Double Majoring in Math and Other Disciplines

In many fields, a strong mathematical background is useful or indispensable for deeper understanding. Roughly one half of the math majors at UMCP also major in another discipline. In addition to the deeper understanding attained, they achieve a significantly broader education and gain an important credential for employment or graduate school.

Below on this page, we have information in the following categories:

Mathematics Double Major & Courses for Students in

• Business
• Chemistry
• Computer Science
• Economics 
• Engineering 
• Philosophy 
• Physics & Astronomy

Double Major vs. Double Degree

A "Double Major'' is a student who plans to receive a single degree, but who will complete the requirements for two majors. In this case the last line of the student's transcript will state that a single Bachelor's Degree was awarded and that the student had two majors. Alternately, a student may complete a "Double Degree'' program. This requires the student to complete the requirements for both majors and also to obtain a total of 150 credits. In this case the student may actually receive two separate Bachelor's Degrees. For more information, consult the Undergraduate Catalog or contact an advisor.

How to Add a Second Major

If you are considering double majoring, you should set up appointments with advisors in both disciplines. For an appointment with a mathematics advisor, send email to ugadvisor@math.umd.edu.

The advisors will help you to complete the "Courses for a Double Major'' Form, which establishes a proposed course of study for satisfying the requirements of both programs. Once this form is complete, you may apply for the addition of a second major by visiting the main office of the college affiliated with the second major. (Mathematics falls under the CMPS college; their office is Room 3400 A.V. Williams Building.)

Graduate and Professional Degree Programs for which double majoring would serve as a particularly good preparation:

Program Name & Indicated Double Major

Chemistry is a fundamental science underlying biomedical, materials, environmental and earth sciences. Research in Chemistry involves mathematical and statistical tools in several essential ways, which are indispensable both to theoreticians and to applied investigators through scientific computing.  Physical chemistry, like physics, is intrinsically mathematical; and the description of chemical reaction kinetics and thermodynamics have always been formulated in terms of differential equations. Molecular simulation can involve Monte Carlo methods to model interaction potentials to determine behaviors of individual molecules or ensembles of molecules. Computational Chemistry employs quantum mechanics in particular for the calculation of "wavefunctions" to model molecular structure and energetics. Even very applied chemistry research nowadays involves sophisticated imaging equipment and computer data-reduction, interpretation, and simulation.  Development of new chemometric techniques requires some understanding of mathematical algorithms and statistical and scientific computing, and a double major with Mathematics is a natural way to develop such understanding.

Calculus (MATH 140-141) is a requirement for UMCP biochemistry and chemistry majors.  Differential equations are fundamental to chemical kinetics and thermodynamics, and all chemistry majors interested in graduate study or research would benefit from a basic differential equations course (MATH 246).

Beyond this level, the mathematical tools of interest vary for specialties within chemistry, but a mathematically strong student who aspires to a solid foundation for graduate work should take multivariable calculus (MATH 241) and linear algebra (MATH 240 or 461), and should seriously consider advanced calculus (MATH 410-411) and other 400-level math courses (see the discussion for PHYSICS majors). Partial differential equations (MATH 462) are the mathematical tools which a chemist would use to describe chemical reaction kinetics, mass transport and thermodynamics. More advanced mathematical analysis (Fourier analysis in MATH 464, plus Hilbert space theory) is crucial to quantum mechanics and to theory within physical chemistry. Group theory (treated with other abstract-algebra topics in MATH 402 or 403) enters in the study of symmetries in crystallography. Geometry enters in the description and visualization of spatial molecular structure through projective imaging techniques (so Chemistry students may be interested in MATH 431, Geometry for Computer Graphics).  Fourier analysis (MATH 464) is a crucial tool in unraveling the information contained in chemometric imaging technologies. Probability theory (STAT 400 or 410) arises in chemistry in several ways, specifically in describing and simulating relative frequencies of sequence patterns in DNA and other biological molecules, in modeling structure of DNA-protein complexes and polymer sequence configurations, and more generally in Monte Carlo simulation methodology. Statistical tools (STAT 401,430) often enter applied chemistry research, in roles ranging from pattern-recognition techniques of multivariate data analysis (e.g., in screening large collections of compounds for specific types of reactive properties), to validation of compound or sequence identifications from sequences of titrations which are not individually conclusive (as in DNA sequencing and chemical forensics), to statistical quality control of chemical processes.  A basic understanding of elementary probability and statistics can be obtained from STAT 400.

