, of course):
| MATH 4500-4510 | Numerical Analysis |
| MATH 4600 | Probability |
| STAT 4510 | Mathematical Statistics I |
| STAT 4280 | Applied Time Series Analysis |
and (some of) the following courses in Risk Management, Finance, and Economics:
| RMIN 4000 | Risk Management and Insurance |
| RMIN 5570 | Insurer Operations and Policy |
| FINA 3000 | Financial Management |
| ECON 4010 | Intermediate Microeconomics |
| ECON 4020 | Intermediate Macroeconomics |
Currently these may be taken as electives in a usual mathematics major; P.O.S. cards for the RMIN courses may be obtained from Ms. Kathy Wilson (kwilson@terry.uga.edu). Note: As long as core, college, and major requirements are met, there is no longer any limit on elective courses outside the College of Arts and Sciences counting towards the 120 hours required for graduation.
A number of students have majored
in mathematics as preparation for medical school and law
school. Medical schools look very carefully at students'
performance
in calculus classes. The analytical and reasoning skills required
to do well in mathematics are important in a medical or legal
career (not to mention the "stick-to-it-iveness"). In
particular, the pre-med office always suggests that students ask
their calculus teachers for letters of recommendation for medical
school.
Why should you consider majoring in mathematics if you intend
to go to medical or law school? First of all, because you enjoy
mathematics, and find the thought processes involved in mathematics
interesting and challenging. Secondly, students applying to medical
or law school often try to distinguish themselves by their academic
(and non-academic) records; completing a rather unusual major
program and coming with a different background gives one distinction.
Last, and most important, medicine itself, like so much of our
technological society, depends increasingly on mathematical ideas
and computational methods. For example,
More generally, sophisticated, highly mathematical computer
graphics systems are the basis for scientific visualization techniques
that are spreading in medical practice.
To be competitive in your performance on the MCAT and in your
medical school applications you will need to take the following
science courses:
| YEAR | FRESHMAN | SOPHOMORE | JUNIOR | SENIOR |
| fall | 2250/2300H | 2500 3200 | 4000 4700/4500 | 4600 4450 |
| spring | 2260/2310H | 3000 3100 | 4xxx 4720/4780/4510 | 4150/4250 |
On the other hand, an aggressive program might look like this:
| YEAR | FRESHMAN | SOPHOMORE | JUNIOR | SENIOR |
| fall | 2400(H) | 3500(H) 4000 | 4200 4670/4450 | 4220 8xxx |
| spring | 2410(H) | 3510(H) 4010 | 4250 xxxx | xxxx xxxx |
You should consult with a mathematics department adviser to figure out a suitable major program for you and discuss any questions you might have.
We hope this material has been of help to you.
A minor consists of 15 hours of MATH courses at the 3000 level or above, and must include either
Please see the Associate Head of the Department of Mathematics for discussion and approval of your minor program.