## Departmental Degree Programs

### What are the requirements for the M.A. degree?

Prerequisites:

Undergraduate coursework equivalent to a major in mathematics from an accredited university. This should include a one-year course in either analysis or abstract algebra, equivalent to Mathematics 501-502 and 521-522 at the University of Louisville. Candidates who have not taken both must complete the second in their M.A. program.

Degree Requirements:

- Candidates must complete a program of study approved by the department. All courses (up to a maximum of 12 semester hours) to be taken outside the Department of Mathematics must have prior departmental approval.
- All students must complete a minimum of 30 semester hours of non-thesis graduate credit, including at least 15 semester hours in the Department of Mathematics, with one full-year sequence, in courses numbered 601 through 689.
- Students must satisfy one of the following two requirements:
- Students choosing the Thesis Option must pass a final oral examination described under "Requirements for the Master's Degree" in the General Information section of the Graduate School Catalog.

### What are the requirements for the Ph.D. degree?

Major: MATH

Degree: PHD

Unit: GA

Departmental Ph.D. Requirements:

All students admitted to the program must complete the following or their equivalent:

- Core Courses (24 semester hours:
- Two sequences, each of six (6) semester hours, chosen from:

Algebra MATH 621-622

Combinatorics MATH 681-682

Real Analysis MATH 601-602 - Two sequences, each of six (6) semester hours, chosen from:

Applied Statistics MATH 665-667

Mathematical Modeling MATH 635-636

Probability & Mathematical Statistics MATH 660-662

- Two sequences, each of six (6) semester hours, chosen from:
- Additional Topics and Area of Specialization - 18 semester hours:

In addition to the core, an application area of 18 hours will be required. The courses may be in a department outside Mathematics. They will be chosen in consultation with the student's advisory committee.

- Qualifying Examinations

Students must pass three written examinations. Two of these will be chosen from the areas Algebra, Combinatorics and Real Analysis. The third will be chosen from the areas of Applied Statistics, Mathematical Modeling and Probability & Mathematical Statistics. Normally, these will be taken within a year of completion of the core coursework. These examinations need not be taken together and each may be attempted at most twice.

- Industrial Internship - six (6) semester hours:

Each student, with prior approval of the Graduate Studies Director and the Industrial Internship Director, has to complete at least six (6) semester hours of an internship in an appropriate industrial or governmental setting, or have equivalent experience.

- Computing Project:

Each student must complete an approved computer project related to the student-s area of concentration.

- Candidacy Examination:

Each student must pass an oral examination in the chosen area of concentration. Usually, at most two attempts at passing this examination will be permitted. Students who wish to make a third attempt must petition the Graduate Studies Committee of the department for permission to do so.

- Dissertation - 18 to 24 semester hours

A doctoral dissertation is required of each student.

### What is the joint M.S.P.H./Ph.D. degree?

Purpose:

Joint degrees in Biostatistics and Mathematics are offered by the School of Arts and Sciences and the School of Public Health. Upon completion of the program, students will receive a Ph.D. in Mathematics and an M.S. in Biostatistics.

Application Procedure:

To be admitted to the program, the student is required to apply to and be accepted by both the Department of Mathematics and the Department of Bioinformatics and Biostatistics.

A student seeking admission into this program must submit letters to both departments stating the intent to take advantage of the joint degree program. Applicants will receive written notification stating whether their admission request has been approved or disapproved.

Degree Requirements:

- Required Courses:

The required courses consist of the non-overlapping core courses for both the Ph.D. in Mathematics and the M.S.P.H. in Biostatistics-Decision Science.

Course requirements derived from the Ph.D. in Mathematics (24 semester hours). These can be used to satisfy the 6 to 9 semester hours of electives required for the M.S.P.H. in Biostatistics-Decision Science.

