Lower Division Courses
M302 Introduction to Mathematics
Prerequisite and degree relevance: Three units of high school mathematics at the level of Algebra I or higher and a passing score on the Mathematics section of the THEA test or equivalent. The placememnt test is not required. It may be used to satisfy Area C requirements for the Bachelor of Arts degree under Plan I or the mathematics requirement for the Bachelor of Arts degree under Plan II.
M302 is intended primarily for general liberal arts students. It may not be included in the major requirement for the Bachelor of Arts or the Bachelor of Science degree with a major in mathematics. In some colleges M302 cannot be counted toward the Area C requirement nor toward the total hours required for a degree. Only one of the following may be counted: M302, 303D, or 303F. A student may not earn credit for Mathematics 302 after having received credit for any calculus course.
Course Description: This is a terminal course satisfying the University's generaleducation requirement in mathematics. Topics may include: number theory (divisibility,prime numbers, the Fundamental Theorem of Arithmetic, gcd, Euclidean Algorithm,modular arithmetic, special divisibility tests), probability (definition, laws, permutations and combinations), network theory (Euler circuits, traveling salesman problem, bin packing), game theory. Some material is of the instructor's choosing.
M303D Applicable Mathematics
Prerequisite and degree relevance: A minimal required score on the ALEKS placement exam. May not be included in the major requirement for the Bachelor of Arts or Bachelor of Science degree with a major in mathematics. Only one of the following may be counted: Mathematics 302, 303D, and 303F. A student may NOT earn credit for Mathematics 303D after having received credit for either Mathematics 305G or M505G, nor any calculus course.
The prerequisite is a Mathematics Level I or IC score of at least 430/400, or Mathematics 301 with a grade of at least C.
Course description: The course treats some of the techniques which allow mathematics to be applied to a variety of problems. It is designed for the nontechnical student who needs an entry level course developing such mathematics skills. Topics include: linear and quadratic equations, systems of linear equations, matrices, probability, statistics, exponential and logarithmic functions, and mathematics of finance.
M305G Preparation for Calculus
Prerequisite and degree relevance: The prerequisite is the minimum required score on the ALEKS placement exam. Credit for M305G may NOT be earned after a student receives credit for any calculus course (e.g. 408C, 308K, M403K, or equivalent) with a grade of at least C. Only one of M305G and any collegelevel trigonometry course may be counted. M301, 305G and equivalent courses may not be counted toward the major requirement for the Bachelor of Arts, Plan I, degree with a major in mathematics or toward the Bachelor of Science in Mathematics degree.
Course Description: M305G is a discussion of the functions and graphs met in calculus. The courses cover logarithms, exponential functions, trigonometric functions, inverse trigonometric functions, polynomials, and the range, domain and graphs of these functions.
M408C Calculus I
Prerequisite and degree relevance: The minimum required score on the Aleks placement exam. 408C may not be counted by students with credit for Mathematics 403K, 408K, 408N, or 408L. M408C and M408D (or the equivalent sequence M408K, M408L, M408M; M408N, M408S, M408M) are required for mathematics majors, and mathematics majors are required to make grades of C or better in these courses.
Course description: M408C is our standard firstyear calculus course. It is directed at students in the natural and social sciences and at engineering students. The emphasis in this course is on problem solving, not on the presentation of theoretical considerations. While the course necessarily includes some discussion of theoretical notions, its primary objective is not the production of theoremprovers. The syllabus for M408C includes most of the elementary topics in the theory of real valued functions of a real variable: limits, continuity, derivatives, maxima and minima, integration, area under a curve, volumes of revolution, trigonometric, logarithmic and exponential functions and techniques of integration. M408C classes meet three hours per week for lectures and two hours per week for problem sessions.
M408DAP Honors Calculus II
Course description: An honors version of the calculus for sciences.This brings together advanced students from all Colleges, and is taught by a senior faculty known for worldclass research as well as superb teaching. It picks up where AP AB leaves off so students do not waste time repeating material.
M408D Calculus II
Prerequisite and degree relevance: A grade of C or better in M408C or M408L or the equivalent. 408D may not be counted by students with credit for Mathematics 403L, 408S, 408M. Both of M408C and M408D (or the equivalent sequences M408K, M408L, M408M; M408N, M408S, M408M) are required for mathematics majors, and mathematics majors are required to make grades of C or better in these courses. Certain sections of this course are reserved as advanced placement or honors sections; they are restricted to students who have scored well on the advanced placement AP/BC exam, or are honors students, or who have the approval of the Mathematics Advisor. Such sections and their restrictions are listed in the Course Schedule for each semester.
Note: The pace of M408C and M408D is brisk. For this reason, transfer students with one semester of calculus at another institution are requested to consult with the Undergraduate Adviser for Mathematics to determine whether M408D or an alternative, M308L, is the appropriate second course.
Course description: M408C, M408D is our standard firstyear calculus sequence. It is directed at students in the natural and social sciences and at engineering students. The emphasis in this course is on problem solving, not on the presentation of theoretical considerations. While the course necessarily includes some discussion of theoretical notions, it s primary objective is not the production of theoremprovers. M408D contains a treatment of infinite series, and an introduction to vectors and vector calculus in 2space and 3space, including parametric equations, partial derivatives, gradients and multiple integrals. M408C and M408D classes meet three hours per week for lectures and two hours per week for problem sessions.
M408K Differential Calculus
Prerequisite and degree relevance: The minimum required score on the ALEKS placement exam. Only one of the following may be counted: M403K, M408C, M408K, M408N.
Course description: Introduction to the theory and applications of differential calculus of one variable; topics include limits, continuity, differentiation, mean value theorem and applications.
M408L Integral Calculus
Prerequisite and degree relevance: One of M408C, M408N, or M408K, with a grade of at least C. Only one of the following may be counted: M403L, M408C, M408L, M408S.
Course description: Introduction to the theory and applications of integral calculus of one variable; topics include integration, the fundamental theorem of calculus, transcendental functions, sequences and infinite series.
M408M Multivariable Calculus
Prerequisite and degree relevance: One of M408L, M408S, with a grade of at least C. Only one of the following may be counted: M408D, M408M.
Course description: Introduction to the theory and applications of integral calculus of several variables; topics include parametric equations, polar coordinates, vectors, vector calculus, functions of several variabls, partial derivatives, gradients, and multiple integrals.
M408N Differential Calculus
Prerequisite and degree relevance:
The minimum required score on the ALEKS placement exam. Only one of the following may be counted: M403K, M408C, M408K, M308K, M408N.
Course description:
Introduction to the theory and applications of differential calculus of one variable; topics include limits, continuity, differentiation, mean value theorem and applications.
M408R Calculus for Biologists
M408R is a 1semester survey of calculus. As such, it covers more ground than the first semester of a 2sememster sequence, but with a very different emphasis. We will cover Chapters 16 of Callahan, and part of Chapter 11.
Goals for the class:
a) Learning the key ideas of calculus, which I call the six pillars.
1. Close is good enough (limits)
2. Track the changes (derivatives)
3. What goes up has to stop before is can come down (max/min)
4. The whole is the sum of the parts (integrals)
5. The whole change is the sum of the partial changes (fundamental theorem)
6. One variable at a time.
b) Learning how to analyze a scientific situation and model it mathematically.
c) Learning to analyze a mathematical model using calculus.
d) Learning how to apply the results of the model back into the real world.
e) Learning enough formulas and calculational methods to make the other goals possible. There are three questions associated with every mathematical idea in existence:
1. What is it?
2. How do you compute it?
3. What is it good for?
Compared to most math classes, we're going to spend a lot more time on the first and third questions, but we still need to address the second. You can't spend all your time looking at the big picture! You need some practice sweating the details, too
M408S Integral Calculus
Prerequisite and degree relevance:
One of M408C, M408K, M408N, with a grade of at least C. Only one of the following may be counted: M403L, M408L, M408C, M408S.
Course description:
Introduction to the theory and applications of integral calculus of one variable; topics include integration, the fundamental theorem of calculus, transcendental functions, sequences and infinite series.
M110 Conference Course
Prerequisite and degree relevance: Consent of instructor. May be repeated for credit when the topics vary. Individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. A signed permit form must be submitted before registration. The forms may be obtained in the mathematics office, RLM 8.100. Some sections are offered on a pass/fail basis only; these are identified in the Course Schedule.
Course description: Supervised study in mathematics, with hours to be arranged.
M210 Conference Course
Prerequisite and degree relevance: Consent of instructor. May be repeated for credit when the topics vary. Individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. A signed permit form must be submitted before registration. The forms may be obtained in the mathematics office, RLM 8.100. Some sections are offered on a pass/fail basis only; these areidentified in the Course Schedule.
Course description: Supervised study in mathematics, with hours to be arranged.
M210E Emerging Scholars Seminar
Prerequisite and degree relevance: Restricted to students in the Emerging Scholars Program. Three twohour laboratory sessions for one semester. May be repeated for credit. Offered on a pass/fail basis only.
Course Description: Supplemental problemsolving laboratory for precalculus, calculus, or advanced calculus courses, for students in the Emerging Scholars Program.
M210T Topics in Math
Prerequisite and degree relevance: Two lecture hours a week for one semester. May be repeated for credit when topics vary.
Course Description: Special topics in mathematics, as determined by the Instructor.
M310 Conference Course
Prerequisite and degree relevance: Consent of instructor. May be repeated for credit when the topics vary. Individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. A signed permit form must be submitted before registration. The forms may be obtained in the mathematics office, RLM 8.100. Some sections are offered on a pass/fail basis only; these areidentified in the Course Schedule.
Course description: Supervised study in mathematics, with hours to be arranged.
M310P Modern Mathematics: Plan II
Prerequisite and degree relevance: Restricted to Plan II students.
Course description: Significant developments in modern mathematics. Topics may include fractals, the fourth dimension statistics and society, and techniques for thinking about quantitative problms.
M310T Honors Applied
The M310T honors courses are designed to prepare students for research in mathematics. They are taught by young faculty who are leaders in their field of research. It aims to both show the excitement of research, as well as provide students with tools necessary to succeed in mathematical research.
The Applied course, taught by Prof. Kui Ren, focuses on techniques of applied mathematics; you can link to a recent syllabus.
