2011-12 General Bulletin

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220 Yost Hall
www.case.edu/artsci/math
Phone: 216-368-2880; Fax: 216-368-5163
Daniela Calvetti, Department Chair

This is an archived copy of the 2011-12 Bulletin. To access the most recent version of the bulletin, please visit http://bulletin.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 or 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 earn a Bachelor of Science in either mathematics or applied mathematics and a master’s degree from the mathematics department 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 or applied 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 PhD 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, PhD
(University of North Carolina)
Armington Professor and Chair
Scientific computing; imaging, inverse problems; modeling and simulation in life science

Christopher Butler, MS
(Case Western Reserve University)
Instructor and Theodore M. Focke Professorial Fellow
Teaching of mathematics

John Duncan, PhD
(Yale University)
Assistant Professor
The structure theory of vertex algebras; applications of vertex algebra in representation theory; algebraic geometry; number theory; topology

Weihong Guo, PhD
(University of Florida)
Assistant Professor
Image processing and analysis; compressive sensing; computational neuroscience; computer vision

David Gurarie, PhD
(Hebrew University, Jerusalem, Israel)
Professor
Applied mathematics: differential equations; geophysical fluid dynamics (turbulence, transport, chemistry); mathematical biology: infectious disease modeling, epidemiology, immunology

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

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

Peter Kotelenez, PhD
(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, PhD
(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, PhD
(Brown University)
Professor and Theodore M. Focke Professioral Fellow
Continuum physics; integral equations; functional analysis; mechanics of materials

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

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

Benjamin Nill, PhD
(Eberhard-Karls-Universitat)
Assistant Professor
Algebraic and toric geometry; convex and discrete geometry; cummutative algebra; geometric combinations; tropical geometry

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

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

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

Peter Thomas, PhD
(University of Chicago)
Assistant Professor
Synchronization and reliability of neural activity; gradient sensing, signal transduction and information theory; pattern formation in the visual cortex; malaria informatics

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

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


Secondary Faculty

Colin McLarty, PhD
(Case Western Reserve University)
Truman P. Handy Professor, Department of Philosophy
Logic; philosophy of mathematics, history of mathematics


Adjunct Faculty

Carsten Schütt, PhD
(Christian-Albrecht Universität, Kiel)
Adjunct Professor
Convex geometry; Banach Space theory; functional analysis

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

Teacher Licensure   I  BS in Mathematics   |  BS in Applied Mathematics  BS in Mathematics and Physics   |  Integrated BS/MS   |  Minors

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

The BA degree requires at least 38 hours of mathematics courses, including:

The following four courses: *14
Calculus for Science and Engineering I
Calculus for Science and Engineering II
Calculus for Science and Engineering III
Elementary Differential Equations
MATH 307Introduction to Abstract Algebra I3
MATH 308Introduction to Abstract Algebra II3
MATH 321Fundamentals of Analysis I3
MATH 322Fundamentals of Analysis II3
MATH 324Introduction to Complex Analysis3
or MATH 425 Complex Analysis I
Three approved technical electives: **9
ENGR 131Elementary Computer Programming ***3
Total Units41

*

 Or an equivalent sequence.

**

 No more than one can be from outside the department.

***

 Or other approved computer science course.

 
Teacher Licensure

Students interested in teaching mathematics at the high school level can prepare for teacher licensure through a joint program with John Carroll University. The requirements are:

(a) Completion of the BA program in mathematics, including the following as the three approved technical electives.:

MATH 150Mathematics from a Mathematician's Perspective3
MATH 304Discrete Mathematics3
STAT 312Basic Statistics for Engineering and Science3

 

(b) 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.


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Bachelor of Science in Mathematics

 The BS degree in mathematics requires at least 50 hours of mathematics courses.

