# Difference between revisions of "Option 2 packages"

(→Cryptography) |
|||

Line 21: | Line 21: | ||

* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531] | * Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531] | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

Line 60: | Line 60: | ||

* [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] | * [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] | ||

* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431] or [http://www.math.wisc.edu/531-probability-theory 531] | |||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

Line 95: | Line 95: | ||

** Math 376 is an honors course. | ** Math 376 is an honors course. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]. | * Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]. | ||

Line 119: | Line 119: | ||

** Math 376 is an honors course. | ** Math 376 is an honors course. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]. | * Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]. | ||

Line 163: | Line 163: | ||

* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415] | * Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415] | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514] | * Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514] | ||

Line 224: | Line 224: | ||

* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415] | |||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Additional courses above the 500 level. | * Additional courses above the 500 level. | ||

Line 249: | Line 249: | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Additional courses above the 500 level. | * Additional courses above the 500 level. | ||

Line 276: | Line 276: | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

Line 307: | Line 307: | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

Line 341: | Line 341: | ||

* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]. | * Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] | * Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521] | ||

Line 363: | Line 363: | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

Line 393: | Line 393: | ||

* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322] | * Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322] | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525] | * Computational Mathematics: At least two of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [https://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525] | ||

Line 425: | Line 425: | ||

'''Additional Courses to Consider''' | '''Additional Courses to Consider''' | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525] | * Additional courses in computational mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets Math 515], [http://www.math.wisc.edu/525-linear-programming-methods Math 525] | ||

Line 457: | Line 457: | ||

* Additional courses above the 500 level listed above. | * Additional courses above the 500 level listed above. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632] | * Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632] | ||

Line 482: | Line 482: | ||

* Additional courses above the 500 level listed above. | * Additional courses above the 500 level listed above. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

Line 506: | Line 506: | ||

* Additional courses above the 500 level listed above. | * Additional courses above the 500 level listed above. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632] | * Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632] | ||

Line 589: | Line 589: | ||

* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis Math 322] | |||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]) | * Two courses above the 500 level. Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521] and [http://www.math.wisc.edu/522-advanced-calculus 522]), and computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515], and [http://www.math.wisc.edu/525-linear-programming-methods 525]) | ||

Line 615: | Line 615: | ||

* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]. | |||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]. | * Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632]. | ||

Line 640: | Line 640: | ||

** Math 341 suggested. | ** Math 341 suggested. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475] | * Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475] | ||

Line 677: | Line 677: | ||

** Math 341 suggested. | ** Math 341 suggested. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435] | * Cryptography: [http://www.math.wisc.edu/435-introduction-cryptography Math 435] | ||

Line 708: | Line 708: | ||

* Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. | * Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461]. | ||

− | ** Math 431 | + | ** Math 431 and 309 are equivalent. |

** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ** Math 531 is advanced probability and may be taken only after Math 421 or Math 521. | ||

* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475] | * Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475] |

## Revision as of 14:21, 16 December 2014

The **Option 2 math major** requires six math courses and four courses in an area of application. These four courses are required to have a certain mathematical content. They should also form a coherent collection of courses that reflect a plan to study some discipline outside of mathematics that uses a fair amount of mathematics. The selection of the four courses, together with the six required math courses must be approved by the student's advisor. This page lists some sample packages in several popular areas.

## Contents

## Economics and Business

### Actuarial Mathematics

Actuaries use techniques in mathematics and statistics to evaluate risk in a variety of areas including insurance, finance, healthcare, and even criminal justice. In recent history the field has been revolutionized by advances in the theory of probability and the ability to access, store, and process very large data sets.

Professional actuaries are currently in great demand, have lucrative pay, and is a growth field [1]. Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [2]. Oftentimes students complete some of these examinations before graduating which allows them to move right into a career (Note: these exams are not required for graduation).

Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.