Students apply for this program by contacting Grace Choy, gchoy@mail.utexas.edu.
M310T Honors Pure
The M310T honors courses are designed to prepare students for research in mathematics. They are aught by young faculty who are leaders in their field of research. It aims to both show the excitement of research, as well as provide students with tools necessary to succeed in mathematical research.
The Pure. taught by Prof. Tim Perutz, covers topics in pure mathematics. A recent syllabus is available.
Students apply for this program by contacting Grace Choy, gchoy@mail.utexas.edu.
M315C Functions and Modeling
Prerequisite and degree relevance: Enrollment in a teaching program or consent of the instructor.
Course description: Students will engage in labbased activities designed to stengthen and expand knowledge of the topics in secondary mathematics, focusing especially on topics from precalculus and the transition to calculus. Students will explore a variety of contexts that can be modeled using families of functions, including linear, exponential, polynomial and trigonometric functions. Topics involving conic sections, parametric equations and polar equations will be included. Explorations will involve the use of multiple representations, transformations, data analysis techniques(such as curve fitting) and interconnections among geometry, probability and algebra. Most labs will include signifcant use of various technologies, including computers, calculators and multimedia materials. The use of quantitative approaches (for example to rate of change, limits and accumulation) and building relationships between discrete and continuous reasoning will be recurrent themes.
M316 Elementary Statistical Methods
Prerequisite and degree relevance: A minimal required score on the ALEKS placement exam. May not be counted toward the major requirement for the Bachelor of Arts with a major in mathematics or toward the Bachelor Science in Mathematics. Only one of the following may be counted: Mathematics 316, 306K (Topic 1: Applications of Probability Theory), 362K, Statistics and Scientific Computation 303, 304, 305, 306.
Course Description: The course is designed to provide the student with a clear understanding of basic statistical techniques. Topics include: Descriptive statistics  measures of central tendency, measures of dispersion. Probability  basic rules of probability, joint and marginal probabilities. Statistical modeling  normal and binomial distributions, sampling distributions. Inferential statistics  estimating means and proportions, hypothesis tests, regression and correlation.
The assignments in the course are usually applied problems. Computer work is also assigned, however no prior computer course or computer use is required.
M316K Foundations of Arithmetic
Prerequisite and degree relevance: M316K is intended for prospective elementary teachers and other students whose degree programs require it; it treats basic concepts of mathematical thought. The prerequisite is Mathematics 302, 303D, 305G, or 316 with a grade of at least C. May not be included in the major requirement for the Bachelor of Arts or Bachelor of Science degrees with a major in mathematics. Credit for Mathematics 316K may not be earned after the student has received credit for any calculus course with a grade of C or better, unless the student is registered in the College of Education or a middle grades teacher certification program.
Course description: An analysis, from an advanced perspective, of the concepts and algorithms of arithmetic, including sets; numbers; numeration systems; definitions, properties, and algorithms of arithmetic operations; and percents, ratios, and proportions. Problem solving is stressed.
M316L Geometry, Statistics, Probability
Prerequisite and degree relevance: The prerequisite is M316K with a grade of at least C. May not be included in the major requirement for the Bachelor of Arts or Bachelor of Science degrees with a major in mathematics. Credit for Mathematics 316L may not be earned after a student has received credit for any calculus course with a grade of C or better, unless the student is registered in the College of Education or a middle grades teacher certification program.
Course description: An analysis, from an advanced perspective, of the basic concepts and methods of geometry, statistics, and probability, including representation and analysis of data; discrete probability, random events, and conditional probability; measurement; and geometry as approached through similarity and congruence, through coordinates, and through transformations. Problem solving is stressed.
M410 Conference Course
Prerequisite and degree relevance: Consent of instructor. May be repeated for credit when the topics vary. Individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. A signed permit form must be submitted before registration. The forms may be obtained in the mathematics office, RLM 8.100, or in the Math, physics and Astrnomy Advising office, RLM 4.101. Some sections are offered on a pass/fail basis only; these are identified in the Course Schedule.
Course description: Supervised study in mathematics, with hours to be arranged.
Upper Division Courses
M325KH Honors Discrete Mathematics
Taught in the Spring semester by Prof. Michael Starbird, the course covers topics in mathemtics rarely seen in undergraduate course, such as countability.
M325K Discrete Mathematics
Prerequisite: One of M408D, M408L, or M408S, with a grade of at least C, or consent of instructor. This is a first course that emphasizes understanding and creating proofs. Therefore, it provides a transition from the problemsolving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow adequate time for students to develop theoremproving skills.
Course Topics:
 Fundamentals of logic
 Elementary Number theory
 Sequences and Mathematical Induction
 Set Theory
M328KH Honors Number Theory
Taught in the fall by Professor James Vick, the course will move at a more rapid pace than M328K, in order to explore some advanced topics. Every Friday, students will also work on the development and presentation of proofs, during "Moore Method Fridays."
Students apply for this course by contacting the Math Advising Center in RLM 4.101, during Fall registration.
M328K Introduction to Number Theory
Prerequisite and degree relevance: Required: M325K or M341, with a grade of at least C. This is a first course that emphasizes understanding and creating proofs; therefore, it must provide a transition from the problemsolving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to concentrate on developing the students theoremproving skills.
Course description: The following subjects are included:
 Divisibility: divisibility of integers, prime numbers and the fundamental theorem of arithmetic.
 Congruences: including linear congruences, the Chinese remainder theorem, Eulers jfunction, and polynomial congruences, primitive roots.
The following topics may also be covered, the exact choice will depend on the text and the taste of the instructor.
 Diophantine equations: (equations to be solved in integers), sums of squares, Pythagorean triples.
 Number theoretic functions: the Mobius Inversion formula, estimating and partial sums z(x) of other number theoretic functions.
 Approximation of real numbers by rationals: Dirichlets theorem, continued fractions, Pells equation, Liousvilles theorem, algebraic and transcendental numbers, the transcendence of e and/or z.
M329F Theory of Interest
Prerequisite and degree relevance: M408D, M308L, M408L, or M408S with a grade of at least C, or consent of instructor. This course covers the interest theory portion of the SOA/CAS Financial Mathematics exam (FM/2); this should be about 7580% of the material on this professional exam, with the balance of the exam testing knowledge of elementary financial derivatives. Topics include nominal and effective interest and discount rates, general accumulation functions and force of interest, yield rates, annuities including those with nonlevel payment patterns, amortization of loans, sinking funds, bonds, duration, and immunization.
M329W Cooperative Mathematics
Prerequisite and degree relevance: Students must first apply through the College of Natural Sciences Career services Office. Applicants must have completed one of the following M408D, M408S, or M408L, as well as earned a grade of at least C in two of the following courses: M325K, M341, M427K, M362K, M378K; and consent of the undergraduate adviser. Forty laboratory hours a week for one semester.
Course Description: This course covers the work period of mathematics students in the Cooperative Education program, which provides supervised work experience by arrangement with the employer and the supervising instructor. The student must repeat the course each work period and must take it twice to receive credit toward the degree; at least one of these registrations must be during a longsession semester. No more than three semester hours may be counted toward the major requirement; no more than six semester hours may be counted toward the degree. The students first registration must be on a pass/fail basis.
M427J Differential Equations with Linear Algebra
Prerequisite and degree relevance: The prerequisite is one of 408D, M408L or M408S, with a grade of at least C.
Course description: M427J is a basic course in ordinary and partial differential equations, emphasizing the connection with linear algebra and systems of equation. Other subject matter will include: introduction to vector spaces, linear operators and eigenvalues, systems of linear differential equations, introduction to partial differential equations and Fourier series. The course meets three times a week for lecture and twice more for problem sessions. Five sessions a week for one semester. Mathematics 427J and 427K may not both be counted
M427KH Honors Advanced Calculus For Applications I
This course is taught in the Spring semester; it provides greater depth than the usual differential equations course, has less emphasis on computation, more conceptual material. Each professor chooses additional topics, based on their research interests.
M427K Advanced Calculus for Applications I
Prerequisite and degree relevance: The prerequisite is one of 408D, M408L or M408S, with a grade of at least C.
Course description: M427K is a basic course in ordinary and partial differential equations, with Fourier series. It should be taken before most other upper division, applied mathematics courses. The course meets three times a week for lecture and twice more for problem sessions. Geared to the audience primarily consisting of engineering and science students, the course aims to teach the basic techniques for solving differential equations which arise in applications. The approach is problemoriented and not particularly theoretical. Most of the time is devoted to first and second order ordinary differential equations with an introduction to Fourier series and partial differential equations at the end. Depending on the instructor, some time may be spent on applications, Laplace transformations, or numerical methods. Five sessions a week for one semester.
M427LAP Honors Advanced Calculus for Applications II
Course description: Rather than cover material of a standard claculus class, this course goes directly into an upperdivision treatment of multivariable calculus, and covers this topic from a more advanced perspective.In addition, M427LAP covers additional material, such as theorems of Stokes, Green, and Gauss, that are not found in general calculus. This material prepares studetns for advanced engineering (eg., fluid dynamics, and aerodynamics), physics (eg., electricity and magnetism, quantum field theory), and mathematics (eg., differential topology, Lie groups, etc.). Together with AP BC credit, this course meets the calculus requirement in all Colleges, and gives the student upperdivision math hours.
M427L Advanced Calculus for Applications II
Prerequisite and degree relevance: The prerequisite is M 408D, M 408L, or M 408S with a grade of at least C.
Course description: Topics include matrices, elements of vector analysis and calculus functions of several variables, including gradient, divergence, and curl of a vector field, multiple integrals and chain rules, length and area, line and surface integrals, Greens theorem in the plane and space. If time permits, topics in complex analysis may be included. This course has three lectures and two problem sessions each week. Five sessions a week for one semester.
M139S Seminar on Actuarial Practice
Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.
M333L Structure of Modern Geometry
Prerequisite and degree relevance: The prerequisite is one of 408D, M408L, M408S or upperdivision standing and consent of the instructor. These requirements are set to ensure the mathematical maturity, rather than for content knowledge. They may be waived by the instructor in some cases, most notably a specialization in mathematics. M333L is required for students seeking certification to teach secondary school mathematics.
