Department of Mathematics


220 Yost Hall
www.case.edu/artsci/math
Phone: 216-368-2880; Fax: 216-368-5163
Daniela Calvetti, Chair
E-mail: daniela.calvetti@case.edu


The Department of Mathematics at Case Western Reserve University is an active center for mathematical research. Faculty members conduct research in algebra, analysis, applied mathematics, convexity, dynamical systems, geometry, imaging, inverse problems, life sciences applications, mathematical biology, modeling, numerical analysis, probability, scientific computing, stochastic systems, and other areas.


The department offers a variety of programs leading to both undergraduate and graduate degrees in traditional and applied mathematics. Undergraduate degrees are Bachelor of Arts, Bachelor of Science in Mathematics, and Bachelor of Science in Applied Mathematics. Graduate degrees are Master of Science and Doctor of Philosophy. The BS/MS program allows a student to obtain a Bachelor of Science in Applied Mathematics with a master’s degree from Mathematics or another department in five years. The department, in cooperation with John Carroll University, offers a program for individuals interested in pre-college teaching. It also offers a specialized program with the Department of Physics.


Mathematics plays a central role in the physical, biological, economic, and social sciences. Because of this, employment prospects are always strong for individuals with degrees in mathematics, and there are excellent career opportunities. A bachelor’s degree in mathematics offers a strong background for graduate school in many areas (including computer science, medicine, and law, in addition to mathematics and science) or a position in the private sector. A master’s degree (in mathematics or applied mathematics, or an undergraduate degree in applied mathematics combined with a master’s in a different area) is an excellent basis for employment in the private sector in a technical field. A Ph.D. degree is usually necessary for college teaching and research.
Students, both undergraduate and graduate, have opportunities to interact personally with faculty and other students, and research and other activities are available. In addition, undergraduates can obtain teaching experience via the Department’s supplemental instruction program.


Department Faculty


Daniela Calvetti, Ph.D.
(University of North Carolina)
Professor and Chair
Scientific computing; imaging, inverse problems; modeling and simulation in life science


James C. Alexander, Ph.D.
(Johns Hopkins University)
Professor
Dynamics; applied mathematics


Christopher Butler, M.S.
(Case Western Reserve University)
Instructor
Teaching of mathematics


David Gurarie, Ph.D.
(Hebrew University, Jerusalem, Israel)
Professor
Infectious diseases; epidemiology; mathematical biology; differential equations; gallery of fluid motions


Michael Hurley, Ph.D.
(Northwestern University)
Professor
Dynamical systems; dynamics of cellular automata; dynamics of numerical methods


Steven H. Izen, Ph.D.
(Massachusetts Institute of Technology)
Professor
Image reconstruction from projections, both theoretically and in applied situations


Peter Kotelenez, Ph.D.
(Universität Bremen)
Professor
Stochastic partial and ordinary differential equations; transitions from microscopic to macroscopic equations for particle systems; correlated Brownian motions and depletion phenomena in colloids; stochastic models in nanotechnology and complex systems


Joel Langer, Ph.D.
(University of California, Santa Cruz)
Professor
Static and dynamics of curves and related physical models; the interplay between geometry and integrable Hamiltonian systems; geometry of finite and infinite dimensional spaces of curves


Marshall J. Leitman, Ph.D.
(Brown University)
Professor
Continuum physics; integral equations; functional analysis; mechanics of materials


Elizabeth Meckes, Ph.D.
(Stanford University)
Assistant Professor
Quantitative limit theorems in probability; Stein’s method; high-dimensional phenomena in probability; geometry; statistics


Mark Meckes, Ph.D.
(Case Western Reserve University)
Assistant Professor
Geometry in high dimensions; random matrix theory; geometry probability


David A. Singer, Ph.D.
(University of Pennsylvania)
Professor
Geometry; dynamical systems; variational problems


Erkki Somersalo, Ph.D.
(University of Helsinki)
Professor
Modeling and simulation of complex biological systems; inverse problems and Bayesian scientific computing; medical imaging


Stanislaw J. Szarek, Ph.D.
(Mathematical Institute, Polish Academy of Science)
Levi Kerr Professor in Mathematics
Geometric functional analysis and its applications to study of high-dimensional phenomena; asymptotic geometric analysis


Peter Thomas, Ph.D.
(University of Chicago)
Assistant Professor
Noise and reliability in neural spike time patterns; gradient sensing, signal transduction and information theory; pattern formation in the visual cortex


Catalin Turc, Ph.D.
(University of Minnesota, Minneapolis)
Assistant Professor
Numerical analysis; scientific computing, computational electromagnetism; partial differential equation


Elisabeth Werner, Ph.D.
(Université Pierre et Marie Curie, Paris IV)
Professor
Convex geometry; analysis; probability; applications to approximation theory, mathematical physics, quantum information theory


Secondary Faculty


Colin McLarty, Ph.D.
(Case Western Reserve University)
Associate Professor of Philosophy
Logic; philosophy of mathematics


Adjunct Faculty


Christophe Geuzaine, Ph.D.
(University of Liege, Belgium)
Adjunct Associate Professor
Numerical analysis; scientific computing; computational electromagnetism


Marvin E. Goldstein, Ph.D.
(University of Michigan)
Adjunct Professor; Chief Scientist, NASA-Lewis Research Center
Fluid mechanics; heat transfer


Carsten Schütt, Ph.D.
(Christian-Albrecht Universität, Kiel)
Adjunct Professor
Convex geometry; analysis; probability; applications to approximation theory, mathematical physics, quantum information theory


Richard Varga, Ph.D.
(Harvard University)
Adjunct Professor
Rational approximation; Riemann hypothesis; Gershgorin disks


Undergraduate Programs


Majors


A Bachelor of Arts in Mathematics, a Bachelor of Science in Mathematics, a Bachelor of Science in Mathematics and Physics, and a Bachelor of Science in Applied Mathematics are available to students at Case Western Reserve University. All undergraduate mathematics degrees are based on a four-course sequence in calculus and differential equations and a five-course mathematics core in analysis and algebra.


Bachelor of Arts in Mathematics


(1) Mathematics Requirements


The B.A. degree requires at least 38 hours of mathematics courses, including:

  1. MATH 121, 122, 223, and 224, or an equivalent sequence
  2. Core mathematics for the B.A.
    (i) MATH 307, 308, 321, 322
    (ii) MATH 324 or 425
  3. Three approved technical electives (9 credit hours), no more than one of which can be from outside the department

(2) Non-Mathematics Requirements
A 3-credit hour course in computer science (ENGR 131 or other approved course)


Teacher Licensure


High school teaching certification is available in the B.A. program in mathematics through a joint program with John Carroll University. The requirements are:

  1. Completion of the B.A. program in mathematics, including MATH 150, MATH 304, and STAT 312 as the three approved technical electives.
  2. The completion of a second major in teacher education. Students interested in this option should consult the description of the Teacher Licensure program elsewhere in this bulletin or contact the director of teacher education.

