# Honors Program in Math

The L&S Honors Program serves over 1,300 students in the College of Letters and Science with an enriched undergraduate curriculum. Students in the program pursue the Honors in the Liberal Arts, Honors in the Major or Comprehensive Honors Degrees. The program began in response to a 1958 petition by students seeking more challenging work and opportunities to "delve more deeply" into their fields of interest.

For more information about the Honors Program at UW-Madison in general, please see http://honors.ls.wisc.edu.

See below for Math Department specific information for completing an Honors in the Major distinction.

#### Honors in the Major: Mathematics Requirements.

In order to earn the Honors degree in Math, in addition to completing the basic degree requirements, the student must

- have a cumulative GPA of at least 3.3,
- be in the Honors program during the last three semesters before graduation,
- honors candidates in Mathematics must complete the following basic math honors curriculum with grades B or better
- Honors sections of 521-522
- Honors sections of 541-542
- At least two more courses in the 5XX or 6XX level (one of them usually 551) decided in consultation with the honors adviser. They can be chosen from the list of optional courses below under “Course description”
- 681H-682H or a sequence of two math courses numbered 700 or above.

The following points should also be noted:

- At least one of the two sequences 521-2, 541-2 must be completed by the end of the student junior year;
- The student should declare his/her candidacy for the Honors Program in Mathematics to the Honors adviser by the start of the junior year;
- Before choosing the Honors thesis option (681-2), the student must consult with the Honors adviser to prepare for and complete a suitable thesis. This option is strongly recommended by the Department, specially for students planning to go into graduate studies in Math. For up-to-date information about research groups for undergraduates within the math department please see "Senior Honors Thesis" below.
- If the student elects to complete the alternative sequence of graduate level courses, the sequence must be approved by the Honors adviser.

If you want information about the scheduled rotation of honors sections, the requirements to enroll in honors sections or for honors credits, and suggested plans for taking the scheduled honors sections, go to Honors in Calculus and other Honors sections below.

#### Honors in Calculus and other Honors sections

The mathematics department has in place a first class group of honors courses designated for the calculus sequences, linear algebra and differential equations.

Math 275-276 is the first group of courses in the sequence. They cover roughly the same material as Math 221-222 but with a much more in-depth approach to each subject, the emphasis placed on the understanding of the subject and the theoretical foundations. Following the same approach, Math 375-376 covers most topics on Math 234, the third calculus course, Math 340 linear algebra, plus many of the topics in a typical course in differential equations like for example Math 319. Math 275 and Math 375 are offered in the first semester only, while 276-376 are offered in Spring.

The initial placement in the sequence is by invitation but we also consider applications from qualified students who may not have received an invitation. Please contact one of the departmental Honors coordinators (Professors Street and Mari-Beffa in 2013/14) or the professor scheduled to teach the course.

The mathematics department also expects to offer honors sections or sections with honors credit available in the advanced undergraduate Math courses according to the following rotation.

Semester I : 341, 521, 522, 541.

Semester II: 341, 521, 522, 541, 542.

In the timetable the symbol "!" will denote a separate honors section and the symbol "%" will denote that honors credit is available in a regular section.

In addition, qualified students may enroll for Honors credit in 551, 552, 561, 567, 621, 623, 627, 629, 632, 641. Other higher level courses may occasionally appear in the timetable for honors credits (with the symbol !) or with honors credit available (with the symbol %). Regular sections can also be taken for Honors credits following the appropriate procedure described by the Letters and Science Honors program. Contact the Professor teaching the course if you need to do so. Any graduate course can be taken for Honors credit by a qualified undergraduate.

### Course descriptions

Below there is a list of descriptions of courses which are offered to students as part of the Honors curriculum. They are either mandatory or optional for an Honors student (see requirements above). For a complete list of math courses, please consult the undergraduate catalogue of the Mathematics department. For a description of graduate courses, please consult the graduate catalogue.

##### MATH 275-276 HONORS CALCULUS I-II

This sequence of five credit courses is the first part of the Calculus Honors sequence developed by the Math Department at the UW. The material covers essentially the same topics as the standard calculus I-II courses but the material is discussed in greater depth, and with much more emphasis on mathematical ideas. The goal of the sequence is to provide highly motivated and well prepared students with an opportunity to go beyond the traditional approach to the subject. The standard texbook is Calculus, V. I and II, by T.M. Apostol, and the prerequisites are a personal invitation or consent of the instructor.

