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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.
'''NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see the named options section in the [https://guide.wisc.edu/undergraduate/letters-science/mathematics/mathematics-bs/#requirementstext Guide page of the major]). The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options.
 
'''
 
 
 
The '''Option 2 math major''' requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. 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 course collections in several popular areas.
 
NOTES:
 
1) '''These course collections do not include course prerequisites.'''  For example, math 310 has stats 302 as a prerequisite.  But stat 302 cannot be used as a focus or major course.
 
2) '''Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments.'''  For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.


== Economics and Business ==
== Economics and Business ==
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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.
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 [http://www.bls.gov/ooh/math/actuaries.htm].  Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/].  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).
Professional actuaries are currently in demand, have lucrative pay, and is a growth field [http://www.bls.gov/ooh/math/actuaries.htm].  Similar to some other fields (law, accounting, etc.) there are professional organizations which administer a series of examinations [http://www.beanactuary.org/exams/].  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.
Students who are interested in actuarial mathematics should consider coursework in probability, statistics, analysis, as well as computational mathematics.


'''Application Courses'''
'''Application Courses'''
* [http://www.math.wisc.edu/node/96 Math 303]
* Act Sci 303


* Act. Sci 650 and 652
* Act. Sci 650 and 652
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'''Core Math Courses'''
'''Core Math Courses'''
* [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]]
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** 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: [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 is preferred over 309.   
** 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.  


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


* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics] or even a dual major.
* Advanced courses offered by the [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Department of Statistics].


* Consider combining the major with a [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.
* A [http://bus.wisc.edu/knowledge-expertise/academic-departments/actuarial-science-risk-management-insurance program] offered by the UW-Madison School of Business.


=== Business ===
=== Business ===
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'''Application Courses'''
'''Application Courses'''
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]
* Linear programming and Optimization: [http://www.math.wisc.edu/525-linear-programming-methods Math 525]
** Note that this course cannot also be used as a core math course.
* Operations Research: OTM 410
* Operations Research: OTM 410
* At least two from the following: Gen Bus 304; OTM 351, 411, 633, 654
* At least two from the following: Gen Bus 306, Gen Bus 307; 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.
** 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'''
'''Core Math Courses'''
* [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]]
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** 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: [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 is preferred over 309.   
** 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.


* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]
* Statistics: [http://www.math.wisc.edu/node/111 Math/Stat 310]


* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]
* Computational Mathematics: [https://www.math.wisc.edu/514-numerical-analysis Math 514] or [http://www.math.wisc.edu/513-numerical-linear-algebra 513]


* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]
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* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].
* [http://www.math.wisc.edu/633-queueing-theory-and-stochastic-modeling Math 633].


'''Also:'''  Consider combining this program with a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].
'''Also:'''  Consider a program in the UW-Madison [http://bus.wisc.edu/bba/academics-and-programs/majors/operations-technology-management School of Business].


=== Economics ===
=== 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.[http://en.wikipedia.org/wiki/Economics_imperialism]
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, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 recommended.
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
* 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 and 309 are equivalent. 
** 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].
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].
'''Additional Courses to Consider'''
* Linear programming: [http://www.math.wisc.edu/525-linear-programming-methods Math 525].
'''Also:'''
Consider a program in the [http://www.econ.wisc.edu/undergrad/Reqs%20for%20Major.html Department of Economics].


=== Finance ===
=== 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 recommended.
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
* 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 and 309 are equivalent. 
** 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].
* Partial Differential Equations: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Mat 619].
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].
'''Additional Courses to Consider'''
* Additional Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522] and [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].
* Linear Programming (optimization): [http://www.math.wisc.edu/525-linear-programming-methods Math 525].
'''Also:'''
Consider a program in [http://bus.wisc.edu/bba/academics-and-programs/majors/finance Finance] at the the Wisconsin School of Business.