**Application Courses**

- Act. Sci 650 and 652

- Act. Sci. 651 or 653

**Core Math Courses**

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Statistics: Math 310
- Has the prerequisite: one of the probability courses mentioned above AND an elementary stats class (Stat 302 is recommended).

- Analysis: Math 521

- Stochastic Processes: Math 632

- Linear Programming: Math 525

**Additional Courses to Consider**

- Combinatorics: Math 475

**Also:** Students interested in the areas of mathematics with applications to actuarial science might consider the following as well:

- Advanced courses offered by the Department of Statistics or even a dual major.

- Consider combining the major with a program offered by the UW-Madison School of Business.

### Business

Applications of mathematics to business is often referred to as Operations Research or Management Science. Specifically, the goal is to use mathematics to make the best decisions in a variety of areas: searching, routing, scheduling, transport, etc.

The modern version of the field grew out of the work mathematicians did in order to aid the Allied war effort during world war II.[3] Since then, the field has grown into a robust and active area of research and scholarship including several journals and professional organizations.[4]

Students interested in applications of mathematics to business can find many employment opportunities in private corporations, government agencies, nonprofit enterprises, and more. Students can also move onto postgraduate programs in mathematics or business.

**Application Courses**

- Linear programming and Optimization: Math 525
- Operations Research: OTM 410
- At least two from the following: Gen Bus 304; OTM 451, 411, 633, 654
- Note that OTM 633 is crosslisted with math. It cannot be used as both an application course AND a core math course.

**Core Math Courses**

- Linear Algebra
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Statistics: Math/Stat 310

- Analysis: Math 521

- Stochastics: Math 632

**Additional Courses to Consider**

- Additional courses in computational mathematics.
- Math 633.

**Also:** Consider combining this program with a program in the UW-Madison School of Business.

### Economics

Economics is perhaps the most mathematical of the social sciences. Specifically economists wish to model and understand the behavior of individuals (people, countries, animals, etc.). Typically this is done by quantifying some elements of interest to the individuals.

Due to the quantitative nature of the field, economic theory has begun to move from the classic areas of markets, products, supply, demand, etc. and into many seemingly unrelated areas: law, psychology, political science, biology, and more.[5]

Regardless, the backbone of economics and economic theory is mathematics. The classical area of mathematics most often related with economics is analysis.

**Application Courses**

- Microeconomics: Econ 301 or 311.
- Macroeconomics: Econ 302 or 312.
- Economic Electives: At least two courses from Econ 410, 475, 503, 521, 525, and 666; Math 310 and Math 415.

**Core Math Courses**

- Linear Algebra
- Math 341 recommended.

- Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Analysis: Math 521 and 522.
- Stochastic Processes: [6].

**Additional Courses to Consider**

- Linear programming: Math 525.

**Also:**
Consider Combining this course of study with one offered by the Department of Economics.

### Finance

Financial mathematics is more popular than ever with financial firms hiring "quants" from all areas of mathematics and the natural sciences. Financial markets are of interest to mathematicians due to the difficult nature of modeling the complex systems. The standard tools involved are evolutionary differential equations, measure theory, and stochastic calculus.

**Application Courses**

- Statistics: Econ 410 or Math/Stat 310.
- Finance core: Finance 300, 320, 330.

**Core Math Courses**

- Linear Algebra
- Math 341 recommended.

- Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Analysis: Math 521.
- Partial Differential Equations: Mat 619.
- Stochastic Processes: Math 632.

**Additional Courses to Consider**

- Additional Analysis and Measure Theory: Math 522 and 629.
- Linear Programming (optimization): Math 525.

**Also:**
Consider combining the major with the program in Finance at the the Wisconsin School of Business.

## Physical Sciences

The physical sciences and mathematics have grown hand-in-hand since antiquity. Students with strong backgrounds in mathematics who are also interested in a branch of the physical sciences can find opportunities in laboratories, engineering firms, education, finance, law, business, and medicine. Those with very strong academic records can find themselves as preferred candidates for graduate study in their choice of field.