Bachelor of Science in Mathematics


(1) Mathematics Requirements
The B.S. degree in mathematics requires at least 50 hours of mathematics courses, including:

  1. MATH 121, 122, 223, and 224, or an equivalent sequence
  2. Core mathematics for the B.S. in mathematics
    (i) MATH 307, 308, 321, 322
    (ii) MATH 324 or 425
  3. 21 hours (normally seven courses) of approved technical electives, no more than 9 hours of which may be from outside the department

(2) Non-Mathematics Requirements
The B.S. degree in mathematics requires the following non-mathematics courses:

  1. PHYS 121, 122, 221, or an equivalent sequence
  2. A two-course science sequence from the following list of physical sciences: ASTR 201-202, CHEM 105-106, CHEM 111-ENGR 145, GEOL 110 and either 115 or 210
  3. A 3-credit hour course in computer science (ENGR 131 or other approved course).
  4. An approved science lab (usually 2 credit hours) (BIOC 314, BIOL 111, CHEM 113, GEOL 119, or PHYS 203)

Bachelor of Science in Applied Mathematics


The B.S. degree in Applied Mathematics requires at least 50 hours of course work in mathematics and related subjects, in addition to a professional core that is specific to the area of application of interest to the student. A student in this degree program must design a program of study in consultation with his or her academic advisor. This program of study must explicitly list the technical electives and the professional core in the area of application.


(1) Mathematics Requirements

  1. MATH 121, 122, 223, and 224, or an equivalent sequence
  2. Core mathematics for applied mathematics
    (i) MATH 304, 307, 321, 322, 330
    (ii) MATH 324 or 425
  3. Technical Electives: 18 credit hours (normally six courses) of technical electives as follows:
    (i) Four approved courses, specific to the concentration area of interest to the student
    (ii) Two other courses of MATH at the 300 level or higher

(2) Professional Core Requirement
The professional core requires 12 credit hours of course work specific to the area of application. This requirement is intended to promote scientific breadth and encourage application of mathematics to other fields.


(3) Non-Mathematics Requirements


The B.S. degree in applied mathematics requires the following non-mathematics courses:

  1. PHYS 121, 122, 221, or an equivalent sequence
  2. A two-course science sequence from the following list of physical sciences: ASTR 201-202, CHEM 105-106, CHEM 107-108, GEOL 110 and either 115 or 210
  3. A 3-credit hour course in computer science (ENGR 131 or other approved course)
  4. An approved science lab (usually 2 credit hours) (BIOC 314, BIOL 111, CHEM 113, GEOL 119, or PHYS 203)

Areas of research in applied mathematics well represented in the department include:

Study plans with emphasis on areas of application closely related to mathematics but centered in other departments will be also considered. Such areas might include engineering applications, biology, cognitive science, or economics.


Bachelor of Science in Mathematics and Physics


In contrast to an applied mathematics degree or the B.S. in physics with a mathematical physics concentration, this is a synergistic, coherent, and parallel education in mathematics and physics. To a close approximation, the challenging course work corresponds to combining the mathematics and physics cores, with the Physics Laboratory cluster replaced by a single, fourth-year laboratory semester. A student in this new program may use either of two official advisors, one available from each department, who would also constitute a committee for the administration of the degree and the approval of curriculum petitions.
The total number of required credits is 126 (35 MATH, 38 PHYS, 6 senior project, 11-13 ENGR and CHEM). There are 14 to 16 credits of open electives.


Course

Year

Credit

PHYS 121 or 123

Mechanics

1

4

PHYS 122 or 124

Electricity & Magnetism

1

4

MATH 121 or 123

Calculus I

1

4

MATH 122 or 124

Calculus II

1

4

MATH 223 or 227

Calculus III

1

3

ENGR 131

CompProg

1

3

CHEM 105 or 111

Intro Chemistry

1

3/4

CHEM 106 or ENGR 145

Intro Chemistry

1

3/4

CHEM 113

Chem Lab

1

2

PHYS 221

Intro to Modern Physics

2

3

PHYS 310

Classical Mechanics

2

3

MP group I***

2

3

MATH 224

Diff Eqs

2

3

MATH 307

Linear Algebra

2

3

MATH 308 or 330

Algebra or Scient.
Comput.

2

3

PHYS 313

Thermodynamics & Stat Mechanics

3

3

PHYS 331 or 481

Quantum I

3

3

PHYS 332 or 482

Quantum II

3

3

MATH 321

Analysis I

3

3

MATH 322

Analysis II

3

3

MATH 324

Complex Analysis

3

3

PHYS 3XX**

3

3

MP group II***

3

3

MP group III***

3

3

PHYS 331 or 481

3

3

PHYS 423

Adv Elec & Mag

4

3

PHYS 472

Grad Lab

4

3

MP group IV***

4

3

PHYS 351 or MATH 351

Sr Project

4

4

SAGES

First and University Seminars

  

10

A&S SAGES

Breadth Requirements

  

12

Open Electives

  

14-16

 

*Course usually taken in this year, offered only in F = fall, S = spring


**An advanced physics course to be selected from the following list: PHYS 315, 316, 326, 328, 336, 365.


***The “M&P group” of four courses corresponds to two physics courses and two mathematics courses. The physics courses would be chosen from P250, P349, and P350. The mathematics courses are subject to approval by the advisory committee and are thereby referred to as ‘approved electives.’ They may be chosen from the general list of mathematics courses at the 300 level or higher. Also subject to approval, students may choose a course from outside the mathematics and physics departments as a substitute in the M&P group.


****If approved by the M&P committee, other science sequence courses may be substituted.


*****The number of open electives will vary depending on whether students choose 3-credit or 4-credit courses to fulfill the chemistry/science requirement


Integrated B.S./M.S. Program in Mathematics and/or Applied Mathematics


The integrated B.S./M.S. program is intended for highly motivated candidates for the B.S in mathematics and applied mathematics who wish to pursue an advanced degree. Application to the B.S./M.S. program must be made after completion of 75 semester hours of course work and prior to attaining senior status (completion of 90 semester hours). Generally, this means that a student will submit the application during his/her sixth semester of undergraduate course enrollment and will have no fewer than two semesters of remaining B.S. requirements to complete. Applicants should consult the dean of undergraduate studies.


A student admitted to the program may, in the senior year, take up to nine hours of graduate courses (400 level and above) that will count towards both B.S. and M.S. requirements. The courses to be doubled-counted must be specified at the time of application. Any undergraduate course work that is to be applied to the M.S. must be beyond that used to satisfy B.S. degree requirements and must conform to university, graduate school, and department rules. Students may petition to transfer graduate course work taken prior to application to the B.S./M.S. Program subject to the rules of the graduate school.


Students for whom the master’s project or thesis is a continuation and development of the senior project should register for MATH 651 Thesis (or the appropriate project course) during the senior year and are expected to complete all other courses for the B.S. before enrolling in further M.S. course work and thesis (continuing the senior project). Students for whom the master’s thesis or project is distinct from the senior project will be expected to complete the B.S. degree before taking further graduate courses for the master’s degree.


Integrated B.S./M.S. in Applied Mathematics and Another Discipline


There is the possibility of an integrated five-year study plan leading to a B.S. in applied mathematics and an M.S. in the area of application. In order to complete the requirements for the B.S./M.S. in five years, students must choose an area outside mathematics that integrates well with mathematics, such as computing and information science, operations research, systems engineering, control theory, biology, or cognitive science. The general academic requirements for Integrated B.S./M.S. programs must be followed. (Since the graduate courses required for the M.S. degree are determined by the respective department, each student in the dual-degree program should have a secondary advisor in that department, starting no later than the junior year, and consult with this advisor concerning requirements for the M.S. degree.)