Course description: The course is designed to familiarize prospective mathematics teachers with the geometrical concepts which relate to two and three dimensional geometry and the mathematical techniques used in the study of geometry. The emphasis is both on the development of understanding of the concepts and the ability to use the concepts in proving theorems. The course includes study of axiom systems, transformational geometry, and an introduction to nonEuclidean geometries, supplemented by other topics as determined by the instructor. While the course is primarily designed for teachers, its content and approach may be of interest to other students of mathematics.
M339J Probability Models with Actuarial Applications
Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.
M339U Actuarial Contingent Payments I
Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.
M339V Actuarial Contingent Payments II
Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.
M339W Financial Mathematics for Actuarial Applications
Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.
M340L Matrices and Matrix Calculations
Prerequisite and degree relevance: The prerequisite is one semester of calculus (either M408C, M408K, or M408N) with grade of at least C, or consent of instructor. Only one of M341 and M340L may be counted.
Course description: The goal of M340L is to present the many uses of matrices and the many techniques and concepts needed in such uses. The emphasis is on concrete concepts and understanding and using techniques, rather than on learning proofs and abstractions. The course is designed for applicationsoriented students such as those in the natural and social sciences, engineering, and business. Topics might include matrix operations, systems of linear equations, introductory vectorspace concepts (e.g., linear dependence and independence, basis, dimension), determinants, introductory concepts of eigensystems, introductory linear programming, and least square problems.
M341H Honors Linear Algebra
Taught in the Spring semester by Prof. Ronny Hadani, the course covers linear algebra from an advanced perspective. You can view the Spring 2009 syllabus here.
Students apply for this course by contacting the Math Advising Center in RLM 4.101, during Fall registration.
M341 Linear Algebra and Matrix Theory
Formerly M311.
Prerequisite and degree relevance: The prerequisite is one of M408D, M408L, M408S or the equivalent, with a grade of at least C, or consent of instructor. (Credit may not be received for both M341 and M340L. Majors with a 'math' advising code must register for M341 rather than for M340L; majors without a 'math' advising code must register for M340L. Math majors must make a grade of at least C in M341.)
Course description: The emphasis in this course is on understanding the concepts and learning to use the tools of linear algebra and matrices. Some time should be devoted to teaching students to do proofs. The fundamental concepts and tools of the subject covered are:
 Matrices: matrix operations, the rules of matrix algebra, invertible matrices.
 Linear equations: row operations and row equivalence; elementary matrices; solving ystems of linear equations by Gaussian elimination; inverting a matrix with the aid of row operations.
 Vector spaces:vector spaces and subspaces; linear independence and span of a set of vectors, basis and dimension; the standard bases for common vector spaces.
 Inner product spaces: CauchySchwarz inequality, orthonormal bases, the GrammSchmidt procedure, orthogonal complement of a subspace, orthogonal projection.
 Linear Transformations: kernel and range of a linear transformation, the Rank Nullity Theorem, linear transformations and matrices, change of basis, similarity of matrices.
 Determinants: the definition and basic properties of determinants, Cramers rule.
 Eigenvalues: eigenvalues and eigenvectors, diagonalizability of a real symmetric matrix, canonical forms.
M343K Introduction to Algebraic Structures
Prerequisite and degree relevance: Either consent of Mathematics Advisor, or two of the following courses, with a grade of at least C in each: M341, 328K, 325K (Philosophy 313K may be substituted for M325K). This course is designed to provide additional exposure to abstract rigorous mathematics on an introductory level. Students who demonstrate superior performance in M311 or M341 should take M373K instead of 343K. Those students whose performance in M311 or M341 is average should take M343K before taking M373K. Credit for M343K can NOT be earned after a student has received credit for M373K with a grade of at least C.
Course description: Elementary properties of groups and rings, including symmetric groups, properties of the integers, polynomial rings, elementary field theory.
M343L Applied Number Theory
Prerequisite and degree relevance: Mathematics 328K or 343K with a grade of at least C.
Topics: Basic properties of integers, including properties of prime numbers, congruences, and primitive roots. introduction to finite fields and their vector spaces with applications to encryption systems and coding theory. Three lecture hours a week for one semester.
M343M ErrorCorrecting Codes
Prerequisite and degree relevance: Mathematics 328K or 341, with a grade of at least C.
Course description: Introduction to applications of algebra and number theory to errorcorrecting codes, including finite fields, errorcorrecting codes, vector spaces over finite fields, Hamming norm, coding, and decoding
M344K Intermediate Symbolic logic
Prerequisite and degree relevance: Philosophy 313, 313K, or 313Q.
Course description: Same as Philosophy 344K. A secondsemester course in symbolic logic: formal syntax and semantics, basic metatheory (soundness, completeness, compactness, and LöwenheimSkolem theorems), and further topics in logic.
M346 Applied Linear Algebra
Prerequisite and degree relevance: The prereqiusite is M341 (or M311) or M340L, with a grade of C or better, or consent of the instructor.
Course description: Emphasis on diagonalization of linear operators and applications to discrete and continuous dynamical systems and ordinary differential equations. Other subjects include inner products and orthogonality, normal mode expansions, vibrating strings and the wave equation, and Fourier series.
M348 Scientific Computation in Numerical Analysis
Prerequisite and degree relevance: Computer Science 303E or 307, and either Mathematics 341 or 340L, with a grade of at least C.
Course description: Introduction to mathematical properties of numerical methods and their applications in computational science and engineering. Introduction to objectoriented programming in an advanced language. Study and use of numerical methods for solutions of linear systems of equations, nonlinear leastsquares data fitting, numerical integration of multidimensional, nonlinear equations and systems of initial value ordinary differential equations.
M349P Actuarial Statistical Estimates I
Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.
M349R Applied Regression and Time Series
Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.
M358K Applied Statistics
Prerequisite and degree relevance: The prerequisite is M362K with grade of C or better. This course is intended for students planning to teach secondary mathematics, students working for a BA in mathematics, and students in the natural sciences. Students preparing for graduate work in mathematical statistics should take M378K instead of or after taking this course.
Course Description: This is a first course in applied statistics, building on previous experience with probability. Emphasis will be on development of statistical thinking and working with real data. Topics include: exploratory data analysis, regression and correlation, introduction to planning and conducting surveys and experiments, sampling distributions, confidence intervals (for proportions, means, differences between proportions, differences between two means paired and unpaired), and tests of significance (for proportions, means, differences between proportions, differences between means; chisquared test; one and two sample t procedures; inference for slope of least squares line; analysis of variance). Students will be expected to apply the statistical ideas they learn in one or more projects. The statistical software Minitab will be used extensively in the course.
M326K Foundations of Number Systems
Prerequisite and degree relevance: One of 408D, M408S, 408L with a grade of at least C.Restricted to students in a teacher preparation program or who have consent of instructor.
Course description: Study of numberrelated topics in middle grade and secondary school mathematics. Topics include place value; meanings of arithmetic operations; analysis of computation methods; historical development of number concepts and notation; and rational, irrational, algebraic, transcendental, and complex numbers. Emphasis is on communicating mathematics, developing pedagogical understanding of concepts and notation, and using both informal reasoning and proof. Three lecture hours a week for one semester.
M360M Mathematics as Problem Solving
Prerequisite and degree relevance: One of M408D, M408S, M408L, with a grade of at least C, and written consent of the instructor.
Course description: Discussion of heuristics, strategies, and methods of problem solving, and extensive practice in both group and individual problm solving. Communicating mathematics, reasoning and connections amngst topics in mathematics are emphasized.
M361 Functions of A Complex Variable
Prerequisite and degree relevance: The prerequisite is M427K or M427L with a grade of at least C, or consent of the instructor.
Course description: M361 consists of a study of the properties of complex analytic functions. Students are mainly from physics and engineering, with some mathematics majors and joint majors. Representative topics are Cauchys integral theorem and formula, Laurent expansions, residue theory and the calculation of definite integrals, analytic continuation, and asymptotic expansions. Rigorous proofs are given for most results, with the intent to provide the student with a reliable grasp of the results and techniques.
M361K Intro to Real Analysis
Prerequisite and degree relevance: Either consent of faculty undergraduate advisor, or two of the following courses, with grade of at least C in each: M341, 328K, 325K (Philosophy 313K may be substituted for M325K), with a grade of at least C. May not be counted by students with credit for M365K with a grade of C or better.
Course description: This is a rigorous treatment of the real number system, of real sequences, and of limits, continuity, derivatives, and integrals of realvalued functions of one real variable.
M362K Probability I
Prerequisite and degree relevance: One of M408D, M408S, 408L, with a grade of at least C.
Course description: An introductory course in the mathematical theory of probability, fundamental to further work in probability and statistics, includes basic probability properties, conditional probability and independence, various discrete and continuous random variables, expectation and variance, central limit theorem, and joint probability distributions.
M362M Introduction to Stochastic Processes
Prerequisite and degree relevance: Mathematics 362K with a grade of at least C.
Course description: Introduction to Markov chains, birth and death processes, and other topics.
M364K Vector and Tensor Analysis
Prerequisite and degree relevance: The prerequisite is M427K or M427L, with a grade of at least C.
Course description: Topics include vector algebra and calculus, integral theorems, general coordinates, invariance, tensor analysis, and perhaps an introduction to differential geometry. It is anticipated that a significant percentage of students will be physics majors.
M365C Real Analysis I
Prerequisite and degree relevance: Either consent of Mathematics Advisor, or two of M341, 328K, 325K (Philosophy 313K may be substituted for M325K), with a grade of at least C. Students who receive a grade of C in M325K or M328K are advised to take M361K before attempting M365C. Students who have received a grade of C or better in Mathematics 365C may not take Mathematics 361K.
Course description: This course is an introduction to Analysis. Analysis, together with Algebra and Topology, form the central core of modern mathematics. Beginning with the notion of limit from calculus and continuing with ideas about convergence and the concept of function that arose with the description of heat flow using Fourier series, analysis is primarily concerned with infinite processes, the study of spaces and their geometry where these processes act and the application of differential and integral to problems that arise in geometry, pde, physics and probability.
A rigorous treatment of the real number system, Euclidean spaces, metric spaces, continuity of functions in metric spaces, differentiation and Riemann integration of realvalued functions of one real variable, and uniform convergence of sequences and series of functions.