The following sample programs in mathematics have strong relationships with a particular area of interest in the natural sciences.

### Atmospheric & Oceanic Sciences

Weather and climate is determined by the interaction between two thin layers which cover the planet: The oceans and the atmosphere. Understanding how these two fluids act and interact allow humans to describe historical climate trends, forecast near future weather with incredible accuracy, and hopefully describe long term climate change which will affect the future of human society.

A student interested in atmospheric and oceanic studies who has a strong mathematics background can find a career working in local, national, and international meteorological laboratories. These include private scientific consulting businesses as well as public enterprises. Students interested in graduate study could find their future studies supported by the National Science Foundation, the Department of Energy, NASA, or others [7]. There is a large amount of funding available in the area due to the relevance research findings have on energy and economic policy.

Mathematicians who work in Atmospheric and oceanic studies are drawn to the complexities of the problems and the variety of methods in both pure and applied mathematics which can be brought to bear on them. Students should take coursework in methods of applied mathematics, differential equations, computational mathematics, and differential geometry and topology.

**Application Courses**

- Physics 208 or Physics 248 [8]
- Both of these classes have prerequisites (Physics 207/247).

- ATM OCN 310, 311, and 330 [9]
- 310 and 330 have Physics 208/248 as a prerequisite.

**Core Mathematics Courses**

- Linear Algebra
- Math 320 is recommended.

- Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Applied Analysis: Math 321 and 322
- Computational Mathematics: At least one of 513 or 514
- Theory of Differential Equations: Math 519

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Computational Mathematics: 513 or 514
- Analysis: Math 521
- Differential Geometry: Math 561
- Partial Differential Equations: Math 619

**Also:** Students who are interested in this area might consider

- Combining the math major with the program offered by the Department of Atmospheric and Oceanic Sciences.
- The AMEP program.

### Chemistry

The applications of mathematics to chemistry range from the mundane: Ratios for chemical reactants; to the esoteric: Computational methods in quantum chemistry. Research in this latter topic lead to a Nobel Prize in Chemistry to John Pople[10] who was a mathematician.

All areas of pure and applied mathematics have applications in modern chemistry. The most accessible track features coursework focusing on applied analysis and computational math. Students with a strong interest in theoretical mathematics should also consider modern algebra (for group theory) and topology.

**Application Courses**

- Physics 208 or Physics 248 [11]
- Both of these classes have prerequisites (Physics 207/247).

- Analytical Chemistry: Chem 327 or Chem 329[12]
- Prerequisite: Chem 104 or 109

- Physical Chemistry: Chem 561 and 562

**Core Math Courses**

- Linear Algebra
- Math 320 recommended.

- Introduction to Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Applied Analysis: Math 321 and Math 322
- Computational Mathematics: Math 443, 513, 514, Math 525
- Math 513 or 514 suggested.

- Theory of Differential Equations: Math 519

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Several higher level courses have connections to theoretical chemistry: Modern Algebra (Math 541), Topology (Math 551), Differential Geometry (Math 561), and Complex Analysis (Math 623)
- Any of these courses are acceptable in lieu of the 500 level courses above.

**Also:** Consider combining this program with one offered by the Department of Chemistry.

### Physics

Perhaps the subject with the strongest historical ties with mathematics is physics. Certainly some of the great physical theories have been based on novel applications of mathematical theory or the invention of new subjects in the field: Newtonian mechanics and calculus, relativity and Riemannian geometry, quantum theory and functional analysis, etc.

Nearly all mathematics courses offered here at UW Madison will have some applications to physics. The following is a collection of courses which would support general interest in physics.

**Application Courses**

- Mechanics, Electricity, and Magnetism: Physics 311 and Physics 322
- Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.

**Core Mathematics Courses**

- Linear Algebra
- Math 320 recommended.