Minors


A minor in mathematics is available to all undergraduates. It consists of 17 credit hours of approved course work in mathematics. No more than two courses can be used to satisfy both minor requirements and the requirements of the student’s major field (meaning departmental degree requirements, including departmental technical electives and common course requirements of the student’s school). The 17 hours must be from among the following MATH courses: 121 or 123 or 125, 122 or 124 or 126, 223 or 227, 224 or 228, 150, 201, 301, 302, 303, 304, 307, 308, 321, 322, 323, 324, 331, 338, 343, 345, 380, or any 400-level MATH course (only one of 201, 307).


Graduate Programs


The department offers programs leading to the Master of Science and Doctor of Philosophy degrees. At the master’s and at the doctoral levels, there are two degrees: the degree of Master of Science in Mathematics and the degree of Master of Science in Applied Mathematics.


A student must satisfy all of the general requirements of the graduate school as well as the more specific requirements of the department to earn either a master’s or doctoral degree. Each graduate student is assigned an advisory committee consisting of faculty members during the first year of study. The committee’s primary responsibility is to help the student plan an appropriate and sufficiently broad program of course work and study, which will satisfy both the degree requirements and the special interests of the student. With the aid of the advisory committee, each student must present a study plan indicating how he or she intends to satisfy the requirements for a graduate degree.


The main requirements are as follows.


Master of Science in Mathematics


A minimum of 27 credit hours of approved course work, at least 18 of which must be at the 400 level or higher, is required for the M.S. degree in mathematics. Courses in two of the following three basic areas must be included among the 27 credit hours required for graduation: Abstract Algebra (MATH 401 and MATH 402), Analysis (MATH 423 and one of MATH 424 or MATH 425), and Topology (MATH 461).


The student must pass a comprehensive oral examination on three areas, two of which must be selected from the basic ones listed above (although no particular courses are specified). The third area for the examination may be any approved subject.


A student in the M.S. program in mathematics may substitute the comprehensive exam examination requirement with an expository or original thesis, which will count as 6 credit hours of course work. The thesis will be defended in the course of an oral examination, during which the student will be questioned about the thesis and related topics. These two variants correspond to plan A and plan B in the graduate school literature.


Master of Science in Applied Mathematics


The department offers specialized programs in applied mathematics. For each of the programs, there is a minimum requirement of 27 credit hours of course work, at least 18 of which must be at the 400 level or higher. Students in the program must complete course work requirement in each of the following disjoint groups:

Although individual programs of course work leading to a master’s degree in applied mathematics cannot have a large common core of requirements because of the great diversity of topics used in applications, all students pursuing a Master of Science in Applied Mathematics are strongly advised to take Introduction to Numerical Analysis (MATH 431) and Mathematical Modeling (MATH 441). In addition, to add breadth to the student’s education, the set of courses taken within the department must include three credit hours of approved course work in at least three of the following seven subjects. The courses listed are examples of suitable courses on the given subject. A course can be used to satisfy only one breadth area.


Applied Mathematics Breadth Areas

Other suitable courses for students in applied mathematics include MATH 413 (Graph Theory), MATH 424 (Functional Analysis), MATH 425 (Complex Analysis), MATH 427 (Convexity), MATH 428 (Fourier Analysis), MATH 444 (Data mining and Pattern Recognition), MATH 469 (Calculus of Variations), MATH 475 (Mathematics of Imaging), MATH 492 (Probability), and MATH 495 (Combinatorics).


The student must pass a comprehensive oral examination on three areas, two of which must be in the list for the breadth requirement (although no particular courses are specified). The third area for the examination may be any approved subject.


A student in the M.S. program in applied mathematics may substitute the comprehensive examination requirement with an expository or original thesis, which will count as 6 credit hours of course work. The thesis will be defended in the course of an oral examination, during which the student will be questioned about the thesis and related topics. These two variants correspond to plan A and plan B in the graduate school literature.


Master of Science in Applied Mathematics—Entrepreneurial Track


The Master of Science in Applied Mathematics—Entrepreneurial Track, obtained through the Entrepreneurial Program in Mathematics and Computation, is a degree designed to provide training in applied mathematics for entrepreneurs who have a business idea that depends heavily on mathematics. They wish to learn enough mathematics to refine their business idea and, at the same time, acquire the business skills needed to bring this idea to the marketplace. The Master of Science in Applied Mathematics—Entrepreneurial Track is also appropriate for industrial mathematicians who need to effectively utilize mathematical tools in a business context. It expands our basic Master of Applied Mathematics program by tightly integrating business training into the curriculum. The Entrepreneurial Track provides instruction and real business-world experience to students who have a background in mathematics and a vision for new and growing ventures.


Candidates for the M.S. in Mathematics—Entrepreneurial Track must complete at least 27 hours of course work and present a master’s thesis. It is expected that a business plan be an integral part of the thesis. The two-year program includes these course requirements:

The New Venture Creation and Technology Entrepreneurship courses will be offered by the Weatherhead School of Management. The Technical Elective is a 400-level or higher mathematics course or other technical elective appropriate to an individual student’s program of study, as approved by the Mathematics Entrepreneurship Program Committee. The Restricted Elective is a course in mathematics, science, engineering, or management appropriate to an individual student’s program of study, as approved by the Mathematics Entrepreneurship Program Committee.


Ph.D. Program


The doctorate is conferred not merely upon completion of a stipulated course of study, but rather upon clear demonstration of scholarly attainment and capability of original research work in mathematics. A doctoral student may plan either a traditional program of studies in mathematics (mathematics track) or a program of studies oriented toward applied mathematics (applied mathematics track). In either case, each student must take 36 credit hours of approved courses with a grade average of B or better. For students entering with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, this requirement is reduced to 18 credit hours of approved courses.


In addition to the course work, all Ph.D. students in both tracks must complete the following specific requirements:

Course work requirements
Mathematics Track: A student in the traditional mathematics program must demonstrate knowledge of the basic concepts and techniques of algebra, analysis (real and complex), and topology. This must be done by taking all courses in the three basic areas: abstract algebra (MATH 401-MATH 402), analysis (MATH 423-MATH 424 and MATH 425), and topology (MATH 461). In addition, the student is required to take a minimum of 18 credit hours of approved course work.


A student with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, must take 18 credit hours of approved courses. The graduate committee will determine which of the specific course work requirements stated above have been satisfied by the master’s course work.


Applied Mathematics Track: A student in the applied mathematics track must demonstrate knowledge of scientific computing, mathematical modeling, and differential equations. This may be done by taking:

In addition, a student in this track must take at least 24 credit hours of approved courses, which must include at least 9 credit hours of courses offered outside the Department of Mathematics, and at least 9 credit hours offered by the Department of Mathematics.


A student with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, must take 18 credit hours of approved courses, which must include at least 6 credit hours of courses offered outside the Department of Mathematics and at least 9 credit hours offered by the Department of Mathematics. The graduate committee will determine which of the specific course work requirements stated above have been satisfied by the master’s course work.


Sample study plans for students with concentrations in scientific computing, imaging, mathematical biology, and stochastics follow. The graduate committee will entertain ideas for other serious study plans or qualifying exam subjects in addition to the most common variants specifically suggested.