== Physical Sciences ==
== Physical Sciences ==
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'''Core Mathematics Courses'''
'''Core Mathematics Courses'''
* [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]]
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
** Math 376 is an honors course.
** Students who take math 320 should instead consider an additional course below.
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]
* Applied Analysis: [http://www.math.wisc.edu/321-applied-mathematical-analysis Math 321] and [http://www.math.wisc.edu/322-applied-mathematical-analysis 322]
* Computational Mathematics:  At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]
* Computational Mathematics:  At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513] or [http://www.math.wisc.edu/514-numerical-analysis 514]
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* 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]
* 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 is preferred over 309.   
** 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]
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'''Also:''' Students who are interested in this area might consider  
'''Also:''' Students who are interested in this area might consider  
* Combining the math major with the program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].
* A program offered by the [http://www.aos.wisc.edu/education/undergrad_program.htm Department of Atmospheric and Oceanic Sciences].
* The [http://www.math.wisc.edu/amep AMEP] program.
* The [http://www.math.wisc.edu/amep AMEP] program.


=== Chemistry ===
=== 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[http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html] who was a mathematician.
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 mathematician [http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1998/index.html John Pople].


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 classical mathematics should also consider modern algebra (for group theory) and topology.
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'''
'''Application Courses'''
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]
* Physics 208 or Physics 248 [http://www.physics.wisc.edu/academic/undergrads/course-descriptions]
** Both of these classes have prerequisites (Physics 207/247).
** Both of these classes have prerequisites (Physics 207/247).
* Analytical Chemistry: Chem 327 or Chem 329[https://www.chem.wisc.edu/content/courses]
* Analytical Chemistry: Chem 327 or Chem 329[http://www.chem.wisc.edu/content/courses]
** Prerequisite: Chem 104 or 109   
** Prerequisite: Chem 104 or 109   
* Physical Chemistry: Chem 561 and 562
* Physical Chemistry: Chem 561 and 562


'''Core Math Courses'''
'''Core Math Courses'''
* Linear Algebra: [http://www.math.wisc.edu/320-linear-algebra-and-differential-equations Math 320] or [http://www.math.wisc.edu/340-elementary-matrix-and-linear-algebra Math 340] or [http://www.math.wisc.edu/341-linear-algebra Math 341] or [http://www.math.wisc.edu/375-topics-multi-variable-calculus-and-linear-algebra Math 375]
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 375 is an honors course.
** Math 320 recommended.
** Students who take Math 320 or Math 340 should consider Math 421 before any course above the 521 level.
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who do not take Math 320 should also complete a course in differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
*** Math 376 is an honors course.
** Students who take math 320 should instead consider an additional course below.
* 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]
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]
** Math 513 or 514 suggested.
* Theory of Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519]
 
'''Additional Courses to Consider'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]
* Several higher level courses have connections to theoretical chemistry: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]
** Any of these courses are acceptable in lieu of the 500 level courses above.
 
'''Also:''' Consider a program offered by the [http://www.chem.wisc.edu/content/undergraduate 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: [http://www.physics.wisc.edu/academics/undergrads/inter-adv-311 Physics 311] and [http://www.physics.wisc.edu/academics/undergrads/inter-adv-322 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 421 is suggested to prepare students for math 521.
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
** Students who take math 320 should instead consider an additional course below.
* 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]
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]
* At least one more 500 level course. See suggested courses below.
'''Additional Courses to Consider'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]
* 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 and 309 are equivalent. 
** Math 531 is advanced probability and may be taken only after Math 421 or Math 521.
* ODEs: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].
* PDEs: [http://www.math.wisc.edu/619-analysis-of-partial-differential-equations Math 619].
* Modern Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541].
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].
* Differential Geometry [http://www.math.wisc.edu/561-differential-geometry Math 561].
* Complex Analysis: [http://www.math.wisc.edu/623-complex-analysis Math 623].
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/514-numerical-analysis 514].
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Introduction to Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
** Students who take math 320 should instead consider an additional course below.
* 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]
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521]
* At least one more 500 level course. Suggested courses are: Modern Algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), Topology ([http://www.math.wisc.edu/551-elementary-topology Math 551]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623)]


'''Additional Courses to Consider'''
'''Additional Courses to Consider'''
'''Also:'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]
* 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 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.