- Introduction to Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Applied Analysis: Math 321 and Math 322
- Analysis: Math 521
- At least one more 500 level course. Suggested courses are: Modern Algebra (Math 541), Topology (Math 551), Differential Geometry (Math 561), and Complex Analysis (Math 623)

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Additional courses above the 500 level.

**Also:** Consider combining this program with one offered by the Department of Physics or Department of Astronomy.

### Astronomy

The Astronomy package has the same mathematics core, but different suggested application courses:

**Application Courses**

- Astronomy core: Choose two courses from Astron 310, 320, or 335.
- Physics Electives: At least two 3-credit physics courses above the 400 level. These cannot include labs. Suggested courses are Physics 415, 448, 449, 525, 531, 535, 545, and 551.

**Core Mathematics Courses**

- Linear Algebra
- Math 320 recommended.

- Introduction to Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Applied Analysis: Math 321 and Math 322
- Analysis: Math 521
- At least one more 500 level course. Suggested courses are: Modern Algebra (Math 541), Topology (Math 551), Differential Geometry (Math 561), and Complex Analysis (Math 623)

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Additional courses above the 500 level.

**Also:** Consider combining this program with one offered by the Department of Physics or Department of Astronomy.

## Biological Sciences

Applications of mathematics to biology has undergone a recent boom. Historically, the biologist was perhaps most interested in applications of calculus, but now nearly any modern area of mathematical research has an application to some biological field[13]. The following lists some possibilities.

### Bio-Informatics

Bioinformatics is the application of computational methods to understand biological information. Of course the most interesting items of biological information is genetic and genomic information. Considering that the human genome has over three billion basepairs [14], it's no wonder that many mathematicians find compelling problems in the area to devote their time.

Students with strong mathematical backgrounds who are interested in bioinformatics can find careers as a part of research teams in public and private laboratories across the world [15]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [16].

Students interested in bioinformatics should have a strong background in computational mathematics and probability. Students should also have a strong programming background.

**Application Courses**

- Computer Science: CS 302 and CS 367
- Bioinformatics: BMI/CS 576
- Genetics: Gen 466
- Note that this class has a prerequisite of a year of chemistry and a year of biology coursework. Please contact the UW-Madison genetics program for more information.

**Core Mathematics Courses**

- Linear Algebra
- Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Combinatorics: Math 475

**Additional Courses to Consider**

**Also**

- Students might consider combining this program with one in Computer Science or Genetics.
- Complete this major with a few additional courses if you are interested in medical school [17].

### Bio-Statistics

Biostatistics is the application of mathematical statistical methods to areas of biology. Historically, one could consider the field to have been founded by Gregor Mendel himself. He used basic principles of statistics and probability to offer a theory for which genetic traits would arise from cross hybridization of plants and animals. His work was forgotten for nearly fifty years before it was rediscovered and become an integral part of modern genetic theory.

Beyond applications to genetics, applications of biostatistics range from public health policy to evaluating laboratory experimental results to tracking population dynamics in the field. Currently, health organizations consider there to be a shortage of trained biostatisticians[18]. Students interested in this area should find excellent job prospects.

Students interested in biostatistics should have strong backgrounds in probability, statistics, and computational methods.

**Application Courses**

- Statistics: Stat 333, 424, and 575 [19]
- Stat 333 has as a prerequisite some experience with statistical software. This can be achieved by also registering for Stat 327. Stat 327 is a single credit course which does not count for the mathematics major

- Biostatistics: at least one of Stat 641 or 642

**Core Mathematics Courses**

- Linear Algebra
- Students who use either Math 320 or Math 340 to fulfill their Linear Algebra requirement must take Math 421 before any mathematics course numbered above 500.

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Statistics: Math 310

- Analysis: Math 521

**Additional Courses to Consider**

- More courses in computational mathematics listed above.
- Math 635

**Also**

- Consider combining this program with Statistics or a program in the College of Agriculture and Life Sciences.
- Compare this major program to requirements for Medical School.

