mathematics

Mathematics bridges the humanities and the sciences. Its position among the humanities is based on the study of mathematics as one of the liberal arts for more than two thousand years. The natural sciences have invariably turned to mathematics for techniques needed to explore the consequences of scientific theories. In the last few decades social scientists have increasingly found higher mathematics of value in their training and research.  Still an expanding subject, mathematics is a part of more new and challenging frontiers than at any time in its long history—with many new fields, from data science to quantum computing, requiring new techniques and inspiring ideas for exploration.

Graduating math majors have obtained employment in a variety of jobs in business, industry, and governmental agencies and also have obtained teaching positions at the secondary school level (such teaching positions normally require teaching certification). Others have continued their education at the graduate level in mathematics and other fields. Departments in a variety of fields which use mathematics, including some in the social and biological sciences as well as in engineering and the physical sciences, are interested in attracting math majors into their graduate programs. Math Ph.D.'s obtain academic positions at the college and university level and nonacademic positions entailing consulting and research. The math major requirements are flexible enough to allow preparation for various goals, interests, and careers.

Students interested in mathematics might also consider the related degree program in applied mathematics, engineering and physics.

DECLARATION

To declare a major in mathematics, a student must have completed the sequence MATH 221, MATH 222, and MATH 234, or the sequence MATH 375 and MATH 376, with a 2.500 GPA or better. Major advisors may waive this requirement for students with alternative coursework and experiences (e.g., transfer students). Students should meet with a math advisor before declaring in order to discuss course selection and major plan. Advising information can be found in the Advising and Careers link.

University General Education Requirements

All undergraduate students at the University of Wisconsin–Madison are required to fulfill a minimum set of common university general education requirements to ensure that every graduate acquires the essential core of an undergraduate education. This core establishes a foundation for living a productive life, being a citizen of the world, appreciating aesthetic values, and engaging in lifelong learning in a continually changing world. Various schools and colleges will have requirements in addition to the requirements listed below. Consult your advisor for assistance, as needed. For additional information, see the university Undergraduate General Education Requirements section of the Guide.

General Education
  • Breadth—Humanities/Literature/Arts: 6 credits
  • Breadth—Natural Science: 4 to 6 credits, consisting of one 4- or 5-credit course with a laboratory component; or two courses providing a total of 6 credits
  • Breadth—Social Studies: 3 credits
  • Communication Part A & Part B *
  • Ethnic Studies *
  • Quantitative Reasoning Part A & Part B *

* The mortarboard symbol appears before the title of any course that fulfills one of the Communication Part A or Part B, Ethnic Studies, or Quantitative Reasoning Part A or Part B requirements.

College of Letters & Science Degree Requirements: Bachelor of Arts (B.A.)

Students pursuing a bachelor of arts degree in the College of Letters & Science must complete all of the requirements below. The College of Letters & Science allows this major to be paired with either a bachelor of arts or a bachelor of science curriculum.

Bachelor of Arts degree requirements

Mathematics Complete the University General Education Requirements for Quantitative Reasoning A (QR-A) and Quantitative Reasoning B (QR-B) coursework.
Foreign Language
  • Complete the fourth unit of a foreign language; OR
  • Complete the third unit of a foreign language and the second unit of an additional foreign language.
L&S Breadth
  • 12 credits of Humanities, which must include 6 credits of literature; and
  • 12 credits of Social Science; and
  • 12 credits of Natural Science, which must include one 3+ credit Biological Science course and one 3+ credit Physical Science course.
Liberal Arts and Science Coursework Complete at least 108 credits.
Depth of Intermediate/Advanced work Complete at least 60 credits at the intermediate or advanced level.
Major Declare and complete at least one major.
Total Credits Complete at least 120 credits.
UW-Madison Experience
  • 30 credits in residence, overall; and
  • 30 credits in residence after the 86th credit.
Quality of Work
  • 2.000 in all coursework at UW–Madison
  • 2.000 in Intermediate/Advanced level coursework at UW–Madison

Non–L&S students pursuing an L&S major

Non–L&S students who have permission from their school/college to pursue an additional major within L&S only need to fulfill the major requirements. They do not need to complete the L&S Degree Requirements above.