=== Physics ===
'''Also:''' Consider a program offered by the [http://www.physics.wisc.edu/ Department of Physics] or [http://www.astro.wisc.edu/ Department of Astronomy].


== Biological Sciences ==
== Biological Sciences ==
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'''Application Courses'''
'''Application Courses'''
* Computer Science: [http://www.cs.wisc.edu/courses/302 CS 302] and [http://www.cs.wisc.edu/courses/367 CS 367]
* Computer Science: CS 300 and CS 400 (or CS 302 and CS 367).
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]
* Bioinformatics: [http://www.biostat.wisc.edu/content/bmi-576-introduction-bioinformatics BMI/CS 576]
* Genetics: Gen 466
* Genetics: Gen 466
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'''Core Mathematics Courses'''
'''Core Mathematics Courses'''
* [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]]
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** 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.
** 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: [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 is preferred over 309.   
** 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.


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'''Also'''   
'''Also'''   
* Students might consider combining this program with one in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].
* Consider a program in [http://www.cs.wisc.edu/academics/Undergraduate-Programs Computer Science] or [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].
* Complete this major with a few additional courses if you are interested in medical school [http://prehealth.wisc.edu/explore/documents/Pre-Med.pdf].


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'''Application Courses'''
'''Application Courses'''
* Statistics: Stat 333, 424, and 575 [http://www.stat.wisc.edu/course-listing]
* Statistics: Any four from Stat 333, 424, 575, 641, and 642 [http://www.stat.wisc.edu/course-listing]
** 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
** 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'''
'''Core Mathematics Courses'''
* [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]]
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** 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.
** 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: [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 is preferred over 309.   
** 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 221: Line 345:


'''Also'''
'''Also'''
* Consider combining this program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].
* Consider a program with [http://www.stat.wisc.edu/undergrad/undergraduate-statistics-program Statistics] or in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences].
* Compare this major program to requirements for Medical School.
* Compare this major program to requirements for Medical School.


=== Ecology ===
=== 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.


=== Forestry ===
'''Application Courses'''
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].
* Computational Methods: [http://www.cs.wisc.edu/courses/412 CS 412].
* Any two courses from [http://zoology.wisc.edu/courses/courselist.htm Zoo 460, 504, and 540]; or [http://forestandwildlifeecology.wisc.edu/undergraduate-study-courses F&W Ecol 300, 410, 460, 531, 652, and 655].
 
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 recommended.
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].
* 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 and 309 are equivalent. 
** 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]
* Stochastic Processes: Either [http://www.math.wisc.edu/math-605stochastic-methods-biology Math 605] or [http://www.math.wisc.edu/632-introduction-stochastic-processes 632].
 
'''Additional Courses to Consider'''
* 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/525-linear-programming-methods Math 525]


=== Genetics ===
=== 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.[http://www.genetics.wisc.edu/UndergraduateProgram.htm]
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 recommended for non-honors students.
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
* 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 and 309 are equivalent. 
** 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]
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].
'''Additional Courses to Consider'''
* Computational Mathematics: [http://www.math.wisc.edu/443-applied-linear-algebra Math 443], [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], [http://www.math.wisc.edu/525-linear-programming-methods Math 525]
'''Also:'''
* Consider a program in the [http://www.cals.wisc.edu/departments/major College of Agriculture and Life Sciences] such as [http://www.genetics.wisc.edu/UndergraduateProgram.htm Genetics].