### Ecology, Forestry, Wildlife Ecology

Applications of advanced mathematics to ecology has resulted in science's improved ability to track wild animal populations, predict the spread of diseases, model the impact of humans on native wildlife, control invasive species, and more. Modeling in this area is mathematically interesting due to the variety of scales and the inherent difficulty of doing science outside of a laboratory! As such the methods of deterministic and stochastic models are particularly useful.

**Application Courses**

- Statistics: Math 310.
- Computational Methods: CS 412.
- Any two courses from Zoo 460, 504, and 540; or F&W Ecol 300, 410, 460, 531, 652, and 655.

**Core Math Courses**

- Linear Algebra
- Math 341 recommended.

- Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Dynamical Systems: Math 415.
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Analysis: Math 521
- Stochastic Processes: Either Math 605 or 632.

**Additional Courses to Consider**

### Genetics

Applications of mathematics in genetics appear on a wide range of scales: chemical processes, cellular processes, organism breeding, and speciation. For applications of mathematics in genetics on the scale of chemical processes you might want to examine the suggested packages for bioinformatics or structural biology. If instead you are interested in the larger scale of organisms you might consider the package in biostatistics or the one below:

**Application Courses**

- Any four courses chosen from: GEN 466, 564, 565, 626, 629, and BMI 563.[20]

**Core Math Courses**

- Linear Algebra
- Math 341 recommended for non-honors students.

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Combinatorics: Math 475

- Analysis: Math 521.

- Stochastic Processes: Math 632.

**Additional Courses to Consider**

**Also:**

- Consider combining the program with one offered by the College of Agriculture and Life Sciences such as Genetics.

### Structural Biology

Structural biologists are primarily interested in the large molecules which are involved in cellular processes: the fundamental chemical building blocks of life. The field lies on the intersection of biology, physics, chemistry, and mathematics and so structural biology is an exciting area of interdisciplinary research.

In general, the mathematics involved in structural biology is focused on computational methods, probability, and statistics. Note that we offer a specialized course in Mathematics Methods in Structural Biology - Math 606.

**Application Courses**

- Analytical Methods in Chemistry: Chem 327 or 329
- Physical Chemistry: Chem 561 and 562
- Mathematical Methods in Structural Biology: Math 606

**Core Math Courses**

- Linear Algebra
- Math 320 recommended.

- Applied Analysis: Math 321 and Math 322
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Computational Mathematics: At least two of 513, 514, Math 515, Math 525

**Additional Courses to Consider**

**Also:**
Consider combining the program with one in Biology, Biochemistry, or Chemistry.

### Systems Biology

Systems biology is the computational and mathematical modeling of biological systems at any scale. The classical example of this may be the predator-prey model of differential equations which describe the relative population dynamics of two species. Other systems examples include disease transmission, chemical pathways, cellular processes, and more.

In general, the mathematics involved in systems biology is focused on computational methods, dynamical systems, differential equations, the mathematics of networks, control theory, and others. Note that we offer a specialized course in Mathematical Methods in Systems Biology - Math 609.

**Application Courses**

- Organic Chemistry: Chem 341 or 343
- Introductory Biochemistry: Biochem 501
- Mathematical Methods in Structural Biology: Math 609
- One Biochem elective: Any Biochem class numbered above 600. Suggested courses are Biochem 601, 612, 620, 621, 624, and 630.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended.

- Applied Analysis: Math 321 and Math 322
- Dynamical Systems: Math 415.
- Computational Mathematics: At least one of 513, 514, or Math 525
- Differential Equations: Math 519.

**Additional Courses to Consider**

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Additional courses in computational mathematics: 513, 514, Math 515, Math 525

**Also:**
Consider combining the program with one in Biology, Biochemistry, or Chemistry.

## Engineering

Engineering is the application of science and mathematics to the invention, improvement, and maintenance of anything and everything. As with many of the sciences, engineers and mathematicians have a symbiotic relationship: Engineers use mathematics to make new things; the new things exhibit novel properties that are mathematically interesting.