Requirements for the Major

The mathematics major requirements include exposure to at least two areas of advanced mathematics.  The program is ideal for any student who has a broad interest in mathematics both pure and applied and functions well as a standalone or complementary program. 

Mathematics Major requirements

At least seven MATH courses for at least 21 credits are required for the major as follows1:

Linear Algebra (complete one): 23
Linear Algebra
Linear Algebra and Differential Equations
Elementary Matrix and Linear Algebra
Topics in Multi-Variable Calculus and Linear Algebra
Analysis, Topology, Algebra (complete two):6
Analysis I
Modern Algebra
Elementary Topology
Advanced MATH Elective (complete one):3
Numerical Linear Algebra
Numerical Analysis
Ordinary Differential Equations
Analysis I
Analysis II
Linear Optimization
Probability Theory
Mathematical Methods in Data Science
Linear Algebra II
Modern Algebra
Modern Algebra
Elementary Topology
Elementary Geometric and Algebraic Topology
Differential Geometry
Modern Number Theory
Fundamentals of Set Theory
Mathematical Logic
Stochastic Methods for Biology
Topics in Mathematics Study Abroad
Mathematical Methods for Physical Modeling in Biology
Mathematical Methods for Systems Biology
Analysis of Partial Differential Equations
Analysis III
Complex Analysis
Introduction to Fourier Analysis
Introduction to Measure and Integration
Introduction to Stochastic Processes
An Introduction to Brownian Motion and Stochastic Calculus
Introduction to Error-Correcting Codes
Senior Honors Thesis
Senior Honors Thesis
Undergraduate Thesis
Undergraduate Thesis
Directed Study
Directed Study
Additional MATH Elective to achieve 7 courses and 21 credits in the major9
Introduction to Probability and Mathematical Statistics I 3
An Introduction to Probability and Markov Chain Models
Introduction to the Theory of Probability
Introduction to Probability and Mathematical Statistics II
Techniques in Ordinary Differential Equations 4
Linear Algebra and Differential Equations
Topics in Multi-Variable Calculus and Differential Equations
Applied Mathematical Analysis
Applied Mathematical Analysis
Topics in Mathematics Study Abroad
Applied Dynamical Systems, Chaos and Modeling
The Theory of Single Variable Calculus
Introduction to Combinatorial Optimization
Introduction to Cryptography
Introduction to Modern Algebra
Applied Linear Algebra
College Geometry I
Introduction to Number Theory
History of Mathematics
Introduction to Combinatorics
Undergraduate Seminar
Topics in Undergraduate Mathematics
Numerical Linear Algebra
Numerical Analysis
Ordinary Differential Equations
Analysis I
Analysis II
Linear Optimization
Probability Theory
Mathematical Methods in Data Science
Linear Algebra II
Modern Algebra
Modern Algebra
Elementary Topology
Elementary Geometric and Algebraic Topology
Differential Geometry
Modern Number Theory
Fundamentals of Set Theory
Mathematical Logic
Stochastic Methods for Biology
Topics in Mathematics Study Abroad
Mathematical Methods for Physical Modeling in Biology
Mathematical Methods for Systems Biology
Analysis of Partial Differential Equations
Analysis III
Complex Analysis
Introduction to Fourier Analysis
Introduction to Measure and Integration
Introduction to Stochastic Processes
An Introduction to Brownian Motion and Stochastic Calculus
Introduction to Error-Correcting Codes
Senior Honors Thesis
Senior Honors Thesis
Undergraduate Thesis
Undergraduate Thesis
Directed Study
Directed Study
Total Credits21

Residence and Quality of Work

  • 2.000 GPA in all MATH and major courses.
  • 2.000 GPA on 15 upper-level major credits, taken in residence.5
  • 15 credits in MATH, taken on the UW–Madison campus.

Named Options

Honors in the Major

Students may declare Honors in the Major in consultation with the Mathematics Honors advisor; this should be done by the start of the junior year.  Honors in the major is not available in any Named Option program.