=== Structural Biology ===
=== 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: [http://www.math.wisc.edu/mathematical-methods-structural-biology Math 606]
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* 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]
* 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 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 [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]
'''Additional Courses to Consider'''
* Differential Geometry: [http://www.math.wisc.edu/561-differential-geometry Math 561]
* Topology: [http://www.math.wisc.edu/551-elementary-topology Math 551].
'''Also:'''
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].


=== Systems Biology ===
=== Systems Biology ===
Systems biology is the computational and mathematical modeling of biological systems at any scale.  The classical example of this may be the [http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation 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: [http://www.math.wisc.edu/609-mathematical-methods-systems-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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* 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]
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].
* Computational Mathematics: At least one of [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]
* Differential Equations: [http://www.math.wisc.edu/519-ordinary-differential-equations Math 519].
'''Additional Courses to Consider'''
* 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 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: [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]
'''Also:'''
Consider a program in [http://biologymajor.wisc.edu/ Biology], [http://www.biochem.wisc.edu/ Biochemistry], or [http://www.chem.wisc.edu/ Chemistry].


== Engineering ==
== 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 [http://www.engr.wisc.edu/current/undergrad.html 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 [http://www.math.wisc.edu/amep Applied Mathematics, Engineering, and Physics].


=== Astronautics ===
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 ===
=== Chemical Engineering ===
Line 243: Line 464:
'''Application Courses'''
'''Application Courses'''
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]
* [http://www.engr.wisc.edu/cmsdocuments/cbe-undergrade-handbook-2009-v7.pdf CBE 320, 326, 426, 470]
** Note:  All of these course are required in for the undergraduate program in chemical engineering.
** 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'''
'''Core Math Courses'''
* Linear Algebra: [http://www.math.wisc.edu/340-elementary-matrix-and-linear-algebra Math 340] or [http://www.math.wisc.edu/341-linear-algebra Math 341] or [http://www.math.wisc.edu/375-topics-multi-variable-calculus-and-linear-algebra Math 375]
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who take math 320 should instead consider an additional course below.
* 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]
* Two courses above the 500 level.  Suggested courses to choose from are more real analysis (Math 521 and 522), algebra (Math 541), complex analysis (Math 623), and computational mathematics (Math 513 and 514).
* 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]), algebra ([http://www.math.wisc.edu/541-modern-algebra Math 541]), complex analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]), and computational mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and [http://www.math.wisc.edu/514-numerical-analysis 514]).


'''Additional Courses to Consider'''
'''Additional Courses to Consider'''
* Additional courses above the 500 level listed above.
* 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 is preferred over 309.   
** 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]


=== Civil Engineering ===
=== 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: [http://courses.engr.wisc.edu/cee/ 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who take math 320 should instead consider an additional course below.
* 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]
* 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 mathematics (in particular [http://www.math.wisc.edu/513-numerical-linear-algebra Math 513], [http://www.math.wisc.edu/514-numerical-analysis 514], and [http://www.math.wisc.edu/515-introduction-splines-and-wavelets 515]).
'''Additional Courses to Consider'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]
* Additional courses above the 500 level listed above.
* 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 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: [http://courses.engr.wisc.edu/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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who take math 320 should instead consider an additional course below.
* 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]
* At least 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]), linear programming [http://www.math.wisc.edu/525-linear-programming-methods Math 525], modern algebra [http://www.math.wisc.edu/541-modern-algebra Math 541], differential geometry [http://www.math.wisc.edu/561-differential-geometry Math 561], and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].
'''Additional Courses to Consider'''
* Additional courses above the 500 level listed above.
* 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 and 309 are equivalent. 
** 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]
* Error correcting codes: [http://www.math.wisc.edu/641-introduction-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: [http://courses.engr.wisc.edu/ema/ 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who take math 320 should instead consider an additional course below.
* 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]
* 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]), 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]), and complex analysis [http://www.math.wisc.edu/623-complex-analysis Math 623].
'''Additional Courses to Consider'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]
* Additional courses above the 500 level listed above.