In general all of mathematics can be applied to some field of engineering. However the programs offered below are not substitutes for engineering degrees. That is, student who are interested in an engineering career upon completion of their undergraduate degree should probably enroll in one of the engineering programs offered by the College of Engineering. Similarly, students who are primarily interested in mathematics might instead choose an option I major and concentrate their upper level coursework in applied mathematics. Students who are truly interested in both areas should consider the degree program in Applied Mathematics, Engineering, and Physics.

So who do the programs below serve? They serve engineering students who wish to take a second major in mathematics. In general such students are excellent candidates for graduate study in engineering.

### Chemical Engineering

The following program details an option 2 package for students in the College of Engineering program in Chemical Engineering who are interested in pursuing a second major in mathematics.

**Application Courses**

- CBE 320, 326, 426, 470
- Note: All of these course are required for the undergraduate program in chemical engineering.
- Several of these courses have as a prerequisite other engineering and science courses.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended.

- Differential Equations: Math 319 or Math 376
- Applied Analysis: Math 321 and Math 322
- Two courses above the 500 level. Suggested courses to choose from are real analysis (Math 521 and 522), algebra (Math 541), complex analysis (Math 623), and computational mathematics (in particular Math 513 and 514).

**Additional Courses to Consider**

- Additional courses above the 500 level listed above.
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Stochastic Processes: Math 632

### Civil Engineering

The following program details an option 2 package for students in the College of Engineering program in Civil and Environmental Engineering who are interested in pursuing a second major in mathematics.

**Application Courses**

- Core Fluid Mechanics and Structural Analysis: CIV ENG 310, 311, 340
- Note: All of these course are required for the undergraduate program in civil engineering.
- Several of these courses have as a prerequisite other engineering and science courses.

- One elective Structural Analysis Course: CIV ENG 440, 442, 445, or 447.
- Each of these courses may be used as an elective in the undergraduate program in civil engineering.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended.

- Differential Equations: Math 319 or Math 376
- Applied Analysis: Math 321 and Math 322
- Two courses above the 500 level. Suggested courses to choose from are real analysis (Math 521 and 522); and computational mathematics (in particular Math 513, 514, and 515).

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Additional courses above the 500 level listed above.
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

### Electrical and Computer Engineering

The following program details an option 2 package for students in the College of Engineering program in Computer and Electrical Engineering who are interested in pursuing a second major in mathematics.

**Application Courses**

- Core ECE: ECE 220, 230, 352
- Note: All of these course are required for the undergraduate program in electrical and computer engineering.
- Several of these courses have as a prerequisite other engineering and science courses.

- One elective: ECE 435, 525, or 533.
- Each of these courses may be used as an elective in the undergraduate program in civil engineering.
- ECE 435 is crosslisted with math. It cannot be used as both an application course AND a core math course.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended.

- Differential Equations: Math 319 or Math 376
- Applied Analysis: Math 321 and Math 322
- At least two courses above the 500 level. Suggested courses to choose from are real analysis (Math 521 and 522), linear programming Math 525, modern algebra Math 541, differential geometry Math 561, and complex analysis Math 623.

**Additional Courses to Consider**

- Additional courses above the 500 level listed above.
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Stochastic Processes: Math 632
- Error correcting codes: Math 641

### Engineering Mechanics and Astronautics

The following program details an option 2 package for students in the College of Engineering program in Engineering Mechanics and Astronautics who are interested in pursuing a second major in mathematics.

**Application Courses**

- Core Engineering Mechanics: EMA 201, 202, 303
- One elective: EMA 521, 542, 545, or 563
- All of the above courses may be used to satisfy the EMA program requirements.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended

- Differential Equations: Math 319 or Math 376
- Applied Analysis: Math 321 and Math 322
- Two courses above the 500 level. Suggested courses to choose from are real analysis (Math 521 and 522), computational methods in mathematics (Math 513, 514, 515, and 525), and complex analysis Math 623.