  Top of the page

MATHEMATICS DOUBLE MAJOR FOR COMPUTER SCIENCE STUDENTS

Computer Science interacts closely with mathematics in several ways, which go far beyond the topics in discrete mathematics which all Computer Science majors learn. The interactions are sketched briefly below by reference to computer science specializations within which mathematical topics arise. For some of these interactions, the relationship with mathematics is primarily conceptual and serves as theoretical background; for others, such as the study of algorithms and error estimates in parallel computing and the statistical methods used in artificial intelligence and image processing, mathematical thinking is essential in practice.

Modeling of computer network performance and demand/traffic streams is a central and growing part of the design and testing of computer architectures, priority schemes, and resource-allocation algorithms. Probability, statistics, and stochastic processes are important mathematical tools for these tasks. STAT 410 (Probability) and STAT 405 (Stochastic Processes and Queueing) are the relevant MATH courses in this direction.

Formal logic appears early and often in theoretical computer science; and as is emphasized in computer science courses on Discrete Mathematics, many computer-science problems related to data-structures are essentially combinatorial. These connections explain why certain types of mathematics necessarily arise in theoretical computer science. Issues concerning the types of questions which can be decided computationally using languages and data-processing of various types are also research topics in Logic. The Mathematics Department's logic courses are MATH 450 at an elementary level, and MATH 446-447 at a deeper level.

There is a close connection between Mathematics and the design and performance of algorithms. For example, questions of efficient large-scale parallel scientific computations are studied via theoretical numerical analysis. Since the performance of algorithms is often assessed in terms of their use of time and resources in larger and larger problems, they require the study of asymptotic rates of growth of combinatorial structures; of the worst or most costly configurations of data to which the algorithms could be applied; and also, of average behavior when the algorithm is applied to randomly configured inputs. Courses of particular relevance to these topics are MATH 475 (Combinatorics) and STAT 410 (Probability). 

Within the areas of Network Design and System Security, the mathematics of Cryptology (MATH 456) --- closely related to Number Theory, which is introduced in MATH 406 --- is especially useful. Computer Scientists encounter this material also in studying coding for data-compression.

Several directions in Computer Science are explicitly statistical: Artificial Intelligence, Pattern Recognition, and Neural Networks all address aspects of adaptivity and automated learning based upon data. By their nature, these approaches to automated learning involve highly parameterized models, and the estimation of the necessary parameters relies on the mathematics of optimization and numerical analysis. Moreover, although formal probabilistic models of data are not always relevant to practice in these fields, a conceptual grounding in probability and statistics is very useful.  The courses offered in these areas are MATH 477 (Optimization), MATH 466 (Numerical Analysis), and in Statistics: STAT 400-401 at an introductory level, and STAT 410-420 (which can be taken with only multivariable-calculus prerequisites) at a deeper level. 

Another important and highly mathematical set of topics is Computer Vision and Image Processing. The relevant mathematics here includes geometry (MATH 431, and at a deeper level Projective and Differential Geometry in MATH 436); differential equations (MATH 246, MATH 414) for the description of motion; and also Statistics (to make formal sense of blurring, distortion, and superpositions of signals). Other topics in mathematical analysis, such as Fourier Analysis (MATH 417) and Wavelets, also play a role in image analysis.

Software Engineering employs statistical tools in several ways. Either in management of the development cycle, or in modeling and testing of  software reliability,  designed data-collection of software "metrics' is used  to characterize difficulty and cost of (portions of) software projects. Basic statistical tools and concepts are covered in STAT 400-401. But Software Engineers should also be educated consumers of statistical methods such as regression and analysis of variance (covered in STAT 450 and, with a more computational focus, in STAT  430), and experimental design.

For Scientific Computation, the study of numerical analysis and differential equations is essential. Numerical analysis is introduced in MATH/AMSC 460 and 466; 466 is a more theoretical and rigorous version of 460.  The introductory differential equations course is MATH 246 and there are several upper-level courses (MATH 414, 415, 462).   Differential equations are also important in the  study of image processing, as mentioned above. 