The following four courses: *14
Calculus for Science and Engineering I
Calculus for Science and Engineering II
Calculus for Science and Engineering III
Elementary Differential Equations
MATH 307Introduction to Abstract Algebra I3
MATH 308Introduction to Abstract Algebra II3
MATH 321Fundamentals of Analysis I3
MATH 322Fundamentals of Analysis II3
MATH 324Introduction to Complex Analysis3
or MATH 425 Complex Analysis I
MATH 330Introduction of Scientific Computing3
Six approved technical electives **18
The following three courses: *9
General Physics I - Mechanics
General Physics II - Electricity and Magnetism
Introduction to Modern Physics
One of the following sequences:6
The Sun and its Planets
   and Stars, Galaxies, and the Universe
Principles of Chemistry I
   and Principles of Chemistry II
Principles of Chemistry for Engineers
   and Chemistry of Materials
Physical Geology
   and Introduction to Oceanography
Physical Geology
   and Historical Geology/Paleontology
Total Units65

*

Or an equivalent sequence.

**

No more than 9 hours may be from outside the department.

***

Or other approved computer science course.


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Bachelor of Science in Applied Mathematics

The BS 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 mathematics electives and the professional core in the area of application.

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

  • Applied dynamical systems
  • Applied probability and stochastic processes
  • Imaging
  • Life science
  • Scientific computing

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

The following four courses: *14
Calculus for Science and Engineering I
Calculus for Science and Engineering II
Calculus for Science and Engineering III
Elementary Differential Equations
MATH 304Discrete Mathematics3
MATH 307Introduction to Abstract Algebra I3
MATH 321Fundamentals of Analysis I3
MATH 322Fundamentals of Analysis II3
MATH 330Introduction of Scientific Computing3
One of the following two courses:3
Introduction to Complex Analysis
Complex Analysis I
Approved mathematics electives:18
Four courses specific to the concentration area of interest to the student (12 units)
Two MATH courses at the 300 level or higher (6 units)
Professional Core requirement12
12 approved credit hours specific to an area of application. This requirement is intended to promote scientific breadth and encourage application of mathematics to other fields.
Non-mathematics requirements
The following three courses: *9
General Physics I - Mechanics
General Physics II - Electricity and Magnetism
Introduction to Modern Physics
One of the following sequences:6-8
The Sun and its Planets
   and Stars, Galaxies, and the Universe
Principles of Chemistry I
   and Principles of Chemistry II
Principles of Chemistry for Engineers
   and Chemistry of Materials
Physical Geology
   and Introduction to Oceanography
Historical Geology/Paleontology
Total Units77-79

*

Or equivalent sequence 


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Bachelor of Science in Mathematics and Physics

In contrast to the BS in applied mathematics or the BS in physics with a mathematical physics concentration, this degree provides 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 for the major is 126, with at least 35 in MATH, at least 38 in PHYS, 6-7 in senior project and departmental seminar, and 9-11 in other science. The remainder of the hours include SAGES requirements, CAS distribution requirements, and open electives. 

A. Mathematics requirements
MATH 121Calculus for Science and Engineering I4
MATH 122Calculus for Science and Engineering II4
or MATH 124 Calculus II
MATH 223Calculus for Science and Engineering III3
or MATH 227 Calculus III
MATH 224Elementary Differential Equations3
or MATH 228 Differential Equations
MATH 307Introduction to Abstract Algebra I3
MATH 308Introduction to Abstract Algebra II3
or MATH 330 Introduction of Scientific Computing
MATH 321Fundamentals of Analysis I3
MATH 322Fundamentals of Analysis II3
MATH 324Introduction to Complex Analysis3
Approved Mathematics electives *6
--- (35 hours in Mathematics)
B. Physics requirements
PHYS 121General Physics I - Mechanics4
or PHYS 123 Physics and Frontiers I - Mechanics
PHYS 122General Physics II - Electricity and Magnetism4
or PHYS 124 Physics and Frontiers II - Electricity and Magnetism
PHYS 221Introduction to Modern Physics3
PHYS 310Classical Mechanics3
PHYS 313Thermodynamics and Statistical Mechanics3
PHYS 331Introduction to Quantum Mechanics I3
or PHYS 481 Quantum Mechanics I
PHYS 332Introduction to Quantum Mechanics II3
or PHYS 482 Quantum Mechanics II
One of the following:3
Introduction to Solid State Physics
Introduction to Nuclear and Particle Physics
Physical Optics
Quantum Electronics
Cosmology and the Structure of the Universe
Modern Cosmology
General Relativity
PHYS 423Classical Electromagnetism3
PHYS 472Graduate Physics Laboratory3
Two of the following: **6
Computational Methods in Physics
Methods of Mathematical Physics I
Methods of Mathematical Physics II
--- (38 hours in Physics)
C. Senior project and seminar; one of two options: ***6-7
C(i) Mathematics option
Senior Project for the Mathematics and Physics Program ****
SAGES departmental seminar in Mathematics
C(ii) Physics option
Advanced Laboratory Physics Seminar
Senior Physics Project
Senior Physics Project Seminar
---
D. Other science requirements (9-11 hours)
CHEM 105Principles of Chemistry I3-04
or CHEM 111 Principles of Chemistry for Engineers
CHEM 106Principles of Chemistry II3-04
or ENGR 145 Chemistry of Materials
ENGR 131Elementary Computer Programming3
Total Units88-91