Honors in the Mathematics Major Requirements

To earn Honors in the Major, students must satisfy both the requirements for the mathematics major (above) and the following additional requirements:

  • Earn a 3.300 University GPA
  • Earn a 3.300 GPA for all MATH courses, and all courses accepted in the major
  • Complete the following courses, with individual grades of B or better:
Analysis I
and Analysis II (Taken for Honors) 6
Modern Algebra
and Modern Algebra (Taken for Honors) 6
Select at least two more courses from MATH 500 through MATH/​E C E  641. These course must be taken for honors. The following will usually be one of the courses: 7
Elementary Topology
Select one of these Capstone projects:
Senior Honors Thesis
and Senior Honors Thesis (For a total of 6 credits)
or
A sequence of two upper-level mathematics courses deemed acceptable by the Mathematics Honors advisor 7

Footnotes

1

A course may only apply once toward the courses/credits required for the major.  Thus, a course used to meet the Analysis, Topology and Algebra requirement may not also be used to meet the requirement for MATH 500-699 requirement and a course used to meet the MATH 500-699 requirement may not also be used in the Additional Math requirement.

2

Only one of these courses will be used to fulfill minimum course/credit requirements for the major:  MATH 320MATH 340MATH 341MATH 375

3

At most one course in Introductory Probability may be used to fulfill the course/credit requirements for the major:  MATH/​STAT  309 and MATH/​STAT  431.

4

 At most one course in Elementary Differential Equations may be used to fulfill the course/credit requirements for the major: MATH 319MATH 320MATH 376.

5

MATH courses numbered 307–699 are considered upper level in the major.

6

At least one of the two sequences (MATH 521MATH 522 or MATH 541MATH 542) must be completed prior to enrolling in the Capstone project.

7

Chosen in consultation with the Mathematics Honors advisor.

University Degree Requirements 

Total Degree To receive a bachelor's degree from UW–Madison, students must earn a minimum of 120 degree credits. The requirements for some programs may exceed 120 degree credits. Students should consult with their college or department advisor for information on specific credit requirements.
Residency Degree candidates are required to earn a minimum of 30 credits in residence at UW–Madison. "In residence" means on the UW–Madison campus with an undergraduate degree classification. “In residence” credit also includes UW–Madison courses offered in distance or online formats and credits earned in UW–Madison Study Abroad/Study Away programs.
Quality of Work Undergraduate students must maintain the minimum grade point average specified by the school, college, or academic program to remain in good academic standing. Students whose academic performance drops below these minimum thresholds will be placed on academic probation.
  1. State, explain, and apply the principal results, definitions, and theorems of a wide collection of mathematical areas including at least one area of advanced undergraduate mathematics.
  2. Construct and evaluate mathematical proofs and arguments.
  3. Acquire a diverse set of skills and strategies in mathematical reasoning/problem solving.
  4. Use mathematics to model and analyze phenomena in other disciplines.
  5. Write, explain, and present mathematics to both experts and non-­experts.

Sample Four-Year Plan

This Sample Four-Year Plan is a tool to assist students and their advisor(s). Students should use it—along with their DARS report, the Degree Planner, and Course Search & Enroll tools—to make their own four-year plan based on their placement scores, credit for transferred courses and approved examinations, and individual interests. As students become involved in athletics, honors, research, student organizations, study abroad, volunteer experiences, and/or work, they might adjust the order of their courses to accommodate these experiences. Students will likely revise their own four-year plan several times during college.

Mathematics Major - Bachelor of Arts/Science Degree

Freshman
FallCreditsSpringCredits
MATH 2211,25MATH 22224
Communication A3Ethnic Studies3
Foreign Language (if needed)4Foreign Language (if needed)4
Literature Breadth3Literature Breadth3
 15 14
Sophomore
FallCreditsSpringCredits
MATH 2344MATH 3413
Communication B3Intermediate MATH33
Humanities Breadth 3Humanities Breadth 3
Physical Science Breadth 3Physical Sciences Breadth 3
Elective3Elective3
 16 15
Junior
FallCreditsSpringCredits
Intermediate MATH33Intermediate MATH33
Advanced MATH43Advanced MATH43
Social Sciences Breadth 3Social Sciences Breadth 3
Biological Sciences Breadth 3Biological Sciences Breadth 3
Elective3Elective3
 15 15
Senior
FallCreditsSpringCredits
Advanced MATH43Social Sciences Breadth 3
Social Science Breadth 3Elective3
Elective3Elective3
Elective3Elective3
Elective3Elective3
 15 15
Total Credits 120
1

Math majors will naturally complete Quantitative Reasoning requirements with the introductory calculus courses required to declare the major.