Many new directions in Computer Science can be seen to rely heavily on calculus-based mathematical analysis. The rigorous foundation for all of the Analysis topics mentioned above (including differential equations, numerical analysis, Fourier analysis, probability and statistics) is taught in MATH 410 and 411 respectively in one and several dimensions. The more deeply one needs to understand any of the mathematical topics listed (for example, the more clearly one needs to understand the performance of algorithms in scientific computation), the more important a grasp of this basic theory becomes. A student inclined toward advanced work (or graduate school) in a related area should consider taking these courses.

MATH courses connected with computer science specializations:

Course & Title/Topic; CMSC Specialization 

MATHEMATICS DOUBLE MAJOR FOR PHILOSOPHY STUDENTS

Philosophers study various aspects of the relation between formal or axiomatic structures and human rationality or understanding. Mathematics can play a fundamental role in elucidating understanding through the study of Logic (MATH 450, 446, and 447), for example through the study of decidability of propositions, types of knowledge accessible to a computer, etc. Problems of Epistemology and the Philosophy of Science can be addressed through the study of Inference (STAT 400, 401, 410, 420, 450), mathematical model-building (MATH 420), and in particular the theory of dynamical models (MATH 246, 414, and 452) elucidating what can and cannot be predicted from such models. In addition, Philosophers are interested in formal models of learning and consciousness through neural networks, artifical intelligence, and the like, many of which concern problems of Optimization (studied in MATH 477). Other formal approaches to the study of human decisions and rationality make formal use of game theory (addressed in MATH 475).

Many of the interfaces between Philosophy and Mathematics also concern topics in Computer Science, but to the extent that these questions are going to be accessible to theory, a strong background in mathematics is likely to be needed. See the page on Computer Science and Math double majors for further discussion and information. 

The upper-level mathematics courses likely to be of greatest interest to Philosophy students are: 

Course Number & Title/Topic

MATHEMATICS DOUBLE MAJOR FOR ECONOMICS STUDENTS

Modern economic theory has become more and more quantitative over the last two generations, and in recent years the mathematical demands and expectations of many graduate schools in economics have escalated significantly. An undergraduate interested in pursuing graduate studies either in either economics or econometrics will be well advised to take upper-level courses in mathematical analysis, differential equations, and statistics. Moreover, many employment opportunities in economics involve analysis and forecasting of economic data, often in the form of time series. 

The behavior over time of formally defined economic systems is often modeled in terms of ordinary differential equations (ODE's). Therefore economists are likely to benefit from courses on solution techniques (MATH 246), qualitative behavior of solutions (such as chaotic behavior, blowup, stability, convergence, etc., in courses like MATH 414 and 452), and numerical analysis of methods for computing solutions (AMSC 460 or 466). 

Other theoretical approaches to economics involve characterizing the equilibria of economic systems. "General Equilibrium Theory" in Economics rests on a mathematical basis of rigorous analysis (such as  would be introduced in the courses MATH 410-411) including properties of convex function and some introductory topology of metric spaces. These topics would be elaborated in a form suitable for economists in a course on Optimization (MATH 477).

Finally, all branches of economics nowadays rest on the extraction of estimates of theoretical parameters from empirical data, and Econometrics makes this a specialization. Useful introductory courses on concepts of Probability and Statistics include STAT 400 at univariate-calculus level, continuing in STAT 401; or STAT 410-420 as the advanced-calculus-level introductory sequence which would be particularly suitable as a preparation for graduate study. A more advanced course on Regression (STAT 450), or a new course (STAT 430) emphasizing Regression topics using the SAS statistical software package could serve economics students as a mathematical capstone. Students with some interest in Insurance (e.g. the Actuarial profession) would take the Actuarial Mathematics course, STAT 470.

Economics majors may also be interested in Mathematical Finance, the theoretical underpinning for the methods of pricing Options and complex derivative securities on Wall Street. Some necessary background for the deeper study of these topics would include partial differential equations (MATH 462), probability (STAT 410), and numerical/computer methods (AMSC 460). 

Upper-level math courses of general interest to Economics majors:

Course Number & Title/Topic

MATHEMATICS DOUBLE MAJOR FOR BUSINESS STUDENTS

Quantitative topics of immediate importance to business majors include: optimal resource-allocation, routing and scheduling, inventory or portfolio management; modeling of risk and uncertainty for business decisions; forecasting of macroeconomic and market conditions; statistical methods of quality control; and statistical methods of sampling and auditing. The primary mathematical topics which lend insight in these areas are (Network and Combinatorial) Optimization (treated in MATH 477), Combinatorics and graph theory (MATH 475), and Probability and Statistics (STAT 400-401, 410-420, 430, and 450). Probabilistic network models and stochastic processes are covered in STAT 405.

