Pure Mathematics Concentration

Pure mathematics is the study of the beauty and power of mathematics. All areas in this dynamic discipline demand high standards and encourage independent, critical, and creative thinking.

Students in the Pure Mathematics concentration explore the whole spectrum of mathematics, including such areas as algebra, number theory, analysis, geometry, and topology. Some topics have been explored thoroughly for centuries, while many new seeds of mathematics are being born around us even today. Advanced courses in the subject areas reveal and explain many interesting mathematical truths that few people ever see, and entice and inspire mathematics enthusiasts to explore even deeper into the mysteries that it has to offer.

In abstract algebra one takes the familiar world of numbers, with the operations of addition, subtraction, multiplication, and division, as a model for developing other more abstract types of mathematical objects, structures, and operations.

Topology and geometry deal with the visual world of shapes and space and investigate ways in which shapes can be classified according to various properties.

Analysis is the branch of mathematics that studies processes that involve continuous change through differentiation and integration.

Number theory is one of the oldest branches of mathematics. While it began as a study of the properties of the integers, in modern times, the subject has moved beyond that into investigations of abstract number systems.

Combinatorics is often simply described as the “mathematics of counting” which indicates the subject’s origins. Today, however, the subject includes a much wider array of concerns involving the existence, enumeration, analysis, and optimization of discrete structures.

Requirements

The Pure Mathematics concentration is organized around a core of three courses covering the three main areas of pure mathematics:

Math 302 Modern Algebra (3)

Math 414 Topology (3)

Math 450 Advanced Calculus II (3)

Along with the core courses, students choose four of the following courses, investigating additional topics in pure mathematics.

Math 407 Abstract Algebra (3)

Math 412 Complex Analysis (3)

Math 425 Differential Geometry (3)

Math 430 Number Theory (3)

Math 471 Combinatorics (3)

Please see the prerequisite diagram of the courses required for this concentration.

Careers

Many pure mathematics graduates go on to graduate school in one of the mathematical, statistical, or computer sciences. Many others enter the workplace and apply their mathematical expertise and skills in abstraction to problems in research, industry, business, or government. For example, the National Security Agency (NSA) hires countless experts in the field of number theory to perform highly specialized activities to protect United States information systems and produce foreign intelligence information. NSA mathematicians design and analyze complex algorithms, express difficult cryptographic problems in mathematical terms, and find solutions or demonstrate that no solution exists, given certain computational limitations and time constraints. More information about the NSA is available at www.nsa.gov.

Other employers of pure math majors include brokerage firms, commercial banks, financial organizations, and companies involved with aerospace, communications, transportation, insurance, pharmaceuticals, manufacturing, and consulting, as well as other government agencies such as the Department of Defense, the CIA, and NASA.

Not surprisingly, many math majors, including pure math majors, choose to become teachers. The demand for good mathematics teachers at all levels of the U.S. educational system is immense, and public schools in particular are often forced to hire teachers who are not ideally qualified for the job. Therefore, the opportunities for math teachers at the K-12 level are great. More information about this can be found in the department's teaching credential page.

Teaching positions at a community college or technical institute usually require a Master's degree in math, while a career in teaching at a 4-year college or university often requires a Ph.D. in mathematics.

Positions at 4-year colleges and universities normally expect the individual to pursue research activities in addition to teaching so as to contribute to the advancement and knowledge of his or her field of expertise. This work may involve collaborative efforts with other professors at the institution or at other institutions, or with students. Moreover, mathematics professors are invariably called upon to provide service to the university in areas ranging from curriculum development to career advisement to admissions protocol.

Finally, earning a degree in pure mathematics is a challenging and impressive accomplishment in its own right. Training in pure mathematics prepares one with the analytical skills needed in a host of careers outside of mathematics, including computer programming, actuarial science, and academic research which occurs at the interface with other scientific fields such as biology, chemistry, and physics. A degree in pure mathematics is an attractive background for potential employers to see, a trait that makes it a worthy goal towards achieving a lifetime of success.

Faculty

The faculty specializing in areas of pure mathematics and their specific areas of research are:

Alfonso Agnew, Professor of Mathematics, Mathematical Physics, Quantum Field Theory

Scott Annin, Professor of Mathematics, Algebra, Noncommutative Ring Theory

Alain Bourget, Professor of Mathematics, Analysis, Mathematical Physics

Riley Casper, Assistant Professor of Mathematics, Noncommutative Algebra, Algebraic Geometry

Adam Glesser, Professor of Mathematics, Algebra, Representation Theory of Finite Groups

Zair Ibragimov, Professor of Mathematics, Analysis

Christopher Lyons, Associate Professor of Mathematics, Number Theory, Algebraic Geometry

Thomas Murphy, Associate Professor of Mathematics, Geometric Analysis, Differential Geometry

Matthew Rathbun, Associate Professor of Mathematics, Low-dimensional Topology, Applications to DNA

Shoo Seto, Assistant Professor of Mathematics, Geometric Analysis, Differential Geometry, PDEs

Bogdan Suceava, Professor of Mathematics, Differential Geometry

Hassan Yousefi, Professor of Mathematics, Analysis, Operator Theory

Research

The pure mathematics faculty in the Department of Mathematics are actively involved in their own research in the areas of algebra, differential geometry, complex and functional analysis, quantum field theory, and cosmology. Pure math faculty are actively engaged in research with undergraduates as well. Undergraduate research experience is increasing in popularity, and may provide a significant advantage when competing for a job or graduate school admission. There are many opportunities for students to get involved in research activities. For example, one may attend a seminar or work on a faculty supervised independent study or research project.

Opportunities for undergraduates to participate in research projects in pure mathematics also include formal Research Experiences for Undergraduates programs, funded by the National Science Foundation. These competitive programs often run for several weeks during the summer months at institutions around the nation.