M365D Real Analysis II
Prerequisite and degree relevance: Mathematics 365C with a grade of at least C.
Course description: Recommended for students planning to undertake graduate work in mathematics. A rigorous treatment of selected topics in real analysis, such as Lebesgue integration, or multivariate integration and differential forms.
Topics:
 Geometry and topology of R^{n}
 Spaces of continuous functions
 Lebesgue integration in onevariable
 L^{p} spaces, Hilbert spaces, linear operators
Text: Davidson & Donsig: Real Analysis and Real Applications PrenticeHall ISBN 0130416479
M365G Curves and Surfaces
Prerequisite and degree relevance: Credit with a grade of at least C or registration for Mathematics 365C. Students should know multivariable calculus and a little linear algebra.
Course Description: Calculus applied to curves and surfaces in three dimensions: graphs and level sets, tangent spaces, vector fields, surfaces, orientation, the Gauss map, geodesics, parallel transport, the second fundamental form and the Weingarten map, length and curvature of plane curves, curvature of surfaces, the exponential map, and the GaussBonnet Theorem.
Textbook: Elementary Topics in Differential Geometry, John A. Thorpe, SpringerVerlag, New York, 1979. (ISBN 0387903577)
INTRODUCTION: Differential geometry is a rich and active area of research in pure mathematics. It also provides powerful tools for disciplines like general relativity and other branches of mathematical physics as well as for applications in engineering and computer graphics. This course will introduce the basic language and methods of differential geometry by studying the geometry of ndimensional hypersurfaces in (n+1)dimensional Euclidean space. The course would be excellent preparation for graduate courses in Differential Topology or Riemannian Geometry, as well as for further study in applications like those listed above.
M367K Topology I
Prerequisite and degree relevance: One of M361K or 365C with a grade of C or better, or consent of instructor.
Course description: An introduction to topology, including sets, functions, cardinal numbers, and the topology of metric spaces
This is a course that emphasizes understanding and creating proofs. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to concentrate on developing the students theorem proving skills. The syllabus below is a typical syllabus. Other collections of topics in topology are equally appropriate. For example, some instructors prefer to restrict themselves to the topology of the real line or metric space topology.
 Cardinality: 11 correspondance, countability, and uncountability.
 Definitions of topological space: basis, subbasis, metric space.
 Countability properties: dense sets, countable basis, local basis.
 Separation properties: Hausdorff, regular, normal.
 Covering properties: compact, countably compact, Lindelof.
 Continuity and homeomorphisms: properties preserved by continuous functions, Urysohns Lemma, Tietze Extension Theorem.
 Connectedness: definition, examples, invariance under continuous functions.
Notes containing definitions, theorem statements, and examples have been developed for this course and are available. The notes include some topics beyond those listed above.
M367L Topology II
Prerequisite and degree relevance: Mathematics 367K with a grade of at least C or consent of instructor.
Course description: Various topics in topology, primarily of a geometric nature.
M368K Numerical Methods for Applications
Prerequisite and degree relevance: Mathematics 348 with a grade of at least C. Only one of the following may be counted: Computer Sciences 367, Mathematics 368K, Physics 329.
Course description: Continuation of Mathematics 348. Topics include splines, orthogonal polynomials and smoothing of data, iterative solution of systems of linear equations, approximation of eigenvalues, twopointboundary value problems, numerical approximation of partial differential equations, signal processing, optimization, and Monte Carlo methods.
M175 Conference Course
Prerequisite and degree relevance: Vary with the topic, and are given in the course schedule.
Course description: It is the responsibility of the student to select a professor and make individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. Signed permit form must be submitted. The forms may be obtained in the mathematics office,RLM 8.100.
M275 Conference Course
Prerequisite and degree relevance: Vary with the topic, and are given in the course schedule.
Course description: It is the responsibility of the student to select a professor and make individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. Signed permit form must be submitted. The forms may be obtained in the mathematics office,RLM 8.100.
M372K PDE and Applications
Prerequisite and degree relevance: M427K, with a grade of at least C. One of M361K or M365C is also recommended.
Course description: Partial differential equations arise as basic models of flows, diffusion, dispersion, and vibrations. Topics include first and second order partial differential equations and classification, particularly the wave, diffusion, and potential equations, their origins in applications and properties of solutions, characteristics, maximum principles, Greens functions, eigenvalue problems, and Fourier expansion methods.
M373K Algebraic Structures I
Prerequisite and degree relevance: Consent of the faculty undergraduate adviser, or two of the following courses with a grade of at least C in each: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. Students who receive a grade of C in one of the prerequisite courses are advised to take Mathematics 343K before attempting 373K. Students planning to take Mathematics 365C and 373K concurrently should consult a mathematics adviser.
Course description: A study of groups, rings, and fields, including structure theory of finite groups, isomorphism theorems, polynomial rings, and principal ideal domains.
M373K is a rigorous course in pure mathematics. The syllabus for the course includes topics in the theory of groups and rings. The study of group theory includes normal subgroups, quotient groups, homomorphisms, permutation groups, the Sylow theorems, and the structure theorem for finite abelian groups. The topics in ring theory include ideals, quotient rings, the quotient field of an integral domain, Euclidean rings, and polynomial rings. This course is generally viewed (along with 365C) as the most difficult of the required courses for a mathematics degree. Students are expected to produce logically sound proofs and solutions to challenging problems.
M373L Algebraic Structures II
Prerequisite and degree relevance: The prerequisite is M373K. M373L is strongly recommended for undergraduates contemplating graduate study in mathematics.
Course description: Topics from vector spaces and modules, including direct sum decompositions, dual spaces, canonical forms, and multilinear algebra.
M373L is a continuation of M373K, covering a selection of topics in algebra chosen from field theory and linear algebra. Emphasis is on understanding theorems and proofs.
M374 Fourier and Laplace Transforms
Prerequisite and degree relevance: Mathematics 427K with a grade of at least C.
Course description: The course covers operational properties and applications of Laplace transforms and covers some properties of Fourier transforms.
M374G Linear Regression Analysis
Prerequisite and degree relevance: Mathematics 358K or 378K with grade of at least C, Mathematics 341 or 340L, and consent of instructor.
Course description: Fitting of linear models to data by the method of least squares, choosing best subsets of predictors, and related materials.
M375 Conference Course
Prerequisite and degree relevance: Vary with the topic, and are given in the course schedule.
Course description: It is the responsibility of the student to select a professor and make individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. Signed permit form must be submitted. The forms may be obtained in the mathematics office,RLM 8.100.
M375C Conference Course
Prerequisite and degree relevance: Vary with the topic, and are given in the course schedule.
Course description: It is the responsibility of the student to select a professor and make individual arrangements with the professor regarding the meeting, time, and course content. One, two, three, or four meetings a week for one semester. Signed permit form must be submitted. The forms may be obtained in the mathematics office, RLM 8.100.
M375T Functions and modeling
 M375T Functions and Modeling runs concurrent with M315C Foundations, Functions, and Regression Models. This course is a mathematics research and report course that is restricted to Mathematics Education graduate students.
M375T Analysis on Manifolds
M375T Intro Grad Std in Math
 This is a mathematics research and report topics course supervised by a Mathematics Faculty member. Students work directly with a faculty member, identifying a research topic focused on effective teaching strategies. Faculty member approval, in writing, is required.
M376C Methods of Applied Mathematics
Prerequisite and degree relevance: Computer Sciences 303E or 307, Mathematics 427K, and Mathematics 341 or 340L, with a grade of at least C in each.
Course description: Variational methods and related concepts from classical and modern applied mathematics. Models of conduction and vibration that lead to systems of linear equations and ordinary differential equations, eigenvalue problems, initial and boundary value probl ems for partial differential equations. Topics may include a selection from diagonalization of matrices, eigenfunctions and minimization, asymptotics of eigenvalues, separation of variables, generalized solutions, and approximation methods. May be repeated for credit when the topics vary.
M378K Intro to Mathematical Statistics
Prerequisite and degree relevance: Mathematics 362K with a grade of at least C. Students taking this course are usually majoring in mathematics, actuarial science, or one of the natural sciences. M362K, 358K, and 378K form the core sequence for students in statistics.
Course description: Sampling distributions of statistics, estimation of parameters (confidence intervals, method of moments, maximum likelihood, comparison of estimators using mean square error and efficiency, sufficient statistics), hypothesis tests (pvalues, power, likelihood ratio tests), and other topics.
This is the first course in mathematical statistics and is taught from a classical viewpoint. The major topics are: estimation of parameters, including maximum likelihood estimation; sufficient statistics, and confidence intervals; testing of hypotheses including likelihood ratio tests and the Neyman Pearson theory; the distributions and other properties of some statistics that occur in sampling from normal populations such as the gamma, beta, chisquared, Students t, and F distributions; and fitting straight lines. The course is designed to give students some insight into the theory behind the standard statistical procedures and also to prepare continuing students for the gradu ate courses. Within the limits of the prerequisites, students are expected to reproduce and apply the theoretical results; they are also expected to be able to carry out some standard statistical procedures.
M379H Honors Tutorial Course
Prerequisite and degree of relevance: Prerequisite: Admission to the Mathematics Honors Program, Mathematics 365C, 367K, 373K, or 374G with a grade of A, and another of these courses with a grade of at least B; and consent of the honors adviser.
Course description: Directed reading, research, and/or projects, under the supervision of a faculty member, leading to an honors thesis.
M474M Mathematical Modeling in Science and Engineering
Prerequisite and degree relevance: Mathematics 427K, and 341 or 340L, with a grade of at least C in each; and some basic programming skills.
Course description: Tools for studying differential equations and optimization problems that arise in the engineering an physical sciences. Includes: dimensional analysis and scaling, regular and singular perturbatin methods, optimization and calculus of variations, and stability.
Graduate Courses
FALL 2019 GRADUATE COURSES
M 380C (Iushchenko) Algebra
It is assumed that students know the basic material from an undergraduate course in linear algebra and an undergraduate abstract algebra course. The first part of the Prelim examination will cover sections 1 and 2 below. The second part of the Prelim examination will deal with section 3 below.