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Additional courses above the 500 level listed above.

### Industrial Engineering

Industrial engineering is the application of engineering principles to create the most effective means of production. In particular, they work to optimize complex systems.

**Application Courses[21]**

- Core Industrial engineering: I SY E 315, 320, and 321.
- Industrial Engineering Elective: At least one of I SY E 516, 526, 558, 615, 620, 624, 635, or 643.

**Core Math Courses**

- Linear Algebra
- Math 341 recommended.

- Differential Equations: Math 319 or Math 376
- Math 376 is an honors course.

- Probability: Math 309 or Math 431 or 531
- Both 309 and 431 are preferred over stat 311.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Computational Mathematics: 513 and Math 525
- Stochastic Processes: Math 632.

**Additional Courses to Consider**

**Also:**
Consider the program in Industrial Engineering offered by the College of Engineering.

### Materials Science

The following program details an option 2 package for students in the College of Engineering program in Materials Science and Engineering who are interested in pursuing a second major in mathematics.

**Application Courses**

- Core Materials Courses: MSE 330, 331, and 351
- One Engineering Elective: CBE 255, CS 302, CS 310, ECE 230, ECE 376, EMA 303, Phys 321, Stat 424].
- All of the above classes may be used to satisfy the program requirements for MS&E BS degree.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended

- Differential Equations: Math 319 or Math 376
- Applied Analysis: Math 321 and Math 322
- Two courses above the 500 level. Suggested courses to choose from are real analysis (Math 521 and 522), and computational methods in mathematics (Math 513, 514, 515, and 525)

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Additional courses above the 500 level listed above.

### Mechanical Engineering

The following program details an option 2 package for students in the College of Engineering program in Mechanical Engineering who are interested in pursuing a second major in mathematics.

**Application Courses**

- Core Mechanical Engineering Courses: ME 340, 361, 363, 364
- All of the above courses are required by the Mechanical Engineering program.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended

- Differential Equations: Math 319 or Math 376
- Applied Analysis: Math 321 and Math 322
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Two courses above the 500 level. Suggested courses to choose from are real analysis (Math 521 and 522), and computational methods in mathematics (Math 513, 514, 515, and 525)

**Additional Courses to Consider**

- Dynamical Systems: Math 415
- Additional courses above the 500 level listed above.

### Nuclear Engineering

The following program details an option 2 package for students in the College of Engineering program in Nuclear Engineering who are interested in pursuing a second major in mathematics.

**Application Courses**

- Core Nuclear Engineering Courses: NE 305, 405, and 408
- One Engineering Elective: Physics 321 or 322, ECE 376, BME 501, or NE 411.
- All of the above classes may be used to satisfy the program requirements for the Nuclear Engineering BS degree.

**Core Math Courses**

- Linear Algebra
- Math 320 recommended

- Differential Equations: Math 319 or Math 376
- Applied Analysis: Math 321 and Math 322
- Two courses above the 500 level. Suggested courses to choose from are real analysis (Math 521), computational methods in mathematics (Math 513 and 514), Differential Geometry (Math 561), and Complex Analysis (Math 623).

**Additional Courses to Consider**

- Dynamical Systems: Math 415.
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Stochastic Processes: Math 632.
- Additional courses above the 500 level listed above.

## Computer Science

Computer science as an independent discipline is rather young: The first computer science degree program offered in the United States was formed in 1962 (at Purdue University). Despite its youth, one could argue that no single academic discipline has had more of an effect on human society since the scientific revolution.

Since computer science is foremost concerned with the theory of computation, its link with mathematics is robust. As examples consider Alan Turing: A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation. John Von Neumann: A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[22] More recently, one of the most famous unsolved problems in mathematics, the P vs NP problem, is considered an open problem in the theory of computation as well.