Scientific Computing Concentrations

MATH 431

Application area

MATH 432

Application area

MATH 433

Application area

MATH 441

Application area

MATH 445

Application area

MATH 439/440

Application area

MATH 448

Application area

MATH 487

Application area

MATH 449/469/478

Application area

Imaging Concentrations

MATH 431

PHYS 431

MATH 432

PHYS 460

MATH 433

EBME 410

MATH 441

PHYS 431

MATH 428

PHYS 460

MATH 475

EBME 410

MATH 445

PHYS 431

MATH 439/440

PHYS 460

MATH 444

EBME 410

Life Science Concentrations

MATH 431

Application area

MATH 432

Application area

MATH 433

Application area

MATH 441

Application area

MATH 449

Application area

MATH 478

Application area

MATH 445

Application area

MATH 439/487

Application area

MATH 440

Application area

Stochastics Concentrations

MATH 431

Application area

MATH 423

Application area

MATH 441

Application area

MATH 481

Application area

MATH 424

Application area

MATH 491

Application area

MATH 487

Application area

MATH 469

Application area

MATH 492

Application area

 

Ph.D. students entering with a bachelor’s degree are also subject to the breadth requirements for students in the program for the M.S. degree in applied mathematics.


Petitions


Any exceptions to departmental regulations or requirements must have the formal approval of the graduate committee of the department. Such exceptions are to be sought by a written petition, approved by the student’s advisory committee or thesis advisor, to the graduate committee.


Any exception to university rules and regulations must be approved by the dean of graduate studies. Such exceptions are to be sought by presenting a written petition to the graduate committee for departmental endorsement and approval prior to forwarding the petition to the dean.


Course Descriptions


MATH 110. Introduction to Mathematical Communication and Software (1)
Mathematical text editors. Mathematical composition and exposition. Posting mathematical material on the Web. Basics of computer symbolic manipulation (Mathematica). Computer vector/matrix manipulation and applications (MATLAB). Basic computer statistical methods (Minitab). Integration of output from computer calculations into text.


MATH 120. Elementary Functions and Analytic Geometry (3)
Polynomial, rational, exponential, logarithmic, and trigonometric functions (emphasis on computation, graphing, and location of roots) straight lines and conic sections. Primarily a precalculus course for the student without a good background in trigonometric functions and graphing and/or analytic geometry. Not open to students with credit for MATH 121 or MATH 125. Prereq: Three years of high school mathematics.


MATH 121. Calculus for Science and Engineering I (4)
Functions, analytic geometry of lines and polynomials, limits, derivatives of algebraic and trigonometric functions. Definite integral, antiderivatives, fundamental theorem of calculus, change of variables. Prereq: Three and one half years of high school mathematics.


MATH 122. Calculus for Science and Engineering II (4)
Continuation of MATH 121. Exponentials and logarithms, growth and decay, inverse trigonometric functions, related rates, basic techniques of integration, area and volume, polar coordinates, parametric equations. Taylor polynomials and Taylor’s theorem.
Prereq: MATH 121 or MATH 123 or MATH 126.


MATH 123. Calculus I (4)
Limits, continuity, derivatives of algebraic and transcendental functions, including applications, basic properties of integration. Techniques of integration and applications. Students must have 31/2 years of high school mathematics.


MATH 124. Calculus II (4)
Review of differentiation. Techniques of integration, and applications of the definite integral. Parametric equations and polar coordinates. Taylor’s theorem. Sequences, series, power series. Complex arithmetic. Introduction to multivariable calculus.
Prereq: MATH 123 or placement by department.


MATH 125. Math and Calculus Applications for Life, Managerial, and Social Sci I (4)
Discrete and continuous probability; differential and integral calculus of one variable; graphing, related rates, maxima and minima. Integration techniques, numerical methods, volumes, areas. Applications to the physical, life, and social sciences. Students planning to take more than two semesters of introductory mathematics should take MATH 121. Prereq: Three and one half years of high school mathematics.


MATH 126. Math and Calculus Applications for Life, Managerial, and Social Sci II (4)
Continuation of MATH 125 covering differential equations, multivariable calculus, discrete methods. Partial derivatives, maxima and minima for functions of two variables, linear regression. Differential equations; first and second order equations, systems, Taylor series methods; Newton’s method; difference equations.
Prereq: MATH 121 or MATH 123 or MATH 125.


MATH 150. Mathematics from a Mathematician’s Perspective (3)
An interesting and accessible mathematical topic not covered in the standard curriculum is developed. Students are exposed to methods of mathematical reasoning and historical progression of mathematical concepts. Introduction to the way mathematicians work and their attitude toward their profession. Should be taken in freshman year to count toward a major in mathematics. Prereq: Three and one half years of high school mathematics.


MATH 201. Introduction to Linear Algebra (3)
Matrix operations, systems of linear equations, vector spaces, subspaces, bases and linear independence, eigenvalues and eigenvectors, diagonalization of matrices, linear transformations, determinants. Less theoretical than MATH 308. May not be taken for credit by mathematics majors. Only one of MATH 201 or MATH 308 may be taken for credit.
Prereq: MATH 122 or MATH 124 or MATH 126.


MATH 223. Calculus for Science and Engineering III (3)
Introduction to vector algebra; lines and planes. Functions of several variables: partial derivatives, gradients, chain rule, directional derivative, maxima/minima. Multiple integrals, cylindrical and spherical coordinates. Derivatives of vector valued functions, velocity and acceleration. Vector fields, line integrals, Green’s theorem.
Prereq: MATH 122 or MATH 124.


MATH 224. Elementary Differential Equations (3)
A first course in ordinary differential equations. First order equations and applications, linear equations with constant coefficients, linear systems, Laplace transforms, numerical methods of solution.
Prereq: MATH 223 or MATH 227.


MATH 227. Calculus III (3)
Vector algebra and geometry. Linear maps and matrices. Calculus of vector valued functions. Derivatives of functions of several variables. Multiple integrals. Vector fields and line integrals.
Prereq: MATH 124 or placement by the department.


MATH 228. Differential Equations (3)
Elementary ordinary differential equations: first order equations; linear systems; applications; numerical methods of solution.
Prereq: MATH 227 or placement by the department.


MATH 301. Undergraduate Reading Course (1-3)
Students must obtain the approval of a supervising professor before registration. More than one credit hour must be approved by the undergraduate committee of the department.


MATH 302. Departmental Seminar (3)
A seminar devoted to understanding the formulation and solution of mathematical problems. SAGES Department Seminar. Students will investigate, from different possible viewpoints, via case studies, how mathematics advances as a discipline--what mathematicians do. The course will largely be in a seminar format. There will be two assignments involving writing in the style of the discipline. Enrollment by permission (limited to majors depending on demand).
SAGES Dept Seminar


MATH 303. Elementary Number Theory (3)
Primes and divisibility, theory of congruencies, and number theoretic functions. Diophantine equations, quadratic residue theory, and other topics determined by student interest. Emphasis on problem solving (formulating conjectures and justifying them).
Prereq: MATH 122 or MATH 124.


MATH 304. Discrete Mathematics (3)
A general introduction to basic mathematical terminology and the techniques of abstract mathematics in the context of discrete mathematics. Topics introduced are mathematical reasoning, Boolean connectives, deduction, mathematical induction, sets, functions and relations, algorithms, graphs, combinatorial reasoning.
Offered as EECS 302 and MATH 304.
Prereq: MATH 122 or MATH 124 or MATH 126.