=== Computer Engineering ===


=== Industrial Engineering ===
=== 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[http://www.engr.wisc.edu/isye/isye-curriculum-documents.html]'''
* Core Industrial engineering: I SY E 315, 320, and 323.
* Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.
** Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 recommended.
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Math 376 is an honors course.
* 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]
** 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: [http://www.math.wisc.edu/513-numerical-linear-algebra 513] and [http://www.math.wisc.edu/525-linear-programming-methods Math 525]
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].
'''Additional Courses to Consider'''
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].
* Numerical Analysis: [http://www.math.wisc.edu/514-numerical-analysis Math 514].
'''Also:'''
Consider the program in [http://www.engr.wisc.edu/isye/isye-academics-undergraduate-program.html 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: [http://www.engr.wisc.edu/cmsdocuments/Degree_requirements_2014.pdf MSE 330, 331, and 351]
* One Engineering Elective: CBE 255, CS 300, 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.


=== Material Science ===
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who take math 320 should instead consider an additional course below.
* 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]
* 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])
 
'''Additional Courses to Consider'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]
* Additional courses above the 500 level listed above.


=== Mechanical Engineering ===
=== 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: [http://www.engr.wisc.edu/cmsdocuments/me-flowchart-spring-2014.pdf ME 340, 361, 363, 364]
** All of the above courses are required by the Mechanical Engineering program.
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who take math 320 should instead consider an additional course below.
* 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]
* 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 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 ([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])
'''Additional Courses to Consider'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415]
* Additional courses above the 500 level listed above.


=== Nuclear Engineering ===
=== 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: [http://www.engr.wisc.edu/cmsdocuments/NE-UGguide2014.pdf 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
* Differential Equations: [http://www.math.wisc.edu/319-techniques-ordinary-differential-equations Math 319] or [http://www.math.wisc.edu/node/83 Math 376]
** Students who take math 320 should instead consider an additional course below.
* 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]
* Two courses above the 500 level.  Suggested courses to choose from are real analysis ([http://www.math.wisc.edu/521-advanced-calculus Math 521]), computational methods in mathematics ([http://www.math.wisc.edu/513-numerical-linear-algebra Math 513] and  [http://www.math.wisc.edu/514-numerical-analysis 514]), Differential Geometry ([http://www.math.wisc.edu/561-differential-geometry Math 561]), and Complex Analysis ([http://www.math.wisc.edu/623-complex-analysis Math 623]).
'''Additional Courses to Consider'''
* Dynamical Systems: [http://www.math.wisc.edu/415-applied-dynamical-systems-chaos-and-modeling Math 415].
* 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 and 309 are equivalent. 
** 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].
* Additional courses above the 500 level listed above.


== Computer Science ==
== 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.  Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[https://web.archive.org/web/20130314123032/http://qss.stanford.edu/~godfrey/vonNeumann/vnedvac.pdf]  There are broad overlaps in reasearch in the two fields.  For example, one of the most famous unsolved problems in mathematics, the [http://www.claymath.org/millenium-problems/p-vs-np-problem P vs NP] problem, is also considered an open problem in the theory of computation.
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, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.
** 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 suggested.
* 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 and 309 are equivalent. 
** 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]
* Advanced Calculus [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] and/or [http://www.math.wisc.edu/521-advanced-calculus 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], [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 to Consider'''
* Additional courses in computational mathematics above.
* Statistics: [http://www.math.wisc.edu/node/114 Math 310].
* Analysis II: [http://www.math.wisc.edu/522-advanced-calculus Math 522].
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542].
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567].
* Logic: [http://www.math.wisc.edu/571-mathematical-logic Math 571].
'''Also:'''
Consider the program in the [http://www.cs.wisc.edu/academics/Undergraduate-Programs 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 ===
=== 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 300 and CS 400 (or 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'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 suggested.
* 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 and 309 are equivalent. 
** 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]
* Algebra: [http://www.math.wisc.edu/541-modern-algebra Math 541] and [http://www.math.wisc.edu/542-modern-algebra 542]
* Number Theory: [http://www.math.wisc.edu/567-elementary-number-theory Math 567]
'''Additional Courses to Consider'''
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].
'''Also:'''
Consider combining the programs offered by [http://www.cs.wisc.edu/academics/Undergraduate-Programs Department of Computer Science] or [http://www.engr.wisc.edu/ece/ece-academics-undergraduate-program.html Computer Engineering].
== Secondary Education ==
The so called STEM fields 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 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 [http://www.uwteach.com/mathematics.html Masters Degree in Secondary Mathematics] which concludes with state certification. 
''Note that a major requires at least two courses at the 500 level. Therefore you should consider the suggestions below carefully.''
'''Application Courses'''
* History and philosophy of mathematics: [http://www.math.wisc.edu/473-history-mathematics Math 473].
* Math education capstone course: [http://www.math.wisc.edu/371-basic-concepts-mathematics Math 471]
* Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.
** Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567