Two sequences, each of six (6) semester hours, chosen from:- Two sequences, each of six (6) semester hours, chosen from:

Algebra MATH 621-622

Combinatorics MATH 681-682

Real Analysis MATH 601-602

- Two sequences, each of six (6) semester hours, chosen from:

Applied Statistics MATH 665-667

Mathematical Modeling MATH 635-636

Probability & Mathematical Statistics MATH 660-662

Course requirements derived from the M.S.P.H. in Biostatistics-Decision Science (17 to 23 semester hours). These courses can be applied to the 18 semester hours of electives, which are required for the Ph.D. in Mathematics.- Ethical Issues in Decision Making - Ph.D.A 605 (2 semester hours)
- Introduction to Pubilc Health and Epidemiology - PHCI 611 (2 semester hours)
- Social and Behavioral Sciences in Health Care - PHCI 631 (2 semester hours)
- Introduction to Environmental Health - PHCI 651 (2 semester hours)
- Health Economics - PHCI 662 (2 semester hours)
- Biostatistics-Decision Science Seminar - Ph.D.A 602 (1 semester hour)
- Biostatistical Methods I and II - Ph.D.A 680 and 681 (6 semester hours)
- Probability and Mathematical Statistics - Ph.D.A 661 and 662 (6 semesters hours). This requirement is waived if the student instead takes the Mathematics 660, 662 sequence listed above.

- Two sequences, each of six (6) semester hours, chosen from:
- Combined Internship, Practicum and Masters Thesis:
The industrial internship required by the Department of Mathematics, and the Public Health Practicum and Masters thesis required for the M.S.P.H. can be satisfied by a single internship and technical report which simultaneously satisfies the requirements for both degrees.

The internship must focus on public health and contain advanced mathematical content.

The technical report must meet two requirements:

- it must satisfy the requirements for a Master's thesis for the M.S.P.H. degree and it must be written at an advanced mathematical level.
- The six (6) semester hours of the internship will be divided as 3 hours for the Department of Mathematics and 3 hours for the Department of Bioinformatics and Biostatistics.

- Dissertation and Qualifying Examinations
In order for the student to fulfil the Ph.D. requirements, the student must satisfy both the qualifying examination and dissertation requirements for the Ph.D. in Mathematics. Failure to complete these requirements will not jeopardize the M.S.P.H. degree, if its requirements have all been satisfactorly completed.

### What is the accelerated M.A. degree for undergraduates?

Students who wish to pursue an accelerated M.A. may apply up to nine (9) hours of coursework taken for graduate credit to the requirements for their baccalaureate degree.

- Students must apply for admission to the program no later than the end of the junior year and must have completed MATH 205, 206, 301, and 325, or equivalent courses, prior to application.

- Applicants must have a minimum overall GPA of 3.5, and a minimum GPA of 3.66 in mathematics courses.

- As a part of the combined degree, students must complete MATH 405 and at least four (4) of the following: MATH 501, 502, 521, 522, 561, 562, or 581, including at least one sequence completed from among these courses.

- The student may take a maximum of nine (9) hours for graduate credit, which will also apply to the requirements for the baccalaureate degree in Mathematics. All 600-level courses numbered 689 or below qualify, as do 500-level courses, when completed in accord with the stipulations for graduate credit outlined in the syllabus.

## M.A. Program Questions

### How do I choose a thesis advisor?

There is no formal mechanism for connecting students with advisors. Usually, a student takes a class from a professor, likes the material or the professor, and asks the professor to become her advisor for a Masters thesis.

The professor is under no obligation to say yes, but most of the time an amicable arrangement can be reached.

When a professor agrees to be your advisor, please tell the Graduate Studies Director.

### What about an internship instead of a thesis or tests?

The three testing areas are chosen by the departmental Graduate Studies Committee in consultation with the student. The easiest way to proceed is to suggest three testing area to the Graduate Studies Director, who will then put them before the Graduate Studies Committee.

### How can I arrange for the testing option?

Students starting the M.A. program before Fall 2003 can choose to do an internship instead of a thesis or tests as a final project. Students starting during Fall 2003 or later may do an internship for credit, but it cannot be used as a final project in lieu of tests or a thesis. Here are Mathematics Department procedures for an M.A. internship.