2

Declaration of the Mathematics major requires a 2.500 cumulative GPA across the introductory calculus sequence. Students that are unable to establish a GPA for any courses in the introductory calculus sequence are encouraged to speak with a math major advisor as soon as possible.

3

An intermediate level math course is any numbered above 306 excluding MATH 320, MATH 340, or MATH 341, or MATH/​CURRIC  471.

4

An advanced level MATH course is any numbered above 500.

Sample Three-Year Plan

This Sample Three-Year Plan is a tool to assist students and their advisor(s). Students should use it —along with their DARS report, the Degree Planner, and Course Search & Enroll tools — to make their own three-year plan based on their placement scores, credit for transferred courses and approved examinations, and individual interests.

Three-year plans may vary considerably from student to student, depending on their individual preparation and circumstances. Students interested in graduating in three years should meet with an advisor as early as possible to discuss feasibility, appropriate course sequencing, post-graduation plans (careers, graduate school, etc.), and opportunities they might forgo in pursuit of a three-year graduation plan.

Departmental Expectations

Historically, students who have successfully complete a three year undergraduate degree with a major in Mathematics have the following qualifications: a minimum of 29 advanced standing credits, which include completion of the following with either course credit or via placement examination:

Therefore the plan below assumes these requirements, but none other. When considering the plan below, students should note the following:

  • Advanced standing credits may satisfy Ethnic Studies, Communication Part B, and/or Letters & Science Breadth degree requirements which are listed in the plan. In this case, students should adjust their plan by reorganizing the remaining degree requirements using the following priorities: 1) Ethnic Studies and Communication Part B (obligatory in the first year); 2) Physical, Biological, and Social Science Breadth (which may be prerequisites for more advanced electives); 3) Humanities and Literature. Remaining schedule space should be considered electives.
  • At least 26 of the non-MATH credits must be at the Intermediate or Advanced level.
  • Consider using the elective space in the plan as follows: additional major or certificate, career readiness, graduate school preparation, and other personal interests.
First Year
FallCreditsSpringCredits
MATH 2344MATH Linear Algebra3
Ethnic Studies3Intermediate MATH3
Communication B3Physical Science Breadth3
Biological Science Breadth3Biological Science Breadth3
Physical Science Breadth3Foreign Language (if needed for the B.A.) or Elective3
 16 15
Second Year
FallCreditsSpringCredits
Intermediate MATH3Advanced MATH3
Advanced MATH3Intermediate MATH3
Literature Breadth3Literature Breadth3
Social Science Breadth3Social Science Breadth3
Elective (Intermediate or Advanced level)3Elective (Intermediate or Advanced level)3
 15 15
Third Year
FallCreditsSpringCredits
Advanced MATH3Humanities Breadth (Intermediate or Advanced level)3
Social Science Breadth3Social Science Breadth (Intermediate or Advanced level)3
Humanities Breadth3Elective (Intermediate or Advanced level)9
Elective (Intermediate or Advanced level)6 
 15 15
Total Credits 91

Advising

Students who are interested in the math major should visit a faculty advisor. During the fall and spring semesters faculty advisors have regular office hours. Students are encouraged to make appointments, but drop in advising is also welcome. During the winter break and the summer semester there are no fixed advising hours. Students should contact one of the advisors directly to set up an appointment. The current list of advisors, their schedule of office hours, and available appointments can be found at the Math advising page.

For advice on college algebra, pre-calculus, and calculus, see the placement advising pages of the department. 

Transition courses

All majors are required to complete at least one of the following. It is suggested that majors (and those interested in the major) complete such a course as soon in their academic career as possible.