=== Theoretical Computer Science ===
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 (or Math 375) suggested.
* College Geometry: [http://www.math.wisc.edu/461-college-geometry-i Math 461].
* Probability: [http://www.math.wisc.edu/node/111 Math 309] or [http://www.math.wisc.edu/431-introduction-theory-probability Math 431]
** Math 431 and 309 are equivalent. 
** [http://www.math.wisc.edu/531-probability-theory Math 531] can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.
* Statistics: [http://www.math.wisc.edu/node/114 Math 310] (Math 310 has a prerequisite of Math 309 or 431.)
* Modern Algebra: [http://www.math.wisc.edu/441-introduction-modern-algebra Math 441] or [http://www.math.wisc.edu/541-modern-algebra 541].
* Analysis: [http://www.math.wisc.edu/421-theory-single-variable-calculus Math 421] or [http://www.math.wisc.edu/521-advanced-calculus 521].
** Math 521 is strongly suggested for students planning to teach AP Calculus in high school


== Education ==
'''Additional Courses to Consider'''
* Math 421 can be a useful course to take before the 500 level coursework.
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475]
* Additional courses at the 500 level in mathematics.
* Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.


== Statistics ==
== Statistics ==
Statistics is the study of the collection, measuring, and evaluation of data.  Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before.  Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.
Statistics has a strong relationship with mathematics.  The areas of mathematics of particular interest are linear algebra, probability, and analysis.
'''Application Courses'''
* Core Statistics: Stat 333 and Stat 424
* Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.
'''Core Math Courses'''
* one [[Undergraduate_Linear_Algebra_Courses|Linear Algebra]] course
** Math 341 suggested.
* Mathematical Statistics Sequence: [http://www.math.wisc.edu/node/111 Math 309] and [http://www.math.wisc.edu/node/114 Math 310]
** Math 431 may be used for Math 309.
* Combinatorics: [http://www.math.wisc.edu/475-introduction-combinatorics Math 475].
* Analysis: [http://www.math.wisc.edu/521-advanced-calculus Math 521].
* Stochastic Processes: [http://www.math.wisc.edu/632-introduction-stochastic-processes Math 632].
'''Additional Courses to Consider'''
* Computational Mathematics: [http://www.math.wisc.edu/513-numerical-linear-algebra 513], [http://www.math.wisc.edu/514-numerical-analysis 514], or [http://www.math.wisc.edu/525-linear-programming-methods Math 525]
* Analysis and Measure Theory: [http://www.math.wisc.edu/522-advanced-calculus Math 522], [http://www.math.wisc.edu/621-analysis-iii-0 621], or [http://www.math.wisc.edu/629-introduction-measure-and-integration 629].
* Advanced Probability Theory [http://www.math.wisc.edu/531-probability-theory Math 531].
* Algebra: [https://www.math.wisc.edu/541-modern-algebra Math 541].
'''Also:'''
A student who wishes to complete a major in statistics offered by the [https://www.stat.wisc.edu/undergrad/undergraduate-major-statistics Department of Statistics] should complete the program above and include:
* Stat 302 and 327.
* A course in programming (e.g. CS 300).
* At least one more course from the statistics electives above.