##### MATH 375-376 HONORS CALCULUS III-IV

This sequence is the second part of five credit courses in the Calculus Honors sequence developed by the Math Department at the UW. The material covers essentially the same topics as the standard third semester calculus course, plus linear algebra in 375 and differential equations in 376. Again, the material is studied with more depth, more rigorously and with higher expectations of students. The standard texbook is Calculus, V. I and II, by T.M. Apostol, and the prerequisites are a personal invitation or consent of the instructor.

##### MATH 521-522 ANALYSIS I-II

This sequence of 3 credit courses (concluded by Math 621-Analysis III) introduces students to fundamental concepts and basic elementary theorems of analysis with emphasis on functions of several variables. The objective is to convey an understanding of the structure of analysis in itself as well as its role as a tool for other disciplines. This sequence is essential for students preparing for graduate studies in mathematics; also it should be taken by students of physics and engineering who intend to do graduate work in their areas.

Topics in 521: The real numbers, Elements of set theory and topological notions, metric spaces, Sequences and series, functions, limits, continuity, differentiation, integration, sequences and series of functions, uniform convergence.

Topics in 522: More on convergence (Fourier series, approximations of the identity, polynomial approximation, infinite products(, Compactness in metric spaces with applications, The contraction principle with applications, Differential calculus in normed spaces, implicit function theorem, and other topics.

Possible texts for Math 521-522: Principles of Mathematical Analysis, 3rd edition, by W. Rudin. Mathematical Analysis, by A. Browder. The way of analysis, by R. Strichartz.

##### MATH 541-542 MODERN ALGEBRA I-II

This is a sequence of 3 credit courses. Math 541 gives an introduction to basic abstract algebra. The coninuation emphasizes linear algebra and field theory. Both courses are essential for students preparing for graduate studies in mathematics.

Topics in 541: group theory: subgroups, homomorphisms, isomorphisms, normal subgroups, permutation groups, class equation, Sylow theorem, finite abelian groups; ring theory: homomorphisms, isomorphisms, ideals, integral domains, polynomial rings.

Topics in 542: vector spaces: subspaces, homomorphisms, quotient spaces, bases, dual spaces, inner product spaces, modules; field theory: extension fields, transcendence of "e", roots of polynomials, construction with straightedge and compass; linear transformations: algebra of linear transformations, eigenvalues, eigenvectors, matrices, canonical forms, determinants.

Possible texts: Math 541: Abstract Algebra by Herstein; Contemporary Abstract Algebra by Gallian. Math 542: Topics in Algebra by I. N. Herstein; Finite Dimensional Vector Spaces by P. Halmos; Linear Algebra by K. Hoffman and R. Kunze.

Prereq for Math 541: Math 340 or equiv. Prereq for Math 542: Math 541.

##### MATH 551 ELEMENTARY TOPOLOGY

This 3 credit course is an introduction to the basic ideas and methods of point set topology. It is a good background for analysis courses and graduate topology courses. Topics: basic intuitive set theory, topological spaces, separation axioms, compactness, connectedness, metric spaces, special topics.

Possible texts: Topology, by Munkres. Principles of Topology by Croom. Basic Topology by Armstrong.

Prereq: Math 234 (however, it is advisable to have at least one previous "abstract" course such as Math 521 or 541 before taking Math 551).

##### MATH 552 ELEMENTARY GEOMETRIC AND ALGEBRAIC TOPOLOGY

This 3 credit course is occasionally taught.

Topics include: Fundamental group and applications: classification of closed 2-manifolds, elementary homotopy theory, the fundamental group of the circle, covering spaces. Simplicial homology: simplexes, triangulation, homology groups, Euler characteristic, simplicial approximation. Selected topics such as fixed point theorems, singular homology, knot theory, group actions.

Prereq: Math 551 and 542 or consent of the instructor.

##### MATH 561 DIFFERENTIAL GEOMETRY

In this 3 cr. course, curves and surfaces in three and higher dimensional spaces are studied using calculus. The course is useful in preparation for the study of differentiable manifolds and for some aspects of applied mathematics and physics. Topics: curves, ac arc length, Serret-Frenet equations, two dimensional surfaces, first and second fundamental forms, geodesics, Gauss-Bonnet theorem.

Possible texts: Elements of Differential Geometry by R.S. Millman and G.D. Parker; Differential Geometry of curves and surfaces by P. do Carmo; Elementary Differential Geometry by O'Neil.

Prereq: Math 320 or 340 (or equiv.) and 521 (Math 321 or 522 also recommended).

##### MATH 567 ELEMENTARY NUMBER THEORY

This course (3cr.) is an undergraduate introduction to number theory.

Topics include: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem. Number-theoretic functions, multiplicative functions, MÃ¶bius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.

Prereq: Math 340 (or equiv.) or concurrent registration.