## Ph.D. Program Questions

### What about the qualifier exams?

All Ph.D. students must pass three examinations in different areas of advanced mathematics or statistics. Following is the relevant part of the Ph.D. program description:

Students must pass three written examinations. Two of these will be chosen from the areas Algebra, Combinatorics and Real Analysis. The third will be chosen from the areas Mathematical Modeling, Applied Statistics and Probability & Mathematical Statistics. Normally, these will be taken within a year of completion of the core coursework. These examinations need not be taken together. Usually, at most two attempts at passing this examination will be permitted. Students who wish to make a third attempt must petition the Graduate Studies Committee of the department for permission to do so.

Every year, the exams will be given in January and August. Each exam will be 3 1/2 hours long.

Topics students are expected to know for each exam are listed below. Past exams and tips are available.

### Algebra

- Groups: homomorphisms and subgroups, cyclic groups, cosets and counting, normality, quotient groups, symmetric, alternating and dihedral groups, direct product and direct sum, finitely generated abelian groups, group action, the Sylow Theorems, nilpotent and solvable groups, normal and subnormal series.
- Rings: rings and homomorphisms, ideals (prime, maximal), factorization in commutative rings, unique factorization domains, principal ideal domains, euclidean domains, polynomial rings, Eisenstein's criterion.
- Fields and Galois Theory: field extensions, splitting fields, Galois group, separability, cyclic extensions, finite fields, cyclotomic extensions, radical extensions.

**Suggested Reference:**

- Thomas W. Hungerford, Algebra, Springer Verlag, 8th ed. (1997) ISBN: 0387905189, Chapters 1-3 and 5.

### Applied Statistics

- Linear Model: Linear model designs such as (a) crossed, (b) split plot, (c) nested, (d) repeated measures; maximum likelihood estimators and least squares; hypothesis testing; confidence intervals; regression diagnostics, and variable transformations.
- Classification: Nearest neighbor discriminant analysis, logistic regression, neural networks, and C-5 rule induction (decision trees).
- Clustering: K-means, and hierarchical.

Suggested References:

- The Analysis of Messy Data, vol. 1 by Miliken and Johnson.
- The Elements of Statistical Learning: Data Mining, Inference, and Prediction by Hastie, Tibshirani, and Friedman.
- Generalized, Linear, and Mixed Models by McCulloch and Searle.

### Combinatorics

- Advanced Counting: sequences, selections, distributions, partitions, lattice paths, Catalan numbers, Stirling numbers, Ferrers diagrams, Ramsey numbers, inclusion-exclusion, derangements, recurrence relations, generating functions (both ordinary and exponential), characteristic polynomial method to solve recurrences, Stirling's approximation, Polya counting
- Elementary Graph Theory: graphs, digraphs, subgraphs, degrees, adjacency matrices, incidence matrices, graph isomorphism, paths, cycles, trees, connectivity, bipartite graphs, edge contraction, subdivisions, linegraphs, independent sets, cliques
- Flows and Related Graph Concepts: network flows, integral flows, Max-Flow/Min-Cut theorem, (edge and vertex) connectivity, vertex cuts, edge cuts, perfect matchings, Menger's theorem, Hall's marriage theorem, Tutte's theorem, factorizations, Petersen graph, Konig-Egervary theorem
- Planar Graphs: drawings in the plane, planar duals, Euler's formula, Kuratowski's theorem, convex embeddings in the plane, coloring planar graphs
- Graph Coloring: vertex and edge coloring, Brooks' theorem, Vizing's theorem, color critical graphs, perfect graphs, Lovasz's Perfect graph theorem, the Strong Perfect Graph Theorem
- Miscellaneous: Ramsey graphs, Hamiltonion cycles, random graphs, Turan's theorem, partially ordered sets, Dilworth's theorem

**Suggested References:**

- Kenneth P. Bogart, Introductory Combinatorics, Harcourt-Academic Press, 3rd ed. (2000) ISBN 0121108309
- Douglas B. West, Introduction to Graph Theory, Prentice Hall, 2nd Edition (2000), ISBN: 0130144002

Also useful are the following Schaum's Outlines:

- V. K. Balakrishnan, Schaum's Outline: Graph Theory, McGraw Hill (1997) ISBN: 0070054894
- V. K. Balakrishnan, Schaum's Outline: Combinatorics (including graph theory), McGraw Hill (1995) ISBN: 007003575X