Latest revision as of 16:43, 9 May 2021

NOTE: in the Fall 2020 semester the Department of Mathematics introduced five new named options (see the named options section in the Guide page of the major). The old "Option 2" math major is not available to students anymore. Those who declared the Option 2 math major before Fall 2020 may finish it with the original rules, or they may switch to one of the new named options.



The Option 2 math major requires six math courses and four courses in an area of focus. These four courses are required to have a certain mathematical content. 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 course collections in several popular areas.

NOTES:

1) These course collections do not include course prerequisites. For example, math 310 has stats 302 as a prerequisite. But stat 302 cannot be used as a focus or major course.

2) Courses offered by departments/schools/colleges outside of mathematics may have restricted enrollments. For example, an L&S student interested in an option 2 program with finance emphasis may not reliably be able to enroll in fin 300 since it is taught by Business.

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 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 303
  • 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).

Additional Courses to Consider

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

  • 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
    • Note that this course cannot also be used as a core math course.
  • Operations Research: OTM 410
  • At least two from the following: Gen Bus 306, Gen Bus 307; 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

  • one Linear Algebra 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.

Additional Courses to Consider

  • Additional courses in computational mathematics.
  • Math 633.

Also: Consider 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, 460, 475, 503, 521, 525, and 666; Math 310 and Math 415.

Core Math Courses

Additional Courses to Consider

Also: Consider a program in 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

  • one Linear Algebra course
    • 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 a 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 [6]. 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 [7]
    • Both of these classes have prerequisites (Physics 207/247).
  • ATM OCN 310, 311, and 330 [8]
    • 310 and 330 have Physics 208/248 as a prerequisite.

Core Mathematics Courses

  • one Linear Algebra course
  • Differential Equations: Math 319 or Math 376
    • Math 376 is an honors course.
    • Students who take math 320 should instead consider an additional course below.
  • 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

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 mathematician John Pople.

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 [9]
    • Both of these classes have prerequisites (Physics 207/247).
  • Analytical Chemistry: Chem 327 or Chem 329[10]
    • Prerequisite: Chem 104 or 109
  • Physical Chemistry: Chem 561 and 562

Core Math Courses

  • one Linear Algebra course
    • Math 320 recommended.
  • Introduction to Differential Equations: Math 319 or Math 376
    • Math 376 is an honors course.
    • Students who take math 320 should instead consider an additional course below.
  • 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 a program 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

  • one Linear Algebra course
    • Math 421 is suggested to prepare students for math 521.
  • Introduction to Differential Equations: Math 319 or Math 376
    • Math 376 is an honors course.
    • Students who take math 320 should instead consider an additional course below.
  • Applied Analysis: Math 321 and Math 322
  • Analysis: Math 521
  • At least one more 500 level course. See suggested courses below.

Additional Courses to Consider


Also: Consider a program 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

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 a program 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[11]. 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 [12], 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 [13]. Moreover, many universities have established interdisciplinary graduate programs promoting this intersection of mathematics, biology, and computer science [14].

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 300 and CS 400 (or 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

  • one Linear Algebra course
    • 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.

Additional Courses to Consider

Also

  • Consider a program in Computer Science or Genetics.
  • Complete this major with a few additional courses if you are interested in medical school [15].

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[16]. 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: Any four from Stat 333, 424, 575, 641, and 642 [17]
    • 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.

Core Mathematics Courses

  • one Linear Algebra course
    • 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.

Additional Courses to Consider

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

Also

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

Core Math Courses

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.[18]

Core Math Courses

  • one Linear Algebra course
    • 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.