##### MATH 621 ANALYSIS III

This is the third course in our analysis sequence (3 credits). It is essential for students preparing for graduate school in mathematics.

Topics include: Integration in Euclidean spaces. Measure and content zero, and characterization of Riemann integrability Fubini's theorem, Partition of unity, Changes of variables. Some multilinear algebra. Manifolds and tangent spaces, vector fields and differential forms, orientation. Integration, Stokes' theorem on manifolds. Euclidean measure for submanifolds. Classical vector analysis. Other optional topics such as Cauchy's integral theorem and formula, homotopy.

Possible text for Math 621: Calculus on manifolds, by M.Spivak, Vector Analysis, by Klaus Jaenich.

Prereq. Math 522.

##### MATH 623 COMPLEX ANALYSIS

This is a 3 credit introduction to the theory of analytic functions of a complex variable. Attention is given to the techniques of complex analysis as well as the theory. It is suitable for math majors as well as students in the physical sciences and engineering. Topics: complex numbers, elementary functions, analyticity, complex integration, Cauchy's theorem and formula, power series, residues, conformal mapping, harmonic functions.

Possible texts: Complex Analysis by E.M. Stein and R. Shakarchi.

Prereq: Math 321 or 521.

##### MATH 627 FOURIER ANALYSIS

This is a 3 credit introductory course in Fourier Analysis which is occasionally taught. It is targeted towards advanced students and should be useful especially for those students who plan to enter graduate studies in mathematics, physics and engineering.

Topics include: Motivation from PDE, Extensive study of Fourier series and their convergence. The Fourier transform on the real line and in Euclidean space. The discrete Fourier transform. More optional topics such as applications to number theory problems.

Possible texts: Fourier Analysis, by E.M. Stein and R. Shakarchi. Fourier Analysis, by T.W. Koerner.

Prereq: Math 521. Math 522 is recommended.

##### MATH 629 INTRODUCTION TO MEASURE AND INTEGRATION

This is a 3 credit introduction to measure and integration theory. It is particularly suitable for further studies in analysis, probability or statistics. Topics: Lebesgue integration, convergence theorems, general measure theory, differentiation, applications to probability.

Possible texts: Real Analysis by Royden; Real Analysis and Probability by Ash; Measure and Integral by Wheeden and Zygmund.

Prereq: Math 522.

##### MATH 632 INTRODUCTION TO STOCHASTIC PROCESSES

(Same as Stat., Industrial Eng., Bus, 632) This is a continuation of the introduction to probability begun in Math 43l with particular emphasis on stochastic processes. Topics: Markov chains (discrete time); stationary distributions of a Markov chain; Markov pure jump processes (continuous time); topics chosen from random walks, renewal theory, semi-Markov processes, Brownian motion and optimal stopping.

Possible texts: A First Course in Stochastic Processes, 2nd Ed. by Karlan and Taylor; Introduction to Probability Models by Ross; Introduction to Stochastic Processes by Hoel, Port & Stone.

Prereq: Math 431, or Stat 309-310 or Stat 311-312 or Stat 313-314.

##### MATH 641 INTRODUCTION TO ERROR-CORRECTING CODES

(Same as ECE 641.) This is a 3-credit first course in coding theory which is occasionally taught. It is of interest to mathematicians, computer scientists, statisticians and electrical engineers. Topics: linear codes, decoding and encoding; Hamming codes, Shannon's theorem on existence of good codes; binary Golay code; finite fields and BCH codes; dual codes and the weight distribution; cyclic codes: generator polynomial and check polynomial; Reed-Soloman codes and burst errors; Euclidean algorithm for decoding BCH codes; Reed-Muller codes.

Possible texts: The Theory for Error-Correcting Codes I by N.J.A. Sloane and F. J. MacWilliams; The Theory of Information and Coding by R. McEliece (only the second part); Introduction to Error Correcting Codes by V. Pless.

Prereq: Math 320 or 340 or equiv., and 541 or consent of instructor.

### Senior Honors Thesis and undergraduate research

**MATH 681-682 SENIOR HONORS THESIS**

This course is used for work on the honors thesis under the supervision of a faculty member.

Prereq: senior standing and enrollment in the math honors program.

If you are interested in participating actively in research activities, please talk to the Professor running a current program or

consult with the Honors adviser (Prof. Gloria Mari Beffa in 2013/14).

### Grants, Awards, etc

Check out the list of Scholarships offered by the University of Wisconsin-Madison for undergraduate math majors.

In addition to these UW awards there are several federal and international competitions. Please consult with the Honors advisor for an up-to-date list.