MATH 307. Introduction to Abstract Algebra I (3)
First semester of an integrated, two-semester theoretical course in abstract and linear algebra, studied on an axiomatic basis. The major algebraic structures studied are groups, rings, fields, modules, vector spaces, and inner product spaces. Topics include homomorphisms and quotient structures, the theory of polynomials, canonical forms for linear transformations and the principal axis theorem. This course is required of all students majoring in mathematics. Only one of MATH 201 or MATH 307 may be taken for credit.
Prereq: MATH 122 or MATH 124.


MATH 308. Introduction to Abstract Algebra II (3)
Continuation of MATH 307.
Prereq: MATH 307.


MATH 319. Applied Probability and Stochastic Processes for Biology (3)
Applications of probability and stochastic processes to biological systems. Mathematical topics will include: introduction to discrete and continuous probability spaces (including numerical generation of pseudo random samples from specified probability distributions), Markov processes in discrete and continuous time with discrete and continuous sample spaces, point processes including homogeneous and inhomogeneous Poisson processes and Markov chains on graphs, and diffusion processes including Brownian motion and the Ornstein-Uhlenbeck process. Biological topics will be determined by the interests of the students and the instructor. Likely topics include: stochastic ion channels, molecular motors and stochastic ratchets, actin and tubulin polymerization, random walk models for neural spike trains, bacterial chemotaxis, signaling and genetic regulatory networks, and stochastic predator-prey dynamics. The emphasis will be on practical simulation and analysis of stochastic phenomena in biological systems. Numerical methods will be developed using both MATLAB and the R statistical package. Student projects will comprise a major part of the course.
Offered as BIOL 319, EECS 319, MATH 319, BIOL 419, EBME 419, and PHOL 419.
Prereq: MATH 224 or MATH 228 or BIOL 300 or BIOL 306.


MATH 321. Fundamentals of Analysis I (3)
Abstract mathematical reasoning in the context of analysis in Euclidean space. Introduction to formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Required for all mathematics majors. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
Offered as MATH 321 and MATH 421.
Prereq: MATH 223 or MATH 227.


MATH 322. Fundamentals of Analysis II (3)
Continuation of MATH 321. Point-set topology in metric spaces with attention to n-dimensional space; completeness, compactness, connectedness, and continuity of functions. Topics in sequences, series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration. Required for all mathematics majors. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
Offered as MATH 322 and MATH 422.
Prereq: MATH 321.


MATH 324. Introduction to Complex Analysis (3)
Properties, singularities, and representations of analytic functions, complex integration. Cauchy’s theorems, series residues, conformal mapping and analytic continuation. Riemann surfaces. Relevance to the theory of physical problems.
Prereq: MATH 224 or MATH 228.


MATH 326. Geometry and Complex Analysis (3)
The theme of this course will be the interplay between geometry and complex analysis, algebra and other fields of mathematics. An effort will be made to highlight significant, unexpected connections between major fields, illustrating the unity of mathematics. The choice of text(s) and syllabus itself will be flexible, to be adapted to the range of interests and backgrounds of pre-enrolled students. Possible topics include: the Mobius group and its subgroups, hyperbolic geometry, elliptic functions, Riemann surfaces, applications of conformal mapping, and potential theory in classical physical models.
Offered as MATH 326 and MATH 426.
Prereq: MATH 324.


MATH 327. Convexity and Optimization (3)
Introduction to the theory of convex sets and functions and to the extremes in problems in areas of mathematics where convexity plays a role. Among the topics discussed are basic properties of convex sets (extreme points, facial structure of polytopes), separation theorems, duality and polars, properties of convex functions, minima and maxima of convex functions over convex set, various optimization problems.
Offered as MATH 327, MATH 427, and OPRE 427.
Prereq: MATH 223 or MATH 227.


MATH 330. Introduction of Scientific Computing (3)
Among the topics which will be covered in the course are solutions of linear systems and least squares, approximation and interpolation, solution of nonlinear systems, numerical integration and differentiation, and numerical solution of differential equations. Projects where the numerical methods are used to solve problems from various application areas will be assigned throughout the semester.
Prereq or Coreq: MATH 224 or MATH 228.


MATH 338. Introduction to Dynamical Systems (3)
Nonlinear discrete dynamical systems in one and two dimensions. Chaotic dynamics, elementary bifurcation theory, hyperbolicity, symbolic dynamics, structural stability, stable manifold theory.
Prereq: MATH 223 or MATH 227.


MATH 342. Introduction to Research in Mathematical Biology (1)
The purpose of this seminar is to introduce students to some of the research being done at Case Western Reserve that explores questions at the intersection of mathematics and biology. Students will explore roughly five research collaborations, spending two weeks with each research group. In the first three classes of each two-week block, students will read and discuss relevant papers, guided by members of that research group, and the two-week period will culminate in a talk in which a member of the research group will present a potential undergraduate project in that area. After the final group’s talk, students will divide themselves into groups of two to four people and choose one project for further exploration. Together, they will write up this project as a research proposal, introducing the problem, explaining how it connects to broader scientific questions, and outlining the proposed work. It is expected that students will use the associated research group as a resource, but the proposal should be their own work. Students will submit a first draft, receive feedback, and then submit a revised draft.
Offered as BIOL 309 and MATH 342.


MATH 343. Theoretical Computer Science (3)
Introduction to mathematical logic, different classes of automata and their correspondence to different classes of formal languages, recursive functions and computability, assertions and program verification, denotational semantics. MATH/EECS 343 and MATH 410 cannot both be taken for credit.
Offered as EECS 343 and MATH 343.
Prereq: MATH 304 and EECS 340.


MATH 351. Senior Project for the Mathematics and Physics Program (2)
A two-semester course (2 credits per semester) in the joint B.S. in Mathematics and Physics program. Project based on numerical and/or theoretical research under the supervision of a mathematics faculty member, possibly jointly with a faculty member from physics. Study of the techniques utilized in a specific research area and of recent literature associated with the project. Work leading to meaningful results which are to be presented as a term paper and an oral report at the end of the second semester. Supervising faculty will review progress with the student on a regular basis, including detailed progress reports made twice each semester, to ensure successful completion of the work.
SAGES Senior Cap


MATH 352. Mathematics Capstone (3)
Mathematics Senior Project. Students pursue a project based on experimental, theoretical or teaching research under the supervision of a mathematics faculty member, a faculty member from another Case Western Reserve department or a research scientist or engineer from another institution. A departmental Senior Project Coordinator must approve all project proposals and this same person will receive regular oral and written progress reports. Final results are presented at the end of the second semester as a paper in a style suitable for publication in a professional journal as well as an oral report in a public Mathematics Capstone symposium.
SAGES Senior Cap


MATH 361. Geometry I (3)
An introduction to the various two-dimensional geometries, including Euclidean, spherical, hyperbolic, projective, and affine. The course will examine the axiomatic basis of geometry, with an emphasis on transformations. Topics include the parallel postulate and its alternatives, isometrics and transformation groups, tilings, the hyperbolic plane and its models, spherical geometry, affine and projective transformations, and other topics. We will examine the role of complex and hypercomplex numbers in the algebraic representation of transformations. The course is self-contained.
Prereq: MATH 224.
SAGES Dept Seminar


MATH 363. Knot Theory (3)
An introduction to the mathematical theory of knots and links, with emphasis on the modern combinatorial methods. Reidemeister moves on link projections, ambient and regular isotopies, linking number tricolorability, rational tangles, braids, torus knots, seifert surfaces and genus, the knot polynomials (bracket, X, Jones, Alexander, HOMFLY), crossing numbers of alternating knots and amphicheirality. Connections to theoretical physics, molecular biology, and other scientific applications will be pursued in term projects, as appropriate to the background and interests of the students.
Prereq: MATH 223 or MATH 227.