Additional Courses to Consider

Also:

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

Additional Courses to Consider

Also: Consider a program 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

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 a program 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

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

  • one Linear Algebra course
  • Differential Equations: Math 319 or Math 376
    • Students who take math 320 should instead consider an additional course below.
  • 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

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

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[19]

  • Core Industrial engineering: I SY E 315, 320, and 323.
  • Industrial Engineering Elective: At least one of I SY E 425, 516, 525, 526, 558, 575, 615, 620, 624, 635, or 643.
    • Note: ISYE 425 and 525 are both crosslisted with math and cannot be used to complete both the application and core math requirements.

Core Math Courses

  • one Linear Algebra course
    • 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 300, 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

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

  • one Linear Algebra course
  • Differential Equations: Math 319 or Math 376
    • Students who take math 320 should instead consider an additional course below.
  • 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

  • one Linear Algebra course
  • Differential Equations: Math 319 or Math 376
    • Students who take math 320 should instead consider an additional course below.
  • 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. Historical examples include Alan Turing, A mathematician and WWII cryptoanalyst who's theory of the Universal Turing Machine forms the central framework of modern computation; and John Von Neumann, A mathematician who's name is ascribed to the architecture still used for nearly all computers today.[20] There are broad overlaps in reasearch in the two fields. For example, one of the most famous unsolved problems in mathematics, the P vs NP problem, is also considered an open problem in the theory of computation.

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, 400, 412, 435, 475, 513, 514, 515, 520, 525, 533, 540, 545, 558, 559, and 577.
    • 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

Additional Courses to Consider

Also: Consider the program 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 300 and CS 400 (or 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

Additional Courses to Consider

Also: Consider combining the programs offered by Department of Computer Science or Computer Engineering.

Secondary Education

The so called STEM fields 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 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. Therefore you should consider the suggestions below carefully.

Application Courses

  • History and philosophy of mathematics: Math 473.
  • Math education capstone course: Math 471
  • Two additional courses from Mathematics, Computer Science, Physics, or Economics at the Intermediate or Advanced Level.
    • Suggested: CS 300, CS 302, Phys 207, Math 421, Math 475, Math 561, Math 567

Core Math Courses

  • one Linear Algebra course
    • Math 341 (or Math 375) suggested.
  • College Geometry: Math 461.
  • Probability: Math 309 or Math 431
    • Math 431 and 309 are equivalent.
    • Math 531 can also be considered. This is a proof based introduction to probability and may be taken only after Math 421 or Math 521.
  • Statistics: Math 310 (Math 310 has a prerequisite of Math 309 or 431.)
  • Modern Algebra: Math 441 or 541.
  • Analysis: Math 421 or 521.
    • Math 521 is strongly suggested for students planning to teach AP Calculus in high school

Additional Courses to Consider

  • Math 421 can be a useful course to take before the 500 level coursework.
  • Combinatorics: Math 475
  • Additional courses at the 500 level in mathematics.
  • Courses in computer programming, statistics, physics, economics, and finance can broaden your content areas and qualify you for more subjects.

Statistics

Statistics is the study of the collection, measuring, and evaluation of data. Recent advances in our ability to collect and parse large amounts of data has made the field more exciting then ever before. Positions in data analysis are becoming common outside of laboratory environments: marketing, education, health, sports, infrastructure, politics, etc.

Statistics has a strong relationship with mathematics. The areas of mathematics of particular interest are linear algebra, probability, and analysis.

Application Courses

  • Core Statistics: Stat 333 and Stat 424
  • Statistics Electives: At least two from: Stat 349, 351, 411, 421, 456, 471, 609, or 610.

Core Math Courses

Additional Courses to Consider

Also: A student who wishes to complete a major in statistics offered by the Department of Statistics should complete the program above and include:

  • Stat 302 and 327.
  • A course in programming (e.g. CS 300).
  • At least one more course from the statistics electives above.
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