*

Subject to the approval of the advisory committee. These may be chosen from the general list of mathematics courses at the 300 level or higher. Subject to approval, students may substitute an appropriate course from outside the mathematics and physics departments.

**

Subject to the approval of the advisory committee, students may substitute an appropriate course from outside the mathematics and physics departments. 

***

 Either option satisfies the SAGES capstone and departmental seminar requirements. 

****

is a year-long course, 2 credits per semester.  

 

In addition to the major coursework listed, there are requirements of 10 hours of SAGES freshman and university seminars, 12 hours of CAS distribution requirements, and enough open electives to bring the total number of hours to at least 126. 

First YearUnits
FallSpring
General Physics I - Mechanics (PHYS 121)
or Physics and Frontiers I - Mechanics (PHYS 123)
4  
Calculus for Science and Engineering I (MATH 121)4  
Elementary Computer Programming (ENGR 131)3  
Principles of Chemistry I (CHEM 105)*
or Principles of Chemistry for Engineers (CHEM 111)
3  
SAGES freshman seminar4  
Principles of Chemistry Laboratory (CHEM 113)2  
General Physics II - Electricity and Magnetism (PHYS 122)
or Physics and Frontiers II - Electricity and Magnetism (PHYS 124)
  4
Calculus for Science and Engineering II (MATH 122)
or Calculus II (MATH 124)
  4
Principles of Chemistry II (CHEM 106)*
or Chemistry of Materials (ENGR 145)
  3
Principles of Chemistry Laboratory (CHEM 113)  2
Other non-major course***  3
Year Total: 20 16
 
Second YearUnits
FallSpring
Introduction to Modern Physics (PHYS 221)3  
Calculus for Science and Engineering III (MATH 223)
or Calculus III (MATH 227)
3  
Introduction to Abstract Algebra I (MATH 307)3  
Non-major courses***9  
Classical Mechanics (PHYS 310)  3
MP Group I**  3
Elementary Differential Equations (MATH 224)
or Differential Equations (MATH 228)
  3
Introduction to Abstract Algebra II (MATH 308)
or Introduction of Scientific Computing (MATH 330)
  3
Year Total: 18 12
 
Third YearUnits
FallSpring
Thermodynamics and Statistical Mechanics (PHYS 313)3  
Introduction to Quantum Mechanics I (PHYS 331)
or Quantum Mechanics I (PHYS 481)
3  
Fundamentals of Analysis I (MATH 321)3  
MP Group II**3  
Non-major courses***9  
Introduction to Quantum Mechanics II (PHYS 332)
or Quantum Mechanics II (PHYS 482)
  3
Fundamentals of Analysis II (MATH 322)  3
Introduction to Complex Analysis (MATH 324)  3
MP Group III**  3
Year Total: 21 12
 
Fourth YearUnits
FallSpring
PHYS 3XX****3  
Graduate Physics Laboratory (PHYS 472)3  
MP Group IV**3  
SAGES Department seminar*****3  
Classical Electromagnetism (PHYS 423)  3
Senior Project*****  3-4
Non-major courses***  12
Year Total: 12 18-19
 
Total Units in Sequence:  129-130

 

*

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

**

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.