MATH 378. Computational Neuroscience (3)
Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems. Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural learning, models of brain systems, and their relationship to artificial and neural networks. Term project required. Students enrolled in MATH 478 will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Recommended preparation: MATH 223 and MATH 224 or BIOL 300 and BIOL 306.
Offered as BIOL 378, COGS 378, MATH 378, BIOL 478, EBME 478, EECS 478, MATH 478 and NEUR 478.


MATH 380. Introduction to Probability (3)
Combinatorial analysis. Permutations and combinations. Axioms of probability. Sample space and events. Equally likely outcomes. Conditional probability. Bayes’ formula. Independent events and trials. Discrete random variables, probability mass functions. Expected value, variance. Bernoulli, binomial, Poisson, geometric, negative binomial random variables. Continuous random variables, density functions. Expected value and variance. Uniform, normal, exponential, Gamma random variables. The De Moivre-Laplace limit theorem. Joint probability mass functions and densities. Independent random variables and the distribution of their sums. Covariance. Conditional expectations and distributions (discrete case). Moment generating functions. Law of large numbers. Central limit theorem. Additional topics (time permitting): the Poisson process, finite state space Markov chains, entropy.
Prereq: MATH 223 or MATH 227.


MATH 381. Introduction to Mathematical Methods in Finance (3)
Mathematical finance in discrete time. Single period market models. Arbitrage. Risk-neutral valuation of contingent claims. Complete markets. Summary of results from probability theory and stochastic processes in discrete time. Conditional expectation. Discrete parameter martingales. Multiperiod market modes. Equivalent martingale measures. Risk-neutral valuation. Hedging strategies. Complete markets. The Cox-Ross-Rubinstein model. European options. American options. The Black-Scholes model. Binomial approximation. The pricing formula for European call options.
Offered as MATH 381 and MATH 481.
Prereq: MATH 380.


MATH 399. Special Topics (3)
Special Topics in Mathematics


MATH 400. Mathematics Teaching Practicum (1)
Practicum for teaching college mathematics. Includes preparation of syllabi, exams, lectures. Grading, alternative teaching styles, use of technology, interpersonal relations and motivation. Handling common problems and conflicts.


MATH 401. Abstract Algebra I (3)
Basic properties of groups, rings, modules and fields. Isomorphism theorems for groups; Sylow theorem; nilpotency and solvability of groups; Jordan-Holder theorem; Gauss lemma and Eisenstein’s criterion; finitely generated modules over principal ideal domains with applications to abelian groups and canonical forms for matrices; categories and functors; tensor product of modules, bilinear and quadratic forms; field extensions; fundamental theorem of Galois theory, solving equations by radicals.
Prereq: MATH 308.


MATH 402. Abstract Algebra II (3)
A continuation of MATH 401.
Prereq: MATH 401.


MATH 406. Mathematical Logic and Model Theory (3)
Propositional calculus and quantification theory; consistency and completeness theorems; godal incompleteness results and their philosophical significance; introduction to basic concepts of model theory; problems of formulation of arguments in philosophy and the sciences.
Offered as PHIL 306, MATH 406 and PHIL 406.


MATH 408. Introduction to Cryptology (3)
Introduction to the mathematical theory of secure communication. Topics include: classical cryptographic systems; one-way and trapdoor functions; RSA, DSA, and other public key systems; Primality and Factorization algorithms; birthday problem and other attack methods; elliptic curve cryptosystems; introduction to complexity theory; other topics as time permits. Recommended preparation: MATH 303.


MATH 413. Graph Theory (3)
Building blocks of a graph, trees, connectedness, transversability connectedness, transversability, matching, coverings, planarity, and NP-complete problems; various applications and algorithms.
Prereq: MATH 201 or MATH 308.


MATH 421. Fundamentals of Analysis I (3)
Abstract mathematical reasoning in the context of analysis in Euclidean space. Introduction to formal reasoning, sets and functions, and the number systems. Sequences and series; Cauchy sequences and convergence. Required for all mathematics majors. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
Offered as MATH 321 and MATH 421.


MATH 422. Fundamentals of Analysis II (3)
Continuation of MATH 321. Point-set topology in metric spaces with attention to n-dimensional space; completeness, compactness, connectedness, and continuity of functions. Topics in sequences, series of functions, uniform convergence, Fourier series and polynomial approximation. Theoretical development of differentiation and Riemann integration. Required for all mathematics majors. Additional work required for graduate students. (May not be taken for graduate credit by graduate students in the Department of Mathematics.)
Offered as MATH 322 and MATH 422.
Prereq: MATH 321 or MATH 421.


MATH 423. Introduction to Real Analysis I (3)
General theory of measure and integration. Measures and outer measures. Lebesgue measure on n-space. Integration. Convergence theorems. Product measures and Fubini’s theorem. Signed measures. Hahn-Jordan decomposition, Radon-Nikodym theorem, and Lebesgue decomposition. SpaceP-integrable function. Lebesgue differentiation theorem in n-space.
Prereq: MATH 322 or MATH 422.


MATH 424. Introduction to Real Analysis II (3)
Measures on locally compact spaces. Riesz representation theorem. Elements of functional analysis. Normed linear spaces. Hahn-Banach, Banach-Steinhaus, open mapping, closed graph theorems. Weak topologies. Banach-Alaoglu theorem. Function spaces. Stone-Weierstrass and Ascoli theorems. Basic Hilbert space theory. Application to Fourier series. Additional topics: Haar measure on locally compact groups.
Prereq: MATH 423.


MATH 425. Complex Analysis I (3)
Analytic functions. Integration over paths in the complex plane. Index of a point with respect to a closed path; Cauchy’s theorem and Cauchy’s integral formula; power series representation; open mapping theorem; singularities; Laurent expansion; residue calculus; harmonic functions; Poisson’s formula; Riemann mapping theorem. More theoretical and at a higher level than MATH 324.
Prereq: MATH 322 or MATH 422.


MATH 426. Geometry and Complex Analysis (3)
The theme of this course will be the interplay between geometry and complex analysis, algebra and other fields of mathematics. An effort will be made to highlight significant, unexpected connections between major fields, illustrating the unity of mathematics. The choice of text(s) and syllabus itself will be flexible, to be adapted to the range of interests and backgrounds of pre-enrolled students. Possible topics include: the Mobius group and its subgroups, hyperbolic geometry, elliptic functions, Riemann surfaces, applications of conformal mapping, and potential theory in classical physical models.
Offered as MATH 326 and MATH 426.


MATH 427. Convexity and Optimization (3)
Introduction to the theory of convex sets and functions and to the extremes in problems in areas of mathematics where convexity plays a role. Among the topics discussed are basic properties of convex sets (extreme points, facial structure of polytopes), separation theorems, duality and polars, properties of convex functions, minima and maxima of convex functions over convex set, various optimization problems.
Offered as MATH 327, MATH 427, and OPRE 427.