1. Groups: Finite groups, including Sylow theorems, pgroups, direct products and sums, semidirect products, permutation groups, simple groups, finite Abelian groups; infinite groups, including normal and composition series, solvable and nilpotent groups, JordanHolder theorem, free groups.
References: Goldhaber Ehrlich, Ch. I except 14; Hungerford, Ch. I, II; Rotman, Ch. IVI, VII (first three sections).
2. Rings and modules: Unique factorization domains, principal ideal domains, modules over principal ideal domains (including finitely generated Abelian groups), canonical forms of matrices (including Jordan form and rational canonical form), free and projective modules, tensor products, exact sequences, WedderburnArtin theorem, Noetherian rings, Hilbert basis theorem.
References: Goldhaber Ehrlich, Ch. II, III 1,2,4, IV, VII, VIII; Hungerford, Ch. III except 4,6, IV 1,2,3,5,6, VIII 1,4,6.
3. Fields: Algebraic and transcendental extensions, separable extensions, Galois theory of finite extensions, finite fields, cyclotomic fields, solvability by radicals.
References: Goldhaber Ehrlich, Ch. V except 6; Hungerford, Ch. V, VI; Kaplansky, Part I.
References:
Goldhaber Ehrlich, Algebra, reprint with corrections, Krieger, 1980.
Hungerford, Algebra, reprint with corrections, Springer, 1989.
Isaacs, Algebra, a Graduate Course, Wadsworth, 1994.
Kaplansky, Fields and Rings, 2nd Edition, University of Chicago Press, 1972.
Rotman, An Introduction to the Theory of Groups, 4th Edition, W.C. Brown, 1995.
M 381C (Maggi) Real Analysis
The objective of this syllabus is to aid students in attaining a broad understanding of analysis techniques that are the basic stepping stones to contemporary research. The prelim exam normally consists of eight to ten problems, and the topics listed below should provide useful guidelines and strategy for their solution. It is assumed that students are familiar with the subject matter of the undergraduate analysis courses M365C and M361. The first part of the Prelim examination will cover Real Analysis. The second part of the prelim examination will cover Complex Analysis.
1. Measure Theory and the Lebesgue Integral
Basic properties of Lebesgue measure and the Lebesgue integral on R^{n} (see [5], Ch. 14) and general measure and integration theory in an abstract measure space (see [5], Ch. 1112; and especially [6], Ch. 12). L^{p} spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in L^{p}norm and L^{p}L^{q} duality; integration in product spaces (see [6], Ch. 8) and convolution on R^{n}; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.
2. Holomorphic Functions and Contour Integration
Basic properties of analytic functions of one complex variable (see [1], Ch. 45; [2], Ch. 47; [4], Ch. 48; or [6], Ch. 1012 and 15). Integration over paths, the local and global forms of Cauchy's Theorem, winding number and residue theorem, harmonic functions, Schwarz's Lemma and the Maximum Modulus theorem, isolated singularites, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.
3. Differentiation
The relationship between differentiation and the Lebesgue integral on a real interval (see [5], Ch. 5), derivatives of measures (see [6], Ch. 5), absolutely continuous functions and absolute continuity between measures, functions of bounded variation.
4. Specific Important Theorems
Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, RadonNikodym theorem, FubiniTonelli theorems about product measures and integration on product spaces, Cauchy's theorem and integral formulas, Maximum Modulus theorem, Rouche's theorem, Residue theorem, and Fundamental Theorem of Calculus for Lebesgue Integrals. Students should be familiar with Minkowski's Inequality, Holder's Inequality, Jensen's Inequality, and Bessel's Inequality.
References:
1. L. Ahlfors, Complex Analysis, McGrawHill, New York, 1979.
2. J.B. Conway, Functions of One complex Variable, second edition, SpringerVerlag, New York, 1978.
3. G.B. Folland, Real Analysis, second edition, John Wiley, New York, 1999.
4. B. Palka, An Introduction to Complex Function Theory, second printing, SpringerVerlag, New York, 1995.
5. H.L. Royden, Real Analysis, Macmillan, New York, 1988.
6. W. Rudin, Real and Complex Analysis, third edition, McGrawHill, New York, 1987.
7. R. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977.
M 382C (Gordon) Algebraic Topology
It is assumed that students have a working knowledge of the equivalent of a one semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K). For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course. The first part of the Prelim examination will deal with Algebraic Topology and the second part will deal with Differential Topology.
Algebraic Topology
1. Manifolds: Identification (quotient) spaces and identification (quotient) maps; topological nmanifolds, including surfaces, S^{n}, RP^{n}, CP^{n}, and lens spaces.
2. Triangulated manifolds: Representation of triangulated, closed 2manifolds as connected sums of tori or projective planes.
3. Fundamental group and covering spaces: Fundamental group, functoriality, retract, deformation retract; Van Kampen's Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, RP^{n}, lens spaces via covering spaces.
4. Simplicial homology: Homology groups, functoriality, topological invariance, MayerVietoris sequence; applications, including Euler characteristic, classification of closed triangulated surfaces via homology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.
References:
Armstrong, Basic Topology, Springer, 1983 (principal text).
Greenberg, Lectures on Algebraic Topology, W.A. Benjamin, 1967.
Massey, Algebraic Topology, an Introduction, 4th corrected printing, Springer, 1977.
Munkres, Elements of Algebraic Topology, AddisonWesley, 1984.
M 383C (Koch) Methods of Applied Mathematics
It is assumed that students are familiar with the subject matter of the undergraduate analysis course M365C (see the Analysis section for a syllabus of that course) and an undergraduate course in linear algebra.
The Applied Math Prelim divides into these six areas. The first three are discussed in M383C and will be covered in the first part of the Prelim examination:
1. Banach spaces:
Normed linear spaces and convexity; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; continuous linear transformations; HahnBanach Extension Theorem; linear functionals, dual and reflexive spaces, and weak convergence; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; Closed Range Theorem; compact sets and AscoliArzelà Theorem; compact operators and the Fredholm alternative.
2. Hilbert spaces: Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and HilbertSchmidt operators; spectral theory for compact, selfadjoint and normal operators; SturmLiouville Theory.
3. Distributions: Seminorms and locally convex spaces; test functions and distributions; calculus with distributions.
These three areas are discussed in M383D and will be covered in the second part of the Prelim examination:
4. The Fourier Transform and Sobolev Spaces: The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for H^{s}.
5. Variational Boundary Value Problems (BVP): Weak solutions to elliptic BVP’s; variational forms; LaxMilgram Theorem; Green’s functions.
6. Differential Calculus in Banach Spaces and Calculus of Variations: The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the EulerLagrange equation.
References:
The first four references cover most of the syllabus for the exam. The other references also cover some topics in the syllabus.
1. C. Carath'eodory, Calculus of Variations and Partial Differential Equations of the First Order, 2nd English Edition, Chelsea, 1982.
2. F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974.
3. M. Reed and B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.
4. R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, available at World Wide Web address http://ejde.math.swt.edu//monotoc.html .
5. A. Avez, Introduction to Functional Analysis, Banach Spaces, and Differential Calculus, Wiley, 1986.
6. L. Debnath and P. Mikusi'nski, Introduction to Hilbert Spaces with Applications, Academic Press, 1990.
7. I.M. Gelfand and S.V. Fomin, Calculus of Variations, PrenticeHall, 1963.
8. E. Kreyszig, Introductory Functional Analysis with Applications, 1978.
9. J.T. Oden and L.F. Demkowicz, Applied Functional Analysis, CRC Press, 1996.
10. W. Rudin, Functional Analysis, McGrawHill, 1991.
11. W. Rudin, Real and Complex Analysis, 3rd Edition, McGrawHill, 1987.
12. K. Yosida, Functional Analysis, SpringerVerlag, 1980.
M 385C (Zitkovic) Theory of Probability
(THE FIRST PART OF THE PRELIM EXAM WILL DEAL WITH THE MATERIAL COVERED IN M385C AND THE SECOND PART OF THE PRELIM EXAM WILL DEAL WITH THE MATERIAL COVERED IN M385D)
1. THEORY OF PROBABILITY I  M385C
 Prerequisites:
 Real Analysis (M365C or equivalent),
 Linear Algebra (M341 or equivalent),
 Probability (M362K or equivalent).
 R. Durrett, Probability: theory and examples, third ed., Duxbury Press, Belmont, CA, 1996. (required)
 D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)
Literature:
 Syllabus:
(Note: all references are to Durrett's book)Foundations of Probability:
 Random variables (Sections 1.1, 1.2): probability spaces, σalgebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, LebesgueStieltjes measures (without proof), random vectors, generation, a.s.convergence
 Expected value (Section 1.3): abstract Lebesgue integration (without proofs), inequalities (Jensen, CauchySchwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), changeofvariables formula,
 Dependence (Section 1.4): independence, pairwise independence, Dynkin's  theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)
 Weak laws of large numbers (Sections 1.5, 1.6): the L^{2} weak law of large numbers, triangular arrays, BorelCantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers
 Central limit theorems (Sections 2.2, 2.3a, 2.3b, 2.3c, 2.4a, 2.9part ): weak convergence of distributions, the continuous mapping theorem, Helly's selection theorem, tightness, characteristic functions, the inversion theorem, continuity theorem, the central limit theorem, multivariate normal distributions
 Conditional expectation (Sections 4.1a, 4.1b): RadonNikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
 Martingales (Sections 4.2, 4.4, 4.5, 4.6part , 4.7): martingale transforms, the optional sampling the orem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp convergence, maxi mum inequalities, L^{2} theory, uniform integrability, backwards martingales and the strong law of large numbers.
M 387C (Engquist) Numerical Analysis: Algebra & Approximations
The Prelim sequence is M387C and M387D. The first part of the Prelim examination will cover algebra and approximation and the second part of the Prelim examination will cover diferential equations.
Principles of discretization of differential equations:
 ODEs: Stability and convergence theory, Stiff problems,Symplectic integrators
 FEM (finite element method) and FDM (finite difference method) for boundary value problems
 FEM for PDEs (main focus on elliptic problems): Basic theory, weak formulations, LaxMilgram theorem, finite element spaces, approximation theory, a priori and a posteriori error estimates, practical algorithms, extensions, mixed methods etc.