***

The number of open electives will vary depending on whether students choose 3-credit or 4-credit courses to fulfill other requirements (chemistry, senior project)

****

An advanced physics course to be selected from the following list: , , , , , , .

*****

 The Senior Project and SAGES departmental seminar should either be the Mathematics option ( and a Mathematics departmental seminar), or the Physics option (, , and ). 


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Integrated BS/MS Program in Mathematics and/or Applied Mathematics

The integrated BS/MS program is intended for highly motivated candidates for the BS in mathematics and applied mathematics who wish to pursue an advanced degree. Application to the BS/MS 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 BS 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 BS and MS 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 MS must be beyond that used to satisfy BS 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 BS/MS 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 (or the appropriate project course) during the senior year and are expected to complete all other courses for the BS before enrolling in further MS 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 BS degree before taking further graduate courses for the master’s degree.


Integrated BS/MS in Applied Mathematics and Another Discipline

There is the possibility of an integrated five-year study plan leading to a BS in applied mathematics and an MS in the area of application. In order to complete the requirements for the BS/MS in five years, students must choose an area outside mathematics that integrates well with mathematics, such as computing/information science, operations research, systems engineering, control theory, biology, or cognitive science. The general academic requirements for Integrated BS/MS programs must be followed. (Since the graduate courses required for the MS 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 MS degree.)


 

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Minor

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:

MATH 121Calculus for Science and Engineering I4
or MATH 125 Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 122Calculus for Science and Engineering II4
or MATH 124 Calculus II
or MATH 126 Math and Calculus Applications for Life, Managerial, and Social Sci II
MATH 223Calculus for Science and Engineering III3
or MATH 227 Calculus III
MATH 224Elementary Differential Equations3
or MATH 228 Differential Equations
MATH 150Mathematics from a Mathematician's Perspective *3
MATH 201Introduction to Linear Algebra3
or MATH 307 Introduction to Abstract Algebra I
MATH 301Undergraduate Reading Course1-3
MATH 302Departmental Seminar3
MATH 303Elementary Number Theory3
MATH 304Discrete Mathematics3
MATH 308Introduction to Abstract Algebra II3
MATH 321Fundamentals of Analysis I3
MATH 322Fundamentals of Analysis II3
MATH 324Introduction to Complex Analysis3
MATH 327Convexity and Optimization3
MATH 330Introduction of Scientific Computing3
MATH 333Mathematics and Brain3
MATH 338Introduction to Dynamical Systems3
MATH 343Theoretical Computer Science3
MATH 363Knot Theory3
MATH 380Introduction to Probability3
Or any 400-level MATH course

 

*

 To count toward a minor in Mathematics, must be taken in the first or second year.

 

 

 

 

Graduate Programs

The department offers programs leading to the Master of Science and the Doctor of Philosophy degrees. At both the master’s and the doctoral levels, students may pursue degrees in mathematics or 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 a faculty advisory committee 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 that 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 MS degree in mathematics. Courses in two of the following three basic areas must be included among the 27 credit hours required for graduation:

Abstract Algebra6
Abstract Algebra I
Abstract Algebra II
Analysis6
Introduction to Real Analysis I
Introduction to Real Analysis II
Complex Analysis I
Topology3
Introduction to Topology
Total Units15

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 MS 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 the graduate school's plan A and plan B.

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 requirements in each of the following groups:

  • At least 15 hours offered by the Department of Mathematics
  • At least 6 hours of courses offered outside the Department of Mathematics
  • 6 hours of thesis work (see below) or successful completion of a comprehensive exam

Given the great diversity of topics used in applications, there cannot be a large common core of requirements for the MS in applied mathematics.  Still, all students pursuing this degree are strongly advised to take and . 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 breadth areas. (The list includes suitable courses for each area. Please note that a course may be used to satisfy only one breadth area requirement.)