### Mathematical Modeling

- Mechanical vibrations: Newton's laws, spring-mass systems, two-mass oscillators, friction, damping, pendulum, linear stability and equilibria, energy analysis, phase plane analysis, nonlinear oscillations, control oscillations, inverse probllem. [1] pp 1-114 and [2] Chapters 0-2.
- Traffic flow: Velocities and velocity fields, trafffiic flow and density, conservation laws, linear and nonlinear car-following models, steady state, first order partial differenntial equations (the method of characteristics), green light mooddels and rarefaction solution, shock waves (effect of rred light annd slower traffic ahead), highway with entrance (inhomogenneous problem), traffic wave propagation, optimiization problem. [1] pp. 259-394.
- Dynamical systems: Nonlinear systems in the plane, interacting species, limit cycles, Hamiltonian systems, Liapunov functions and stability, bifurcation theory, three-dimensional autonomous systems and chaos, Poincare maps and nonautonomous systems in the plane, linear discrete dynamical systems. [2] Chapters 3-9, 13.

**Suggested References:**

- [1] Richard Haberman, Mathematical Models, Mechanical Vibrations, Population Dynamics and Traffic Flow, SIAM Classics in Applied Mathematics (1987) ISBN: 978-0898714081.
- [2] Stephen Lynch, Dynamical Systems with Applications Using Maple, Springer-Verlag (2000) ISBN: 0-8176-4150-5.

### Probability & Mathematical Statistics

- Probability Spaces: basic axioms, conditional probability and independence
- Denumerable Probabilities: limit sets, Borel-Cantelli Lemma
- Existence and Extension: fields, sigma-fields
- Simple and Discrete Random Variables: expectation, Chebyshev-Markov inequalities, convergence of random variables
- The Law of Large Numbers: The Strong Law, The Weak Law
- Abstract Measures and Integration Theory: basic definitions and theorems, Fatou’s Lemma, Lebesgue Dominated Convergence Theorem, Monotone Convergence Theorem
- Random Variables and Expected Values: Random variables and distribution functions, densities, expected values, independence
- Sums of Independent Random Variables: Kolmogorov’s 0-1 Law
- Distribution on R: weak convergence, Mapping Theorem, convergence in probability and almost sure convergence, Skorokhod Representation Theorem, Helly’s Selection Theorem, characteristic functions and moments, Central Limit Theorems (classical CLT,Lindeberg’s CLT, Lyapounov’s CLT)
- Distributions on R^k: weak convergence, Cramer-Wold Device Theorem, characteristic functions, multivariate normal distributions, Multivariate Central Limit Theorem
- Conditional Expectations: conditional expectation given a sigma-field, properties, Smoothing-Towering Theorem
- Stochastic Processes: Martingales, Brownian motion

#### Suggested References

[1] Billingsley, Patrick. *Probability and Measure*.Wiley, Anniversary Edition, 2012.

### Real Analysis

- differentiation and Riemann integration of functions of one real variable, sequences of

functions, uniform convergence, Lebesgue's characterization of Riemann integrability - topology of the line, countable and uncountable sets, Borel sets, Cantor sets and Cantor

functions, Baire category theorem - Lebesgue measure and integration on the line, measurable functions, convergence theorems
- AC and BV functions, fundamental theorem of calculus, Lebesgue differentiation theorem
- Hilbert spaces, L
_{p}spaces, l_{p}spaces, Hölder and Minkowski inequalities, completeness.

**Suggested References:**

- H. L. Royden, Real Analysis, Prentice Hall, 3rd ed. (1988) ISBN: 0024041513 Chapters 1-7, 11, 12.
- Walter Rudin, Real and Complex Analysis, McGraw-Hill Science/Engineering/Math, 3rd ed. (1986) ISBN: 0070542341, Chapters 4-5.

Another book that contains more elementary background information is the following.

- Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd ed., (1976) ISBN: 007054235X.