 FDM for PDEs (main focus on hyperbolic and parabolic problems): Lax equivalence theorem, Von Neumann and other stability analysis, nonlinear conservation laws, shocks, entropy, practical algorithms
Brief survey of other methods for PDEs:
 FVM, DG, Spectral and particle methods
 Applications: Elasticity (FEM), Fluids (FVM), and Waves (FDM)
 Solution of linear and nonlinear equations
 Solution of integral equations
 Eigenvalues
 Optimization
 Monte Carlo methods
 Fast Fourier, wavelet transforms, approximation theory
 Basic undergraduate numerical methods
 Interpolation, fixed point iterations, Newton's method for root finding
 Direct and iterative methods for solving linear equations
 Quadratures
Recommended texts:
 Dahlquist and Bjorck, Numerical methods. Dover
 Lambert, Numerical methods for ordinary differential systems. Wiley
 Gustafsson, Kreiss, and Oliger, Time dependent problems and difference methods
 Iserles, A first course in the numerical analysis of differential equations, Cambridge
 Claes Johnson, Numerical solution of partial differential equations by the finite element method. Cambridge University Press
M 390C (Ciperiani) Algebraic Number Theory
This will be an introductory course. We will study the ring of integers of a number field: prove the finiteness of the class group, prove Dirichlet's unit theorem, and analyze the decomposition of prime ideals when lifted to a bigger field. We will continue with a brief discussion of local fields and analytic methods in number theory. Finally we will conclude with an introduction to class field theory without proofs.
M 392C (Siebert) Algebraic Geometry
This is a lecture course on aspects of algebraic geometric related to mirror symmetry. Depending on the interest of the participants, the course may cover topics such as toric geometry, GromovWitten theory and virtual classes, localization techniques, DonaldsonThomas invariants, logarithmic geometry and tropicalization, toric degenerations and wall structures, derived categories and their dg enhancements, homological mirror symmetry.
Prerequisites: More specific background will be provided as needed, but some acquaintance with the language of schemes will be assumed.
M 392C (BenZvi) Geometric Representation Thry
This course will provide an introduction to the Geometric Langlands Correspondence from the point of view of Topological Field Theory, as pioneered by Kapustin and Witten.
The Langlands correspondence provides a nonabelian generalization of the Fourier transform. Likewise Sduality in fourdimensional gauge theory provides a nonabelian generalization of electricmagnetic duality. We will explore the parallels between these structures, and more generally how topological gauge theories in low dimensions can be used to capture structures in representation theory. Some familiarity with Lie theory and algebraic geometry is required.
M 393C (Burton) Ergodic Theory Group Actions
This course will be an introduction to ergodic theory, with a focus on geometric flows. We will begin by presenting the mean and pointwise ergodic theorems in abstract measure spaces. We will then discuss ergodicity and equidistribution of geodesic and horocycle flows on surfaces of negative curvature, before moving to a more general study of Ratner theory. If there is time, we will discuss number theoretic consequences of equidistribution results.
M 393C (Stuvard) Geometric Harmonic Maps
This is a topic course on the regularity theory of minimizing harmonic maps, namely maps minimizing the Dirichlet energy under the constraint of taking values in a prescribed Riemannian manifold. After a brief introduction aimed at explaining the relevance of harmonic maps in Physics (liquid crystals, GinzburgLandau equations,...), we will study in details the key tools needed to tackle the regularity theory: monotonicity identities, variational equations, epsilonregularity, analysis of tangent maps. This road will eventually lead us to the proof of the celebrated partial regularity result by R. Schoen and K. Uhlenbeck, which provides a sharp dimension estimate on the size of the set of possible singularities of a minimizing harmonic map. Other topics that may be discussed if time permits include: analysis of stationary harmonic maps, rectifiability of the singular set, harmonic heat flow.
M 393C (Maggi) Partl Differential Equatns I
This is part one of a sequence of two topics courses on Partial Differential Equations.We will focus on the theory of nonlinear elliptic and parabolic PDE, with a particular focus on variational methods.
M 393C (Baccelli) Point Processes & Stochastic Geom
The course will be structured in 3 basic blocks
I. Point Processes,
II. Random Geometric Graphs,
III. Stochastic Geometry.
The instructional objective is that students having completed it be in a position:
• to start mathematical research on the domains in question;
• to use the described methodology in some applications, i.e. to develop new models comingfrom these applications, analyze and solve these models.
I. Point Processes
Point processes are a fundamental object in probability theory, at the same level of generalityand use as e.g. second order stochastic processes. They were used as early as the first half ofthe 19th century (Poisson) and are now ubiquitous in physics, computer science (image analysis,information theory, networks), engineering (electrical, material), life sciences (biology, ecology),earth science (seismology), sociology, etc. The objective of this block is that students master thebasic formalism of point process on Euclidean spaces. The basic notions and tools to be coveredare:
• Point processes as random measures;
• Stationarity of a point process;
• Marks of a point process;
• Moment measures;
• Palm calculus;
• Poisson point processes;
• Generating functionals.
II. Random Graphs
This block will discuss both random graphs in the Erd˝os–Renyi sense and graphs defined on pointprocesses of the Euclidean space. Both types are commonly used in all the domains listed above,and are particular important in theoretical computer science. The main emphasis will be on theuse of the methodology developed in I. to analyze the latter class of random graphs  often referredto as random geometric graphs. The following notions will be covered:
• The Erd˝os–Renyi model;
• Random geometric graphs;
• Discrete and continuum percolation;
• Mass transport;
• Point maps and point shifts;
• Palm probability and mass transport.
III. Stochastic Geometry
Stochastic geometry is focused on the study of random geometric objects of e.g. the Euclideanspace such as random sets, random tessellations. Kolmogorov is credited for having built thefoundations of the field – the Boolean model and the PoissonVoronoi tessellation – for analyzingthe growth of crystals in materials. Nowadays it is also widely used in computer science andelectrical engineering (image analysis, information theory, wireless communications), cosmology,hydrology, ecology, cell biology, to quote a few. The main objective of this block is that thestudents master the definition of the basic objects of stochastic geometry and the computation oftheir law. The basic notions and tools to be covered are:
• The Boolean model, particle processes and random closed sets;
• Processes of flats (line processes and hyperplane processes);
• Random tessellations (Voronoi, Delaunay);
• Shot noise fields.
M 394C (Zariphopoulou) Stochastic Processes I
In this class the following topics will be covered:
 Part I
o Ito integral and stochastic calculus
o Stochastic Differential Equations (SDE)
o SDE and linear partial differential equations
o Applications to boundary value problems
o Applications to optimal stopping
o Introduction to filtering
 Part II
o Stochastic control of controlled diffusion processes
o The HamiltonJacobiBellman equation
o Viscosity solutions
o Introduction to risk sensitive control
o Introduction to singular stochastic control
o Applications in Mathematical Finance and other areas will be also presented.
BACKGROUND: The course will build on material covered in Probability I and Probability II. While these courses are not prerequisites, familiarity with their content is strongly recommended. The students must have taken an advanced course of Real Analysis and/or Probability Theory.
SPRING 2019 GRADUATE COURSES
M 380D (Neitzke) Algebra
It is assumed that students know the basic material from an undergraduate course in linear algebra and an undergraduate abstract algebra course. The first part of the Prelim examination will cover sections 1 and 2 below. The second part of the Prelim examination will deal with section 3 below.
1. Groups: Finite groups, including Sylow theorems, pgroups, direct products and sums, semidirect products, permutation groups, simple groups, finite Abelian groups; infinite groups, including normal and composition series, solvable and nilpotent groups, JordanHolder theorem, free groups.
References: Goldhaber Ehrlich, Ch. I except 14; Hungerford, Ch. I, II; Rotman, Ch. IVI, VII (first three sections).
2. Rings and modules: Unique factorization domains, principal ideal domains, modules over principal ideal domains (including finitely generated Abelian groups), canonical forms of matrices (including Jordan form and rational canonical form), free and projective modules, tensor products, exact sequences, WedderburnArtin theorem, Noetherian rings, Hilbert basis theorem.
References: Goldhaber Ehrlich, Ch. II, III 1,2,4, IV, VII, VIII; Hungerford, Ch. III except 4,6, IV 1,2,3,5,6, VIII 1,4,6.
3. Fields: Algebraic and transcendental extensions, separable extensions, Galois theory of finite extensions, finite fields, cyclotomic fields, solvability by radicals.
References: Goldhaber Ehrlich, Ch. V except 6; Hungerford, Ch. V, VI; Kaplansky, Part I.
References:
Goldhaber Ehrlich, Algebra, reprint with corrections, Krieger, 1980.
Hungerford, Algebra, reprint with corrections, Springer, 1989.
Isaacs, Algebra, a Graduate Course, Wadsworth, 1994.
Kaplansky, Fields and Rings, 2nd Edition, University of Chicago Press, 1972.
Rotman, An Introduction to the Theory of Groups, 4th Edition, W.C. Brown, 1995.
M 381D (Koch) Complex Analysis
The objective of this syllabus is to aid students in attaining a broad understanding of analysis techniques that are the basic stepping stones to contemporary research. The prelim exam normally consists of eight to ten problems, and the topics listed below should provide useful guidelines and strategy for their solution. It is assumed that students are familiar with the subject matter of the undergraduate analysis courses M365C and M361. The first part of the Prelim examination will cover Real Analysis. The second part of the prelim examination will cover Complex Analysis.
1. Measure Theory and the Lebesgue Integral
Basic properties of Lebesgue measure and the Lebesgue integral on R^{n} (see [5], Ch. 14) and general measure and integration theory in an abstract measure space (see [5], Ch. 1112; and especially [6], Ch. 12). L^{p} spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in L^{p}norm and L^{p}L^{q} duality; integration in product spaces (see [6], Ch. 8) and convolution on R^{n}; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.
2. Holomorphic Functions and Contour Integration
Basic properties of analytic functions of one complex variable (see [1], Ch. 45; [2], Ch. 47; [4], Ch. 48; or [6], Ch. 1012 and 15). Integration over paths, the local and global forms of Cauchy's Theorem, winding number and residue theorem, harmonic functions, Schwarz's Lemma and the Maximum Modulus theorem, isolated singularites, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.