Applied Mathematics Breadth Areas

Analysis and Linear Analysis:
Advanced Engineering Mathematics *
Introduction to Real Analysis I
Advanced Matrix Analysis
Probability and its Applications:
Integrated Numerical and Statistical Computations
Probability I
Numerical Analysis and Scientific Computing:
Introduction to Numerical Analysis I
Numerical Differential Equations
Numerical Solutions of Nonlinear Systems and Optimization
Differential Equations:
Ordinary Differential Equations
Introduction to Partial Differential Equations
Dynamical Models for Biology and Medicine
Inverse Problems and Imaging:
Integrated Numerical and Statistical Computations
Computational Inverse Problems
Mathematics of Imaging in Industry and Medicine
Logic and Discrete Mathematics:
Mathematical Logic and Model Theory
Introduction to Cryptology
Life Science:
Mathematical Modeling
Dynamical Models for Biology and Medicine
Computational Neuroscience

*

Not suitable for credit towards the PhD requirements.

Other suitable courses for students in applied mathematics include:

MATH 424Introduction to Real Analysis II3
MATH 425Complex Analysis I3
MATH 427Convexity and Optimization3
MATH 428Fourier Analysis3
MATH 444Mathematics of Data Mining and Pattern Recognition3
MATH 475Mathematics of Imaging in Industry and Medicine3
MATH 492Probability II3

 

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

A student in the MS 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 the graduate school's plan A and plan B.

PhD 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 PhD students in both tracks must complete the following specific requirements:

  • Pass the PhD qualifying examination, which consists of examinations on three different subjects. All examinations are general proficiency examinations which may or may not be connected to specific courses. The topics for each subject are spelled out in a syllabus, periodically updated, which is available to the student. Students are expected to take the qualifying examination by the end of the second year of study and to successfully pass all parts of it by the beginning of their sixth semester in the PhD program. The subject requirements are different in the two tracks; see below. 
  • Write an acceptable thesis that constitutes an original contribution to mathematical knowledge. It is the responsibility of the student to find a thesis advisor who is willing to help plan a program and guide his or her research. This should be done immediately after passing the qualifying examination. A copy of a student’s thesis is to be available no later than 10 days prior to the final oral examination (see below), and the student is required to deliver an expository lecture on the subject of his or her thesis sometime prior to the final oral examination. This lecture is open to all students and faculty.
  • Pass a final oral examination consisting of a defense of the thesis. The examination committee, which consists of not fewer than four members of the faculty, including one whose appointment is outside the mathematics department, is responsible for certifying that the material presented in the thesis meets acceptable scholarly standards. The examination may also include an inquiry into the student’s competence in the major and related fields. All faculty members are welcome to attend.

Requirements specific to the different tracks

 
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 includes taking all courses in the three basic areas, and successfully completing qualifying examinations in algebra and analysis.

Qualifying Examination

A doctoral student in the mathematics track must take examinations on abstract algebra and real analysis. The third subject is to be selected from the following list: complex analysis, control and calculus of variations, differential equations, dynamical systems, functional analysis, geometry, probability, and topology. The choice of the examination subjects should be finalized by the end of the first year of study.

Abstract Algebra:6
Abstract Algebra I
Abstract Algebra II
Analysis:9
Introduction to Real Analysis I
Introduction to Real Analysis II
Complex Analysis I
Topology:3
Introduction to Topology
18 credit hours of approved course work18
Total Units36

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 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 includes taking qualifying examinations in the areas of computational mathematics and mathematical modeling, and taking certain courses in these three areas, as specified below. 

Qualifying Examination

A doctoral student in the applied mathematics track must take examinations in an area of computational mathematics and in an area of mathematical modeling. The third area of examination may be a more applied subject, including but not restricted to fluid mechanics, statistical mechanics, epidemiology, neuroscience, or a more traditional field of mathematics.


MATH 431Introduction to Numerical Analysis I3
One of the following:3
Numerical Differential Equations
Numerical Solutions of Nonlinear Systems and Optimization
MATH 441Mathematical Modeling3
One of the following:3
Ordinary Differential Equations
Introduction to Partial Differential Equations
24 hours of approved courses *24
Total Units36

*

Must include at least 9 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 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.