### Mathematical Physics

- Particles and fields, recapitulation of quantum mechanics of 1930's, general principles of quantum mechanics including: linear operators, states and observables, fundamental principles of quantum mechanics, compatibility of observations, representations and transformations, time-dependent equation, and the density matrix. Central forces: orbital angular momentum, and motion under central forces. Approximation methods: the method of variation, the method of perturbation, the method of perturbation-variation, and the method of Wentzel-Kramers-Brillouin.
- Rotation and angular momentum, coupling of two angular momenta, scalar, spinor, and vector fields, spin-dependent interactions, polarization of particles with spin, system of particles, LS-coupling and jj-coupling, two-fermion system in LS-coupling, the helium atom, configuration mixing, systems of more than two fermions, calculation of single-particle wave function (Hartree-Fock approximation), time-dependent perturbation theory, scattering, and the method of partial waves.

**Suggested References:**

- R. Shankar, Principles of Quantum Mechanics (2nd Ed.), Kluwer Academic/Plenum Publishers, 1994, ISBN: 0306447908.
- N. Zettili, Quantum Mechanics: Concepts and Applications , John Wiley & Sons, 2001, ISBN: 0471489441.

### What's an advisory committee?

Upon admission to the Ph.D. program, every student is assigned an advisory committee of three faculty members to help with curriculum decisions. This advisory committee need not, and probably will not, contain the student's eventual dissertation advisor. Its purpose is to make sure the student is well informed about early choices in her graduate career.

## GTA Questions

### What if I have to miss a class that I teach?

If the absense is planned ahead of time, notify your course coordinator of your absense, so that someone can be lined up to substitute for you. Regular absenses are not acceptable.

Emergencies happen, so it is possible you might miss class unexpectedly. If this happens, you should call the Mathematics Department Chair as soon as possible to notify him of your problem.

### What should I do if students ask me to let them into my class when it is closed?

Only the Chair or Assistant Chair of the Department have the authority to allow students into closed sections. Explain this to the student and send him or her to see the Assistant Chair.

## General Questions

### What about GradList?

Math Graduate Student E-mail List

**How do I unsubscribe from GradList?**

To do so, you must send a command to, Majordomo, the software that serves the list. This is done by sending it an e-mail message. If you are unsubscribing from GradList, send your message to

GradList-request@erdos.math.louisville.edu

Sending your message to the standard list address will do nothing. The only thing to appear in the message should be the single word

unsubscribe

Make sure your mailer does not automatically append anything else to the message, and it's best to send the message as plain text e-mail. The return address on your message must be in the database of subscribers, so send the message from the same account you use to receive mail from the list. You should get an acknowledgement from the Majordomo software a short time after you unsubscribe. If this acknowledgement does not appear, something went wrong. More information about unsubscribe and other commands is available by sending e-mail containing the single word "help" to majordomo@erdos.math.louisville.edu.

If you change your e-mail address, please remember to unsubscribe from your old one and subscribe under your new address.

**How do I subscribe to GradList?**

Anyone can subscribe to the GradList. To do so, you must send a command to, Majordomo, the software that serves the list. This is done by sending it an e-mail message. If you are subscribing to GradList, send your message to

GradList-request@erdos.math.louisville.edu

In either case, the only thing to appear in the message should be the single word

subscribe

You should get an acknowledgement from the Majordomo software a short time after you subscribe. If this acknowledgement does not appear, something went wrong. More information about subscribe and other commands is available by sending e-mail containing the single word "help" to majordomo@erdos.math.louisville.edu.

**How do I post messages to the list?**

To post a message to the discussion lists, just send an ordinary email message to GradList@erdos.math.louisville.edu. This message will be automatically distributed to all subscribers. If the return address on the message is not in the subscriber list, the message will be rejected.

There is no moderation on the list, so be careful what you post. When your message is sent, it will be immediately forwarded to all list subscribers.

There are a few nuances to posting that are sometimes nuisances.

First, the list management software, Majordomo, is set to look for certain key words in the subject of messages to avoid administrative commands being posted to the list. These include words such as subscribe and unsubscribe.

Second, only those subscribed to the list may post to the list. All other messages are rejected. The way Majordomo determines membership is from the sender's e-mail address. So, if you want to post to the list, you should probably send the message from the same account at which you receive mail from the list. (Or set your mail software to have an appropriate return address.) The reason for this requirement is to keep the spammers off the list.