MATH 428. Fourier Analysis (3)
Introduction to the mathematical aspects of Fourier analysis and synthesis. Accessible to upper level undergraduates and graduate students in the sciences and engineering. Periodic functions. Fourier series. Convergence theorems. The classical sine and cosine series. General orthogonal systems. Multiple Fourier series. Applications. Fourier integrals and Fourier Transforms. Integrable and square integrable function theory. Inversion theorems. Classical sine and cosine transforms. Multiple Fourier Transform. Spherical symmetry. Other important transforms. Applications.
Prereq: MATH 224 or MATH 228.


MATH 431. Introduction to Numerical Analysis I (3)
Numerical linear algebra for scientists and engineers. Matrix and vector norms, computer arithmetic, conditioning and stability, orthogonality. Least squares problems: QR factorization, normal equations and Singular Value Decomposition. Direct solution of linear system: Gaussian elimination and Cholesky factorization. Eigenvalues and eigenvectors: the QR algorithm, Rayleigh quotient, inverse iteration. Introduction to iterative methods. Students will be introduced to MATLAB.
Prereq: MATH 201 or MATH 307.


MATH 432. Numerical Differential Equations (3)
Numerical solution of differential equations for scientists and engineers. Solution of ordinary differential equations by multistep and single step methods. Stability, consistency, and convergence. Stiff equations. Finite difference schemes. Introduction to the finite element method. Introduction to multigrid techniques. The diffusion equation: numerical schemes and stability analysis. Introduction to hyperbolic equations. MATLAB will be used in this course.
Prereq: MATH 224 or MATH 228.


MATH 433. Numerical Solutions of Nonlinear Systems and Optimization (3)
The course provides an introduction to numerical solution methods for systems of nonlinear equations and optimization problems. The course is suitable for upper-undergraduate and graduate students with some background in calculus and linear algebra. Knowledge of numerical linear algebra is helpful. Among the topics which will be covered in the course are Nonlinear systems in one variables; Newton’s method for nonlinear equations and unconstrained minimization; Quasi-Newton methods; Global convergence of Newton’s methods and line searches; Trust region approach; Secant methods; Nonlinear least squares.
Prereq: MATH 223 or MATH 227, and MATH 431 or permission.


MATH 434. Optimization of Dynamic Systems (3)
Fundamentals of dynamic optimization with applications to control. Variational treatment of control problems and the Maximum Principle. Structures of optimal systems; regulators, terminal controllers, time-optimal controllers. Sufficient conditions for optimality. Singular controls. Computational aspects. Selected applications. Recommended preparation: EECS 408.
Offered as EECS 421 and MATH 434.


MATH 435. Ordinary Differential Equations (3)
MATH 435 is a second course in ordinary equations. The course will begin with a quick review of one-dimensional o.d.e.’s with the idea of introducing tools for studying general systems that will be developed later in the course (this is the get-acquainted part of the course; topics will be selected from Parts I and II of the text). After this introduction we will turn to general features of abstract o.d.e.’s existence and uniqueness of solutions, dependence of solutions on parameters, and the idea of a flow. Next we will consider linear systems. The remainder of the semester will be spent on nonlinear systems: topics that will be discussed are linearization, stability, the Poncare-Bendixson theory and bifurcations.
Prereq: MATH 224 and either MATH 201 or MATH 307.


MATH 439. Integrated Numerical and Statistical Computations (3)
This course will embed numerical methods into a Bayesian framework. The statistical framework will make it possible to integrate a prori information about the unknowns and the error in the data directly into the most efficient numerical methods. A lot of emphasis will be put on understanding the role of the priors, their encoding into fast numerical solvers, and how to translate qualitative or sample-based information--or lack thereof--into a numerical scheme. Confidence on computed results will also be discussed from a Bayesian perspective, at the light of the given data and a priori information. The course should be of interest to anyone working on signal and image processing statistics, numerical analysis and modeling.
Recommended Preparation: Math 431.
Offered as MATH 439 and STAT 439.


MATH 440. Computational Inverse Problems (3)
This course will introduce various computational methods for solving inverse problems under different conditions. First the classical regularization methods will be introduced, and the computational challenges which they pose, will be addressed. Following this, the statistical methods for solving inverse problems will be studied and their computer implementation discussed. We will combine the two approaches to best exploit their potentials. Applications arising from various areas of science, engineering, and medicine will be discussed throughout the course.


MATH 441. Mathematical Modeling (3)
Mathematics is a powerful language for describing real world phenomena and providing predictions that otherwise are hard or impossible to obtain. The course gives the students prerequisites for translating qualitative descriptions given in the professional non-mathematical language into the quantitative language for mathematics. While the variety in the subject matter is wide, some general principles and methodologies that a modeler can pursue are similar in many applications. The course focuses on these similarities. The course is based on representative case studies that are discussed and analyzed in the classroom, the emphasis being on general principles of developing and analyzing mathematical models. The examples will be taken from different fields of science and engineering, including life sciences, environmental sciences, biomedical engineering and physical sciences. Modeling relies increasingly on computation, so the students should have basic skills for using computers and programs like Matlab or Mathematica.
Prereq: MATH 224 or MATH 228.


MATH 444. Mathematics of Data Mining and Pattern Recognition (3)
This course will give an introduction to a class of mathematical and computational methods for the solution of data mining and pattern recognition problems. By understanding the mathematical concepts behind algorithms designed for mining data and identifying patterns, students will be able to modify to make them suitable for specific applications. Particular emphasis will be given to matrix factorization techniques. The course requirements will include the implementations of the methods in MATLAB and their application to practical problems.
Prereq: MATH 201 or MATH 307.


MATH 445. Introduction to Partial Differential Equations (3)
Method of characteristics for linear and quasilinear equations. Second order equations of elliptic, parabolic, type; initial and boundary value problems. Method of separation of variables, eigenfunction expansions, Sturm-Liouville theory. Fourier, Laplace, Hankel transforms; Bessel functions, Legendre polynomials. Green’s functions. Examples include: heat diffusion, Laplace’s equation, wave equations, one dimensional gas dynamics and others. Appropriate for seniors and graduate students in science, engineering, and mathematics.
Prereq: MATH 201 or MATH 308 and MATH 224 or MATH 228.


MATH 448. Applied Partial Differential Equations (3)
Continuation of MATH 445. Linear and nonlinear partial differential equations, with emphasis on applications. Variational methods; asymptotic and perturbation methods: regular and singular perturbations; boundary layer, multiple scales, method of geometric optic and stationary phase. Applications to fluid dynamics, elasticity; optics; wave propagation. Topics depend upon instructor and may vary from year to year. Appropriate for seniors and graduate students in science, engineering and mathematics.
Prereq: MATH 445.


MATH 449. Dynamical Models for Biology and Medicine (3)
Introduction to discrete and continuous dynamical models with applications to biology and medicine. Topics include: population dynamics and ecology; models of infectious diseases; population genetics and evolution; biological motion (reaction-diffusion and chemotaxis); Molecular and cellular biology (biochemical kinetics, metabolic pathways, immunology). The course will introduce students to the basic mathematical concepts and techniques of dynamical systems theory (equilibria, stability, bifurcations, discrete and continuous dynamics, diffusion and wave propagation, elements of system theory and control). Mathematical exposition is supplemented with introduction to computer tools and techniques (Mathematica, Matlab).
Prereq: MATH 224 or MATH 228, or BIOL/EBME 300, and MATH 201.