Scientific Computing Concentration
MATH 431Introduction to Numerical Analysis I3
MATH 432Numerical Differential Equations3
MATH 433Numerical Solutions of Nonlinear Systems and Optimization3
MATH 439Integrated Numerical and Statistical Computations3
or MATH 440 Computational Inverse Problems
MATH 441Mathematical Modeling3
MATH 445Introduction to Partial Differential Equations3
MATH 449Dynamical Models for Biology and Medicine3
or 
or MATH 478 Computational Neuroscience
Application area9

 

Imaging Concentration
MATH 428Fourier Analysis3
MATH 431Introduction to Numerical Analysis I3
MATH 432Numerical Differential Equations3
MATH 433Numerical Solutions of Nonlinear Systems and Optimization3
MATH 439Integrated Numerical and Statistical Computations3
or MATH 440 Computational Inverse Problems
MATH 441Mathematical Modeling3
MATH 444Mathematics of Data Mining and Pattern Recognition3
MATH 445Introduction to Partial Differential Equations3
MATH 475Mathematics of Imaging in Industry and Medicine3
EBME 410Medical Imaging Fundamentals3
PHYS 431Physics of Imaging3
PHYS 460Advanced Topics in NMR Imaging3

 

Life Science Concentration
MATH 431Introduction to Numerical Analysis I3
MATH 432Numerical Differential Equations3
MATH 433Numerical Solutions of Nonlinear Systems and Optimization3
MATH 439Integrated Numerical and Statistical Computations3
or 
MATH 440Computational Inverse Problems3
MATH 441Mathematical Modeling3
MATH 445Introduction to Partial Differential Equations3
MATH 449Dynamical Models for Biology and Medicine3
MATH 478Computational Neuroscience3
Application area9

 

Stochastics Concentration
MATH 423Introduction to Real Analysis I3
MATH 424Introduction to Real Analysis II3
MATH 431Introduction to Numerical Analysis I3
MATH 441Mathematical Modeling3
MATH 491Probability I3
MATH 492Probability II3
Application area9

 

PhD students entering with a bachelor’s degree are subject to the same breadth requirements as students pursuing the MS degree in applied mathematics.

Petitions

Any exceptions to departmental regulations or requirements must have the formal approval of the department's graduate committee. 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.

Courses

MATH 110. Introduction to Mathematical Communication and Software. 1 Unit.

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 Units.

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 Units.

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 Units.

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 124. Calculus II. 4 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Unit.

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 Units.

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).

MATH 303. Elementary Number Theory. 3 Units.

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 Units.

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 305. Introduction to Advanced Mathematics. 3 Units.

A course on the theory and practice of writing, and reading mathematics. Main topics are logic and the language of mathematics, proof techniques, set theory, and functions. Additional topics may include introductions to number theory, group theory, topology, or other areas of advanced mathematics. Prereq: MATH 122 or MATH 124 or MATH 126.

MATH 307. Introduction to Abstract Algebra I. 3 Units.

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 Units.

Continuation of MATH 307. Prereq: MATH 307.

MATH 319. Applied Probability and Stochastic Processes for Biology. 3 Units.

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 223 and BIOL 300 or BIOL 306 and MATH 201 or MATH 307 or consent of instructor.

MATH 321. Fundamentals of Analysis I. 3 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

An introductory survey to Scientific Computing from principles to applications. Topics which will be covered in the course include: solution 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 333. Mathematics and Brain. 3 Units.

This course is intended for upper undergraduate students in Mathematics, Cognitive Science, Biomedical Engineering, Biology or Neuroscience who have an interest in quantitative investigation of the brain and its functions. Students will be introduced to a variety of mathematical techniques needed to model and simulate different brain functions, and to analyze the results of the simulations and of available measured data. The mathematical exposition will be followed - when appropriate - by the corresponding implementation in Matlab. The course will cover some basic topics in the mathematical aspects of differential equations, electromagnetism, Inverse problems and Imaging related to brain functions. Validation and falsification of the mathematical models in the light of available experimental data will be addressed. This course will be a first step towards organizing the different brain investigative modalities within a unified mathematical framework. A final presentation and written report are part of the course requirements. Prereq: MATH 224 or MATH 228.