Business majors may have interests in Actuarial Science, the mathematical subject describing the calculations of fair premiums and probabilities of loss in Insurance. The Mathematics Department offers courses relevant to actuarial science (in particular, STAT 470), as well as  advising related to preparation for the Actuarial Examinations. 

BMGT majors may also be interested in Mathematical Finance, the theoretical underpinning for the methods of pricing Options and complex derivative securities on Wall Street. Some necessary background for the deeper study of these topics would include partial differential equations (MATH 462), probability (STAT 410), and numerical/computer methods (AMSC 460).

Upper-level math courses of general interest to Business majors:

Course Number & Title/Topic

MATHEMATICS DOUBLE MAJOR FOR ENGINEERING STUDENTS

Virtually all engineering undergraduates at UMCP move through the freshman-sophomore level of the mathematics major -- MATH 140, 141, 241, 246, 240 (often with a comparable linear algebra course, such as MATH 461, in place of 240). From there the great range of engineering activities is reflected by a great variation in further mathematics coursework.  Generally speaking, the most compelling reason for engineering undergraduates to attain a strong mathematical background, such as a double major, is the range of advanced techniques of engineering analysis and algorithm design which require advanced mathematics, such as: the numerical analysis of differential equations for the strength of materials, automatic tracking and control of system trajectories, signal analysis and filtering, reliability of engineering systems, analysis of the behavior of engineering systems under stochastic fluctuations of demand or loading, and stochastic simulation.

Certainly, any engineering student who intends to go on to graduate engineering coursework in mathematically demanding specialties should consider a double major with mathematics.

ANALYSIS

The mathematics used in engineering (differential equations, probability, statistics, Fourier analysis, etc.) belongs largely to Analysis, the branch of mathematics growing out of calculus. We begin with some perspective on analysis.

Roughly speaking, there are three levels of command in analysis.  The first level is what you have after your freshman-sophomore coursework. Then there is a big jump to the second level, in proof-oriented courses like MATH 410-411, where the curtain is pulled back and you have the opportunity to understand clearly what calculus in one and several dimensions is about. The jump to the third level of Math 630-631 (the graduate analog of MATH 410-411, covering measure theory and Lebesgue integration) is not the end of analysis, but arguably there are no jumps to follow which are as dramatic as the jumps to the first three levels.

For many engineering activities, a level 1 foundation is adequate: techniques are developed in solving specific types of problems in engineering courses. Mathematics courses in the "460" series broaden the range of problem solution techniques available at this level, in areas ranging from numerical computing (MATH 460), linear algebra (MATH 461), partial differential equations (MATH 462), complex analysis and power series (MATH 463), through MATH 464 (Fourier series and transform methods).  Level 2, in which the rigorous foundation goes beyond techniques for problem solving, is the gateway and prerequisite for more advanced mathematical topics, but is initiated in MATH 410-411.  (By the way, the Math Department is initiating a new course, MATH 412, which may be more appropriate for some engineers than MATH 411.) A large body of immensely useful material for engineering, from the more advanced study of differential equations and numerical analysis, to probability and stochastic processes, to control theory, cannot be properly understood without level 2 and (in graduate work) level 3.

Mathematical Topics for Engineering Specialties

What follows is an indication of the main upper-division mathematical topics used in different engineering specialties. In addition to MATH 410-411 (or 412), which supplies a rigorous foundation for the
upper-level mathematics used in all areas. In addition,  every field of engineering nowadays makes heavy use of Optimization (treated in MATH 477),  Simulation (with relevant courses STAT 410 and 405 in the probabilistic context), and statistical data analysis (introduced in STAT 400-401 and STAT 430). All engineers confront issues of Mathematical Modeling, and so may benefit from MATH 420; and almost all must understand something about Reliability, which relies on Probability (STAT 400 or 410).

AEROSPACE ENGINEERING (ENAE) 

Vehicle design requires analysis of stress, material strength and elasticity.  The mathematical models involve partial differential equations (MATH 462 or 415).  Fourier analysis (MATH 464 or 417) is used to approximate solutions to equations.  Computer simulations involve numerical analysis (AMSC 460 or 466).  Navigation, tracking, and control also heavily involve differential equations and numerical analysis. Automatic-control methods involving filtering rely on probability (STAT 410) and stochastic processes.
 