3. Differentiation
The relationship between differentiation and the Lebesgue integral on a real interval (see [5], Ch. 5), derivatives of measures (see [6], Ch. 5), absolutely continuous functions and absolute continuity between measures, functions of bounded variation.
4. Specific Important Theorems
Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, RadonNikodym theorem, FubiniTonelli theorems about product measures and integration on product spaces, Cauchy's theorem and integral formulas, Maximum Modulus theorem, Rouche's theorem, Residue theorem, and Fundamental Theorem of Calculus for Lebesgue Integrals. Students should be familiar with Minkowski's Inequality, Holder's Inequality, Jensen's Inequality, and Bessel's Inequality.
References:
1. L. Ahlfors, Complex Analysis, McGrawHill, New York, 1979.
2. J.B. Conway, Functions of One complex Variable, second edition, SpringerVerlag, New York, 1978.
3. G.B. Folland, Real Analysis, second edition, John Wiley, New York, 1999.
4. B. Palka, An Introduction to Complex Function Theory, second printing, SpringerVerlag, New York, 1995.
5. H.L. Royden, Real Analysis, Macmillan, New York, 1988.
6. W. Rudin, Real and Complex Analysis, third edition, McGrawHill, New York, 1987.
7. R. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977.
M 382D (Perutz) Differential Topology
It is assumed that students have a working knowledge of the equivalent of a one semester course in general topology (for example, see the appended syllabus for the undergraduate course M367K). For the semester in differential topology, it will also be assumed that students know the basic material from an undergraduate linear algebra course. The first part of the Prelim examination will deal with Algebraic Topology and the second part will deal with Differential Topology.
Algebraic Topology
1. Manifolds: Identification (quotient) spaces and identification (quotient) maps; topological nmanifolds, including surfaces, S^{n}, RP^{n}, CP^{n}, and lens spaces.
2. Triangulated manifolds: Representation of triangulated, closed 2manifolds as connected sums of tori or projective planes.
3. Fundamental group and covering spaces: Fundamental group, functoriality, retract, deformation retract; Van Kampen's Theorem, classification of surfaces by abelianizing the fundamental group, covering spaces, path lifting, homotopy lifting, uniqueness of lifts, general lifting theorem for maps, covering transformations, regular covers, correspondence between subgroups of the fundamental group and covering spaces, computing the fundamental group of the circle, RP^{n}, lens spaces via covering spaces.
4. Simplicial homology: Homology groups, functoriality, topological invariance, MayerVietoris sequence; applications, including Euler characteristic, classification of closed triangulated surfaces via homology and via Euler characteristic and orientability; degree of a map between oriented manifolds, Lefschetz number, Brouwer Fixed Point Theorem.
References:
Armstrong, Basic Topology, Springer, 1983 (principal text).
Greenberg, Lectures on Algebraic Topology, W.A. Benjamin, 1967.
Massey, Algebraic Topology, an Introduction, 4th corrected printing, Springer, 1977.
Munkres, Elements of Algebraic Topology, AddisonWesley, 1984.
Differential Topology
1. Smooth mappings: Inverse Function Theorem, Local Submersion Theorem (Implicit Function Theorem).
2. Differentiable manifolds: Differentiable manifolds and submanifolds; examples, including surfaces, S^{n}, RP^{n}, CP^{n} and lens spaces; tangent bundles; Sard's Theorem and its applications; differentiable transversality; orientation.
3. Vector fields and differential forms: Integrating vector fields; degree of a map, Brouwer Fixed Point Theorem, No Retraction Theorem, PoincareHopf Theorem; differential forms, Stokes Theorem.
References:
Guillemin Pollack, Differential Topology, PrenticeHall, 1974 (basic reference).
Hirsch, Differential Topology, Springer, 1976.
Milnor, Topology from the Differentiable Viewpoint, University of Virginia Press, 1965.
Spivak, Calculus on Manifolds, Benjamin, 1965 (differentiation, Inverse Function Theorem, Stokes Theorem).
For the examples indicated we refer to the books of Greenberg, Hirsch and Munkres.
M 383D (Arbogast) Methods of Applied Mathematics
It is assumed that students are familiar with the subject matter of the undergraduate analysis course M365C (see the Analysis section for a syllabus of that course) and an undergraduate course in linear algebra.
The Applied Math Prelim divides into these six areas. The first three are discussed in M383C and will be covered in the first part of the Prelim examination:
1. Banach spaces:
Normed linear spaces and convexity; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; continuous linear transformations; HahnBanach Extension Theorem; linear functionals, dual and reflexive spaces, and weak convergence; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; Closed Range Theorem; compact sets and AscoliArzelà Theorem; compact operators and the Fredholm alternative.
2. Hilbert spaces: Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and HilbertSchmidt operators; spectral theory for compact, selfadjoint and normal operators; SturmLiouville Theory.
3. Distributions: Seminorms and locally convex spaces; test functions and distributions; calculus with distributions.
These three areas are discussed in M383D and will be covered in the second part of the Prelim examination:
4. The Fourier Transform and Sobolev Spaces: The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for H^{s}.
5. Variational Boundary Value Problems (BVP): Weak solutions to elliptic BVP’s; variational forms; LaxMilgram Theorem; Green’s functions.
6. Differential Calculus in Banach Spaces and Calculus of Variations: The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the EulerLagrange equation.
References:
The first four references cover most of the syllabus for the exam. The other references also cover some topics in the syllabus.
1. C. Carath'eodory, Calculus of Variations and Partial Differential Equations of the First Order, 2nd English Edition, Chelsea, 1982.
2. F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974.
3. M. Reed and B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.
4. R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, available at World Wide Web address http://ejde.math.swt.edu//monotoc.html .
5. A. Avez, Introduction to Functional Analysis, Banach Spaces, and Differential Calculus, Wiley, 1986.
6. L. Debnath and P. Mikusi'nski, Introduction to Hilbert Spaces with Applications, Academic Press, 1990.
7. I.M. Gelfand and S.V. Fomin, Calculus of Variations, PrenticeHall, 1963.
8. E. Kreyszig, Introductory Functional Analysis with Applications, 1978.
9. J.T. Oden and L.F. Demkowicz, Applied Functional Analysis, CRC Press, 1996.
10. W. Rudin, Functional Analysis, McGrawHill, 1991.
11. W. Rudin, Real and Complex Analysis, 3rd Edition, McGrawHill, 1987.
12. K. Yosida, Functional Analysis, SpringerVerlag, 1980.
M 385D (Sirbu) Theory of Probability
(THE FIRST PART OF THE PRELIM EXAM WILL DEAL WITH THE MATERIAL COVERED IN M385C AND THE SECOND PART OF THE PRELIM EXAM WILL DEAL WITH THE MATERIAL COVERED IN M385D)
1. THEORY OF PROBABILITY I  M385C
 Prerequisites:
 Real Analysis (M365C or equivalent),
 Linear Algebra (M341 or equivalent),
 Probability (M362K or equivalent).
 R. Durrett, Probability: theory and examples, third ed., Duxbury Press, Belmont, CA, 1996. (required)
 D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)
Literature:
 Syllabus:
(Note: all references are to Durrett's book)Foundations of Probability:
 Random variables (Sections 1.1, 1.2): probability spaces, σalgebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, LebesgueStieltjes measures (without proof), random vectors, generation, a.s.convergence
 Expected value (Section 1.3): abstract Lebesgue integration (without proofs), inequalities (Jensen, CauchySchwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), changeofvariables formula,
 Dependence (Section 1.4): independence, pairwise independence, Dynkin's  theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)
 Weak laws of large numbers (Sections 1.5, 1.6): the L^{2} weak law of large numbers, triangular arrays, BorelCantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers
 Central limit theorems (Sections 2.2, 2.3a, 2.3b, 2.3c, 2.4a, 2.9part ): weak convergence of distributions, the continuous mapping theorem, Helly's selection theorem, tightness, characteristic functions, the inversion theorem, continuity theorem, the central limit theorem, multivariate normal distributions
 Conditional expectation (Sections 4.1a, 4.1b): RadonNikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
 Martingales (Sections 4.2, 4.4, 4.5, 4.6part , 4.7): martingale transforms, the optional sampling the orem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp convergence, maxi mum inequalities, L^{2} theory, uniform integrability, backwards martingales and the strong law of large numbers.
2. THEORY OF PROBABILITY II  M385D
 Prerequisites:
 Graduatelevel probability (M385C or equivalent).
 Literature:
 I. Karatzas and S. Shreve, Brownian motion and stochastic processes, second ed., Springer, 1991 (required)
 D. Revuz and M. Yor, Continuous martingales and stochastic processes, third ed., Springer, 1999 (recommended)
 Syllabus:
(Note: all references are to the book of Karatzas and Shreve)ContinuousTime Martingale Theory:
 General theory of processes (Sections 1.1, 1.2) : Continuoustime processes and filtrations, types of measurability (optional, predictable, progressive), continuous stopping/optional times
 Path regularity of martingales (Section 1.3 A): existence of RCLL modifications, usual conditions for filtrations
 Convergence and optional sampling (Section 1.3 AC): martingale inequalities, convergence theorems, optional sampling, uniform integrability and martingale with a last element
 Quadratic variation (Section 1.5 or Section IV.1 in RevuzYor): quadratic variation for continuous martingales, local martingales and localization, spaces of martingales
 DoobMeyer decomposition (Section 1.4): no proof
 Definition, construction and basic properties (Sections 2.1, 2.2): construction via Kolomogorov extension theorem, Hölder regularity of paths (KolmogorovCentsov), Gaussian processes
 The canonical space (Section 2.4): weak convergence on C[0, infinity), invariance principle, Wiener measure
 Markov and strong Markov property of Brownian motion (Sections 2.52.8, selected topics): reflexion principle, density of hitting times, Brownian filtrations, Blumenthal zeroone law
 Construction of the Stochastic Integral (Sections 3.1, 3.2): stochastic integration with respect to continuous local martingales, quadratic variation and Itô isometry
 Itô formula (Section 3.3): Itô formula, exponential martingales, linear stochastic differential equations
 Paul Léavy's characterization of Brownian motion (Section 3.3 B):
 Changes of measure (Section 3.5): Girsanov theorem, Brownian motion with drift Representations of martingales (Section 3.4): predictable representation property and KunitaWatanabe decomposition, timechanged Brownian motions (DambisDubinsSchwarz), Knight's theorem on orthogonal martingales
 Local time (Sections 3.6, 3.7): local time for Brownian motion and continuous semimartingales, Tanaka's formula, generalized Itô's formula for convex functions.