MATH 338. Introduction to Dynamical Systems. 3 Units.

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 Unit.

The purpose of this seminar is to introduce students to some of the research being done at Case 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 Units.

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 Units.

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.

MATH 352. Mathematics Capstone. 3 Units.

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 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.

MATH 361. Geometry I. 3 Units.

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.

MATH 363. Knot Theory. 3 Units.

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 376. Dynamics of Biological Systems II: Tools for Mathematical Biology. 3 Units.

Building on the material in Biology 300, this course focuses on the mathematical tools used to construct and analyze biological models, with examples drawn largely from ecology but also from epidemiology, developmental biology, and other areas. Analytic "paper and pencil" techniques are emphasized, but we will also use computers to help develop intuition. By the end of the course, students should be able to recognize basic building blocks in biological models, be able to perform simple analysis, and be more fluent in translating between verbal and mathematical descriptions. Offered as BIOL 306 and MATH 376. Prereq: BIOL 300 or MATH 224 or consent of instructor.

MATH 378. Computational Neuroscience. 3 Units.

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 Units.

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 400. Mathematics Teaching Practicum. 1 Unit.

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 Units.

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 Units.

A continuation of MATH 401. Prereq: MATH 401.

MATH 405. Advanced Matrix Analysis. 3 Units.

An advanced course in linear algebra and matrix theory. Topics include variational characterizations of eigenvalues of Hermitian matrices, matrix and vector norms, characterizations of positive definite matrices, singular value decomposition and applications, perturbation of eigenvalues. This course is more theoretical than MATH 431, which emphasizes computational aspects of linear algebra Prereq: MATH 307.

MATH 406. Mathematical Logic and Model Theory. 3 Units.

Propositional calculus and quantification theory; consistency and completeness theorems; Gödel 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 Units.

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 421. Fundamentals of Analysis I. 3 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

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 Units.

A second course in ordinary differential equations. Existence, uniqueness, and continuation of solutions of ODE. Linear systems, fundamental matrix, qualitative methods (phase plane). Dependence on initial data and parameters (Gronwall's inequality, nonlinear variation of parameters). Stability for linear and nonlinear equations, linearization, Poincare-Bendixson theory. Additional topics may include regular and singular perturbation methods, autonomous oscillations, entrainment of forced oscillators, and bifurcations. Prereq: MATH 224 and either MATH 201 or MATH 307.

MATH 439. Integrated Numerical and Statistical Computations. 3 Units.

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 Units.

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 Units.

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 pre-requisites 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 Units.

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 Units.

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 449. Dynamical Models for Biology and Medicine. 3 Units.

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. 3 Units.

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 Units.

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 Units.

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 Units.

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 471. Advanced Engineering Mathematics. 3 Units.

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 Units.

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 Units.

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 491. Probability I. 3 Units.

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 Units.

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 499. Special Topics. 3 Units.

Special topics in mathematics.

MATH 528. Analysis Seminar. 1 - 3 Unit.

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 Unit.

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 549. Mathematical Life Sciences Seminar. 1 - 3 Unit.

Continuing seminar on areas of current interest in the applications of mathematics to the life sciences. Allows graduate and advanced undergraduate students to become involved in research. Topics will reflect interests and expertise of the faculty and may include mathematical biology, computational neuroscience, mathematical modeling of biological systems, models of infectious diseases, computational cell biology, mathematical ecology and mathematical biomedicine broadly constructed. May be taken more than once for credit.

MATH 601. Reading and Research Problems. 1 - 18 Unit.

Presentation of individual research, discussion, and investigation of research papers in a specialized field of mathematics.

MATH 651. Thesis (M.S.). 1 - 18 Unit.


MATH 701. Dissertation (Ph.D.). 1 - 18 Unit.

Prereq: Predoctoral research consent or advanced to Ph.D. candidacy milestone.