Linear Algebra
Applied Mathematical Analysis
and Applied Mathematical Analysis
Topics in Multi-Variable Calculus and Linear Algebra
The Theory of Single Variable Calculus
Introduction to Number Theory

Graduate Study

Students preparing for graduate work in mathematics should take the following courses:

MATH 341 Linear Algebra3
or MATH 375 Topics in Multi-Variable Calculus and Linear Algebra
MATH 521 Analysis I3
MATH 522 Analysis II3
MATH 541 Modern Algebra3
MATH 542 Modern Algebra3
MATH 551 Elementary Topology3
or MATH 561 Differential Geometry
Select at least two other courses at the 500 level or higher

Students who plan to enter a mathematics Ph.D. program should acquire a reading knowledge of at least one foreign language as early as possible. For mathematics study, the most useful languages are French, German, and Russian.

Careers

In recent years graduating math majors have obtained employment in a variety of jobs in business, industry, and governmental agencies and also have obtained teaching positions at the secondary school level (such teaching positions normally require teaching certification). Others have continued their education at the graduate level in mathematics and other fields. Departments in a variety of fields which use mathematics, including the social and biological sciences as well as in engineering and the physical sciences, are interested in attracting math majors into their graduate programs. Math Ph.D.'s obtain academic positions at the college and university level and nonacademic positions entailing consulting and research. The math major requirements are flexible enough to allow preparation for various goals.

L&S career resources

Every L&S major opens a world of possibilities.  SuccessWorks at the College of Letters & Science helps students turn the academic skills learned in their major, certificates, and other coursework into fulfilling lives after graduation, whether that means jobs, public service, graduate school or other career pursuits.

In addition to providing basic support like resume reviews and interview practice, SuccessWorks offers ways to explore interests and build career skills from their very first semester/term at UW all the way through graduation and beyond.

Students can explore careers in one-on-one advising, try out different career paths, complete internships, prepare for the job search and/or graduate school applications, and connect with supportive alumni and even employers in the fields that inspire them.

Professors:

Anderson, David F (Undergraduate Director)
Angenent, Sigurd B.
Arinkin, Dima
Caldararu, Andrei
Craciun, Gheorghe
Denisov, Sergey
Ellenberg, Jordan
Feldman, Mikhail
Gong, Xianghong
Gurevich, Shamgar
Kent, Autumn Exum
Lempp, Steffen
Mari-Beffa, Gloria
Maxim, Laurentiu
Miller, Joseph S
Paul, Sean T
Poltoratski, Alexei
Roch, Sebastien
Seeger, Andreas
Seppalainen, Timo (Chair)
Smith, Leslie M.
Soskova, Mariya
Stechmann, Sam
Stovall, Betsy (Graduate Director)
Street, Brian Thomas
Terwilliger, Paul M.
Thiffeault, Jean-Luc
Valko, Benedek (Associate Chair)
Waleffe, Fabian
Yang, Tonghai

Associate Professors:

Andrews, Uri
Dymarz, Tullia Maria
Erman, Daniel M
Gorin, Vadim
Ifrim, Mihaela
Kim, Chanwoo
Li, Qin
Marshall, Simon Lindsay
Spagnolie, Saverio
Tran, Hung Vinh

Wang, Botong

Assistant Professors:

Chen, Nan
Cochran, Amy
Guo, Shaoming
Kemeny, Michael L J
Lyu, Hanbaek
Rodriguez, Jose Israel
Shankar, Ananth
Shcherbyna, Tetyana
Shen, Hao
Uyanik, Caglar
Waldron, Alex
Wu, Chenxi
Zepeda-Nunez, Leonardo
Zimmer, Andrew

Academic staff:

Benguria Andrews, Soledad (Calculus Coordinator)
Grizzard, Robert (Associate Director for Instructional Programs)
Jackson, Billy (Pre-Calculus Coordinator)
Kwon, Oh Hoon (Math Education Coordinator)
Lindsey, Melissa (Associate Director for Instructional Support)
Rineck, Leah (Math Learning Center Director)
Work, Grace (Associate Director of Undergraduate Research)

Enrollment Coordinator:

Kyle Martinez