BIOLOGICAL RESOURCES ENGINEERING (ENBE) 

Beyond sophomore coursework, statistical methods (STAT 401, STAT 430) are probably most useful.
 

CHEMICAL ENGINEERING (ENCH)

Mass-transport models involve partial differential equations (MATH 462 or 415).  Solutions of these involve Fourier analysis (MATH 464 or 417).  Numerical analysis (AMSC 460 or 466) is used for computer simulations. Optimization techniques are fundamental. Probability and statistics --- particularly quality control and experimental design (introduced in STAT 430) --- are used to design, track and evaluate
systems.

CIVIL ENGINEERING (ENCE) 

Civil engineers also analyze stress on materials. The models involve partial differential equations (MATH 462 or 415), their solution involves Fourier analysis (MATH 464 or 417), and the computer simulations involve finite element methods and eigenvalue problems in numerical analysis (AMSC 460 or 466).  The calculus of variations is used in advanced structural analysis. This is not a subject in the undergraduate curriculum, but MATH 410-411 provides a good preparation for this material.  Probability and statistics (STAT 400-401 or 410) are important for modeling and evaluation and for the reliability aspects of ENCE.  Traffic flow is one of the more mathematical areas of ENCE, involving partial differential equations (MATH 462), queueing theory (STAT 405), and computer simulations (AMSC 460, STAT 405).

ELECTRICAL ENGINEERING (ENEE) 

Overall, ENEE may be the most mathematical area of engineering. Looking over just UMCP's core graduate courses in ENEE, one sees for example information theory, random processes,  control theory and signal processing.  A student in ENEE needs a strong undergraduate background in mathematics. 

In particular, a strong ENEE undergraduate is well advised to take MATH 410-411. It is also advisable to take probability (STAT 410), needed for information theory, random processes, queueing theory in STAT 405, etc.), statistics (STAT 401 or 420, the essential mathematics for understanding real data probabilistically), Fourier analysis (the sine qua non of signal processing and an essential mathematical tool), complex analysis ("i's" are everywhere in EENE, you might as well take the course which gives you a basic understanding of complex vs. real analysis), MATH 464 (Laplace and Fourier transform methods; the course may contain basic "wavelet theory"; check with the Math advisor) and numerical analysis (AMSC 460 or 466).  At this point a math double-major could be completed with a useful algebra course, MATH 403 (abstract algebra) or MATH 405 (advanced linear algebra). 
 

FIRE PROTECTION ENGINEERING (ENFP)

Probability and statistics (e.g. STAT 400) are used in fire risk assessment and computer models.  Queueing theory (STAT 405) is used for resource allocation which is relevant to prioritizing fire protection investments.  Burning rate and heat transfer are modeled via partial differential equations (MATH 462) and computer simulations and numerical analysis (AMSC 460 or 466, MATH 420).

MATERIALS ENGINEERING (ENMA)

Some basic group theory (covered in greater depth as part of MATH 403) is essential in the diffraction analysis of materials. Thermodynamics and kinetics involve differential equations (MATH 246 or 414), partial differential equations (MATH 462) and their solutions with Fourier analysis (MATH 464). Numerical analysis (AMSC 460 or 466) is used in computer simulations. Stastistical experimental-design techniques (STAT 430) are important in optimal design, as is Probability (STAT 400 or 410) in Reliability.

MECHANICAL ENGINEERING (ME)

The importance of mathematics varies in ME with the particular track followed, but differential equations, numerical analysis, and stability/control theory appear everywhere: the relevant undergraduate math courses are MATH 246, 462 and 464, AMSC 460 or 466, MATH 414 or 417. In addition, engineering Reliability involves probability (STAT 400 or 410), and statistical experimental design (STAT 430) is also important.

NUCLEAR ENGINEERING (ENNU) 

Heat transfer and computer simulations are of particular importance in this branch of engineering. The mathematics involved includes probability (STAT 400 etc.), multi linear algebra (tensors), perturbation theory and variational  methods.  A good foundation in analysis (MATH 410, 411, 462, 464) can be helpful. 