Since computer science is a full field enveloping philosophy, mathematics, and engineering there are many possible areas of interest which a student of mathematics and computer science might focus on. Below are several examples.

### Computational Methods

Computational methods are the algorithms a computer follows in order to perform a specific task. Of interest besides the algorithms is methods for evaluating their quality and efficiency. Since computational mathematics is on the interface between pure and applied methods students who concentrate in this area can find many exciting research opportunities available at the undergraduate level.

The mathematical coursework focuses on combinatorics, analysis, and numerical methods.

**Application Courses**

- Any four courses from: CS 352, 367, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 570.
- Note that 435, 475, 513, 514, 515, and 525 are crosslisted with math. They may not be used as both application courses and core mathematics courses

**Core Math Courses**

- Linear Algebra
- Math 341 suggested.

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Combinatorics: Math 475
- Advanced Calculus Math 421 and/or Math 521
- Computational Mathematics: At least two of 513, 514, Math 515, Math 525

**Additional Courses to Consider**

- Additional courses in computational mathematics above.
- Statistics: Math 310.
- Analysis II: Math 522.
- Algebra: Math 541 and 542.
- Number Theory: Math 567.
- Logic: Math 571.

**Also:**
Consider combining this program with one in the Department of Computer Science.

### Theoretical Computer Science

If you are interested in a more theoretical bend to your studies, follow the program above but with the following changes:

- Include both CS 520 and CS 577 into your core applied courses.
- Replace the two computational methods courses with Math 567 and Math 571.

### Cryptography

Due to the widespread use of computer storage, platforms, and devices; security is now of singular interest. Students with expertise in the mathematics associated with cryptography can find interesting opportunities after graduation in public and private security sectors.

The mathematics associated to secure messaging and cryptography is typically centered on combinatorics and number theory.

**Application Courses**

- Programming: CS 302 and 367.
- One of the following two pairs:
- The CS track: Operating systems (CS 537) and Security (CS 642)
- The ECE track: Digital Systems: (ECE 352) and Error Correcting Codes (ECE 641).

**Core Math Courses**

- Linear Algebra
- Math 341 suggested.

- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Cryptography: Math 435
- Algebra: Math 541 and 542
- Number Theory: Math 567

**Additional Courses to Consider**

**Also:**
Consider combining this program with one in the Department of Computer Science or in Computer Engineering.

## Secondary Education

The so called STEM fields are continue to be a major area of interest and investment for education policy makers. In particular secondary education instructors with strong mathematics backgrounds are in demand across the nation in public, private, and charter school environments.

The following program was designed for a math major who is interested in becoming an educator at the secondary (and above) level. Note that successful completion of the coursework outlined below would make a strong candidate for graduate work in mathematics and education at the masters level. Our own School of Education offers a Masters Degree in Secondary Mathematics which concludes with state certification.

Note that a major requires at least two courses at the 500 level, but changing any of the courses listed below may prevent you from covering all of the content areas required for Wisconsin state certification. Therefore you should consider the recommendations below carefully.

**Application Courses**

- Statistics: Math 310
- History and philosophy of mathematics: Math 473.
- At least two additional courses from Computer Science, Physics, or Mathematics at the Intermediate or Advanced Level.
- Suggested: two of Math 371, CS 302, or Phys 207.

**Core Math Courses**

- Linear Algebra
- Math 341 suggested.

- Geometry: Math 461.
- Probability: Math 309 or Math 431 or 531
- Math 431 and 309 are equivalent.
- Math 531 is advanced probability and may be taken only after Math 421 or Math 521.

- Combinatorics: Math 475
- Modern Algebra: Math 441 or 541.
- Math 541 preferred.

- Analysis: Math 421 or 521.
- Math 521 preferred.

**Additional Courses to Consider**

- Additional courses at the 500 level in mathematics.