MATH 461. Introduction to Topology
Metric spaces, topological spaces, and continuous functions. Compactness, connectedness, path connectedness. Topological manifolds; topological groups. Polyhedra, simplical complexes. Fundamental groups.
Prereq: MATH 224 or MATH 228.


MATH 462. Algebraic Topology (3)
The fundamental group and covering spaces; van Kampen’s theorem. Higher homotopy groups; long-exact sequence of a pair. Homology theory; chain complexes; short and long exact sequences; Mayer-Vietoris sequence. Homology of surfaces and complexes; applications.
Prereq: MATH 461.


MATH 465. Differential Geometry (3)
Manifolds and differential geometry. Vector fields; Riemannian metrics; curvature; intrinsic and extrinsic geometry of surfaces and curves; structural equations of Riemannian geometry; the Gauss-Bonnet theorem.
Prereq: MATH 321.


MATH 467. Differentiable Manifolds (3)
Differentiable manifolds and structures on manifolds. Tangent and cotangent bundle; vector fields; differential forms; tensor calculus; integration and Stokes’ theorem. May include Hamiltonian systems and their formulation on manifolds; symplectic structures; connections and curvature; foliations and integrability.
Prereq: MATH 322.


MATH 469. Calculus of Variations (3)
Examples of variational problems; variation of a functional; linear spaces; Frechet derivative; Euler Lagrange equations; Lagrange multipliers; Hamiltonian formulation; canonical coordinates; Noether’s theorem; second variation; conjugate points; direct methods. Other topics such as existence and regularity of solutions; Sobolev spaces; depending on audience.
Prereq: MATH 224 or MATH 228.


MATH 471. Advanced Engineering Mathematics (3)
Vector analysis, Fourier series and integrals. Laplace transforms, separable partial differential equations, and boundary value problems. Bessel and Legendre functions. Emphasis on techniques and applications.
Prereq: MATH 224 or MATH 228.


MATH 475. Mathematics of Imaging in Industry and Medicine (3)
The mathematics of image reconstruction; properties of radon transform, relation to Fourier transform; inversion methods, including convolution, backprojection, rho-filtered layergram, algebraic reconstruction technique (ART), and orthogonal polynomial expansions. Reconstruction from fan beam geometry, limited angle techniques used in MRI; survey of applications. Recommended preparation: PHYS 431 or MATH 471.


MATH 478. Computational Neuroscience (3)
Computer simulations and mathematical analysis of neurons and neural circuits, and the computational properties of nervous systems. Students are taught a range of models for neurons and neural circuits, and are asked to implement and explore the computational and dynamic properties of these models. The course introduces students to dynamical systems theory for the analysis of neurons and neural learning, models of brain systems, and their relationship to artificial and neural networks. Term project required. Students enrolled in MATH 478 will make arrangements with the instructor to attend additional lectures and complete additional assignments addressing mathematical topics related to the course. Recommended preparation: MATH 223 and MATH 224 or BIOL 300 and BIOL 306.
Offered as BIOL 378, COGS 378, MATH 378, BIOL 478, EBME 478, EECS 478, MATH 478 and NEUR 478.


MATH 481. Introduction to Mathematical Methods in Finance (3)
Mathematical finance in discrete time. Single period market models. Arbitrage. Risk-neutral valuation of contingent claims. Complete markets. Summary of results from probability theory and stochastic processes in discrete time. Conditional expectation. Discrete parameter martingales. Multiperiod market modes. Equivalent martingale measures. Risk-neutral valuation. Hedging strategies. Complete markets. The Cox-Ross-Rubinstein model. European options. American options. The Black-Scholes model. Binomial approximation. The pricing formula for European call options.
Offered as MATH 381 and MATH 481.


MATH 487. Stochastic Processes in Engineering and Sciences (3)
Review of basic probability concepts. Discrete-time Markov chains. Transition probability matrices. Classification of states. Stationary distributions. Limiting behavior. Random walk; application to the gambler’s ruin problem. Branching processes; application to population growth models. Examples of continuous time Markov chains. Poisson and compound Poisson processes. Birth and death processes. Limiting behavior. Renewal processes. Examples are drawn from queuing theory, reliability theory, population growth processes and other biological models.
Prereq: MATH 322.


MATH 491. Probability I (3)
Probabilistic concepts. Discrete probability, elementary distributions. Measure theoretic framework of probability theory. Probability spaces, sigma algebras, expectations, distributions. Independence. Classical results on almost sure convergence of sums of independent random variables. Kolmogorov’s law of large numbers. Recurrence of sums. Weak convergence of probability measures. Inversion, Levy’s continuity theorem. Central limit theorem. Introduction to the central limit problem.
Prereq: MATH 423.


MATH 492. Probability II (3)
Conditional expectations. Discrete parameter martingales. Stopping times, optional stopping. Discrete parameter stationary processes and ergodic theory. Discrete time Markov processes. Introduction to continuous parameter stochastic processes. Kolmogorov’s consistency theorem. Gaussian processes. Brownian motion theory (sample path properties, strong Markov property, Martingales associated to Brownian motion, functional central limit theorem).
Prereq: MATH 491.


MATH 495. Combinatorics (3)
Permutations, combinations and variations. Principle of inclusion and exclusion. Generating functions. Difference equations. Partitions. Stirling numbers. Eulerian numbers. Ballot problems. Ramsey’s theorem. Finite groups. Polya’s theorem. Debruijn’s theorem. Graphs. Trees. Finite fields. Finite geometries. Orthogonal Latin squares. Hadamard matrices. Block designs. Coding theory.
Prereq: MATH 307.


MATH 499. Special Topics (3)
Special topics in mathematics.


MATH 501. Topics in Algebra (3)
Selected topics from fields, rings, and modules.
Prereq: MATH 402.


MATH 527. Functional Analysis (3)
Selected topics in Functional Analysis. Recommended preparation: MATH 425.
Prereq: MATH 424.


MATH 528. Analysis Seminar (1-3)
Continuing seminar on areas of current interest in analysis. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include functional analysis, convexity theory, and their applications. May be taken more than once for credit. Consent of department required.


MATH 535. Applied Mathematics Seminar (1-3)
Continuing seminar on areas of current interest in applied mathematics. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include topics in applied probability and stochastic processes, continuum mechanics, numerical analysis, mathematical physics or mathematical biology. May be taken more that once for credit.


MATH 563. Topology Seminar (1-3)
Continuing seminar on areas of current interest in topology and geometry. Topics may include: minimal submanifolds; hyperbolic geometry and diffeomorphisms of surfaces; global analysis; discrete dynamical systems; gauge theory; symplectic geometry; closed geodesics. May be taken more than once for credit.


MATH 601. Reading and Research Problems (1-18)
Presentation of individual research, discussion, and investigation of research papers in a specialized field of mathematics.


MATH 651. Thesis (M.S.) (1-18)


MATH 701. Dissertation (Ph.D.) (1-18)
Prereq: Predoctoral research consent or advanced to Ph.D. candidacy milestone.