RELIABILITY ENGINEERING (ENRE)

Probability and statistics (STAT 400-401, 410, and 430) are of particular importance in ENRE. 

SYSTEMS ENGINEERING (ENSE)

ENSE uses optimization (MATH 477), probability and statistics (STAT 400 etc.)  and numerical analysis (AMSC 460 or 466) in computer simulations.

MATHEMATICS DOUBLE MAJOR FOR PHYSICS AND ASTRONOMY STUDENTS 

No science is more deeply mathematical than physics. 
   Nature speaks to us in the language of mathematics.
       -Richard Feynmann, Physics Nobel Laureate

Mathematics courses not only support undergraduate work in physics. They provide a foundation for understanding the diverse higher mathematics with which a good physicist becomes comfortable.

For physics, most areas of mathematics are important.  Differential Equations are pervasive since Newton, with Ordinary Differential Equations arising in Classical Mechanics and Partial Differential Equations in Electromagnetic Theory, Continuum Mechanics and Field Theories.  Fourier Analysis is a key tool for solution of differential equations and analyzes of spectra, and in problems where explicit solution is not possible, Topology and Differential Geometry aid in understanding the qualitative behavior of solution trajectories.  The rapidly developing field of Nonlinear Dynamics --- of great interest both to mathematicians and physicists --- concerns the search for general qualitative structure of complicated trajectories of dynamical systems. Probability is fundamental in dynamics, statistical mechanics and quantum mechanics.  Probability is also the basis of Statistics, which is essential to deciphering the messiness of real data.  Lie Groups provide the setting for exploiting symmetry in problems of mechanics. Differential Geometry is essential for general relativity. Complex Analysis is fundamental from quantum mechanics through string theory. Functional Analysis provides the framework for quantum mechanics.

One may surmise that a physicist needs to know more math than a mathematician. This might be true!  The mathematician studies what can be proved rigorously. The physicist does too, but may often proceed by 
approximation, analogy, and numerical conjecture: all of these activities require a strong mathematical foundation.

The undergraduate math requirements in physics and astronomy (MATH 140, 141, 240, 241, 246) provide some initial ideas and vocabulary, and of course more mathematics is taught in the physics courses themselves.  We discuss next some of the other mathematics courses of particular relevance for physics (including astronomy).  Appropriate preparation and timing for some of these courses (especially MATH 410-411) varies with the student and should be discussed with an advisor. 

MATH 463 is a course about calculus, power series with complex numbers and potential theory.

MATH 452 is an undergraduate introduction to nonlinear dynamics and chaos. College Park is one of the world's major research centers in this area.

MATH 410-411 is the advanced calculus sequence. This sequence is heavily oriented to proof and theory and it is a watershed in the student's mathematical development. The material is prerequisite to further mathematics in differential equations, functional analysis, dynamics, geometry, etc.  This is the key optional sequence a mathematically strong student aiming at physics graduate school should consider.  It is one of the most difficult undergraduate mathematics courses, but not bad in comparison to graduate school in mathematics or physics.

MATH 403 introduces the basic abstract algebraic structures: groups, modules, rings, fields. This course is less crucial for a physicist than 410-411, but these fundamental structures (especially groups) arise in physics and a course like this is a good place to get clear about them. 

MATH 414 and MATH 415 are introductions (requiring MATH 410 and 411) to ordinary and partial differential equations at a deeper mathematical level, respectively going beyond the solution-techniques
in MATH 246 (ODE) and MATH 462 (PDE) to study rigorously qualitative theory of solutions.

MATH 417 is a mathematical introduction to Fourier analysis. MATH 464 (Transform Methods) addresses at a lower mathematical level some practical computational uses of  the Fourier and Laplace transforms. 

MATH 436 is a rigorous course in Differential Geometry.

STAT 410 is an introduction to Probability at advanced-calculus level, proving limit theorems like the Central Limit Theorem and Law of Large Numbers.

There are several undergraduate courses in applied and computational mathematics : AMSC 460 (Computational Methods) (emphasizing computation); AMSC 466 (like 460, but at a more rigorous mathematical level); and MATH 472 and MATH 473 (Methods and Models in Applied Mathematics I and II) (still more challenging).

Finally: some of the very best undergraduates take graduate courses in mathematics, especially MATH 630-631, which is to graduate mathematical study what MATH 410-411 is to undergraduate.

Top of the page