M 387D (Hughes) Numerical Analysis: Algebra & Approximations
The Prelim sequence is M387C and M387D. The first part of the Prelim examination will cover algebra and approximation and the second part of the Prelim examination will cover diferential equations.
Principles of discretization of differential equations:
 ODEs: Stability and convergence theory, Stiff problems,Symplectic integrators
 FEM (finite element method) and FDM (finite difference method) for boundary value problems
 FEM for PDEs (main focus on elliptic problems): Basic theory, weak formulations, LaxMilgram theorem, finite element spaces, approximation theory, a priori and a posteriori error estimates, practical algorithms, extensions, mixed methods etc.
 FDM for PDEs (main focus on hyperbolic and parabolic problems): Lax equivalence theorem, Von Neumann and other stability analysis, nonlinear conservation laws, shocks, entropy, practical algorithms
Brief survey of other methods for PDEs:
 FVM, DG, Spectral and particle methods
 Applications: Elasticity (FEM), Fluids (FVM), and Waves (FDM)
 Solution of linear and nonlinear equations
 Solution of integral equations
 Eigenvalues
 Optimization
 Monte Carlo methods
 Fast Fourier, wavelet transforms, approximation theory
 Basic undergraduate numerical methods
 Interpolation, fixed point iterations, Newton's method for root finding
 Direct and iterative methods for solving linear equations
 Quadratures
Recommended texts:
 Dahlquist and Bjorck, Numerical methods. Dover
 Lambert, Numerical methods for ordinary differential systems. Wiley
 Gustafsson, Kreiss, and Oliger, Time dependent problems and difference methods
 Iserles, A first course in the numerical analysis of differential equations, Cambridge
 Claes Johnson, Numerical solution of partial differential equations by the finite element method. Cambridge University Press
M 392C (Schmidt) Bridgeland Stability Conditions
One of the fundamental goals in pure mathematics is classification. An active area of research in algebraic geometry is to classify vector bundles on algebraic varieties. Work by Gieseker, Maruyama, and Mumford introduced the notion of a stable vector bundle. They form the building block of all vector bundles, but are easier to understand. The theory of Bridgeland stability conditions vastly generalizes this concept. The course will give an introduction to triangulated and derived categories, and explain the basic theory of Bridgeland stability conditions. We will explore two themes of applications, one in algebraic geometry related to moduli spaces and one in the representation theory of quivers.
M 392C (McCoy) Knots/3Manifolds
This course is about the structure of 3manifolds with applications to knot theory. The course will begin with basic 3manifold topology and move on to discuss more recent structural results, including geometrization. There will be a particular focus on hyperbolic 3manifolds and related topics.
M 392C (Freed) Mathematical Gauge Theory
This is a lecture course on advanced topics in mathematical gauge theory. It
M 392C (Sadun) Symbolic Dynamics and Tiling Theory
M 393C (Israel) Extensions of Smooth Functions
Given a function defined on a subset of Euclidean space, how can we construct an extension of this function on the entire space which belongs to a prescribed Banach space of smooth functions? Can we take this extension to depend linearly on the initial data? If the initial data is defined on a finite set, are there efficient algorithms to compute an extension with nearoptimal norm? These questions and their variants arise in diverse fields of mathematics, such as PDEs and Algebraic Geometry. The finite version of the problem is a question about the interpolation of data by smooth functions. This is related to the more difficult problem of manifold learning in data analysis, in which one attempts to pass a smooth surface with reasonable geometry through a finite set of points. One can also study generalizations of these problems involving additional types of global constraints. For instance, how can one determine whether there exists a smooth *convex* hypersurface, with prescribed bounds on its sectional curvatures, which passes through a given set of points?
M 393C (Martinsson) Fast Methods in Scientific Computing
M 393C (Vasseur) Fluid Mechanics
Mathematical treatment of fluid mechanics
This course is an introduction to the mathematical study of partial differential equations applied to fluid mechanics. We will consider both compressible and incompressible models, and study the properties of their solutions. A special focus will be given on the questions of wellposedness, stability, and regularity.
M 393C (Bajaj) Statistical and Discrete Methods for Scientific Computing. (Advanced Machine Learning Methods for Data Sciences)
M 394C (Taillefumier) Mathematical Neuoroscience (topics in neural dynamics, information theory, and machine learning)
This course is intended for mathematicians interested in neuroscience and mathematicallyinclined computational neuroscientists. The emphasis will be primarily on the analytical treatment of neuroscience inspired models and algorithms. The objectives of the course is to equip students with a solid technical
and conceptual background to tackle research questions in mathematical neuroscience.
The course will be structured in three blocks: neural dynamics, information theory, and machine learning.
1 Neural dynamics
Neural computations emerge from myriads of neuronal interactions occurring in intricate networks that have evolved over eons of time. Due to the obscuring complexity of these networks, we can only hope to uncover principles for neural computations through the lens of mathematical model ing and analysis. The main theoretical challenge is to relate quantitatively structure and activity in a tractable way, i.e. to uncover hierarchies of lowdimensional representations for the activity of highdimensional neural systems. In this block, we will present attempts made in that direction while introducing the mathematical formalisms associated to classical models of neural dynamics. Specifi cally: i) We will characterize distinct dynamical regimen of neural activity in deterministic singlecell models and in deterministic population models. ii) We will analyze neural variability in stochastic neural networks modeled via pointprocesses (i.e. intensitybased models) or via diffusion processes (i.e. integrateandfire models). iii) We will examine network dynamics in various simplifying mean field limits, including the traditional thermodynamics meanfield limits but also the replica meanfield limit. To complete this program, we will mostly rely on tools from the theory of dynamical systems and stochastic calculus (bifurcation theory, Markovian and stationary analysis).
References
[1] Eugene M. Izhikevich. Bifurcations in brain dynamics. ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–Michigan State University.
[2] Paul C. Bressloff. Waves in neural media. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, New York, 2014. From single neurons to neural fields.
[3] D.J.DaleyandD.VereJones.Anintroductiontothetheoryofpointprocesses.Vol.I.Probability and its Applications (New York). SpringerVerlag, New York, second edition, 2003. Elementary theory and methods.
[4] D. J. Daley and D. VereJones. An introduction to the theory of point processes. Vol. II. Probabil ity and its Applications (New York). Springer, New York, second edition, 2008. General theory and structure.
[5]I oannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus, volume 113 ofGraduate Texts in Mathematics. SpringerVerlag, New York, second edition, 1991.
2 Information theory
To elucidate brain structure conceptually, it is tempting to look for “design principles” that would guide the development and the evolution of neural systems. Such a putative design principle is of fered by the “efficient coding hypothesis”, which states that sensory systems have evolved to opti mally transmit information about the natural world given limitations on their biophysical components and constraints on energy use. In this block, we will introduce the theoretical framework suitable for investigating the efficient coding hypothesis from a mathematical standpoint. i) We will start by reviewing the foundations of Shannon’s information theory and its modern application to information processing in neuralnetwork models: a) We will present classical informationtheoretic optimization results, e.g. ratedistortion theory, and some of their more recent variants, e.g. information bottle neck, as well as the corresponding optimization algorithms. b) We will introduce maximumentropy methods for statistical inference about neural networks in the framework of information geometry.ii) Then, we will explore some outstanding informationtheoretical problems in neuroscience (chan nel optimization, entropy production). This block will rely on results from constrained optimization theory (essentially the KKT conditions) and will require some elementary notions of variational and differential calculus.
References

[1] Thomas M. Cover and Joy A. Thomas. Elements of information theory. WileyInterscience [John Wiley & Sons], Hoboken, NJ, second edition, 2006.

[2] Fred Rieke, David Warland, Rob de Ruyter van Steveninck, and William Bialek. Spikes. A Bradford Book. MIT Press, Cambridge, MA, 1999. Exploring the neural code, Computational Neuroscience.

[3] E. T. Jaynes. Probability theory. Cambridge University Press, Cambridge, 2003. The logic of science, Edited and with a foreword by G. Larry Bretthorst.

[4] Shunichi Amari and Hiroshi Nagaoka. Methods of information geometry, volume 191 of Trans lations of Mathematical Monographs. American Mathematical Society, Providence, RI; Oxford University Press, Oxford, 2000. Translated from the 1993 Japanese original by Daishi Harada.
3 Machine learning
Machine learning has allowed the realization of speech recognition, language translation, natural object recognition, and selfdriving cars. These achievements, which rival human performance, are performed by neural networks that mimic many structural features of the brain and learn how to per form tasks via biologically inspired rules, such as reinforcement learning. However, the mathematical theory underlying this computational feats is still in its infancy. This block will present the mathe matical theory supporting a few machine learning methods in supervised learning, in reinforcement learning, and in unsupervised learning. Specifically: i) We will present the theory of reproducing Hilbertkernel spaces (RHKSs) underlying support vector machines (SVMs). ii) We will introduce the theory of Markov decision processes (MDPs) in the context of reinforcement learning (successor representation). ii) We will discuss the probabilistic framework of recent generative models in artifi cial intelligence, namely the autoencoder networks and the generativeadversarial networks (GANs).
References
[1] Bernhard Scholkopf and Alexander J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA, USA, 2001.
[2] Richard S. Sutton and Andrew G. Barto. Introduction to Reinforcement Learning. MIT Press, Cambridge, MA, USA, 1st edition, 1998.
M 394C (Tran) Tropical Mathematics (Stochastic tropical geometry: theory, applications and open problems)
M 394C (Treisman) Math Education Policy Practicum
This advanced seminar is designed to provide students with practical experience in policy development and a broad understanding of the dynamics of American education policymaking. The course is organized around a substantive consulting project for the Education Commission of the States (ECS). ECS is a nonpartisan, nonprofit policy organization (www.ecs.org) created by the states in 1965. It tracks state policy trends, translates academic research, provides unbiased advice and creates opportunities for state leaders to learn from one another.
The content focus of the course––and the consulting project––is on the shifting landscape of college readiness policy. We will begin with an analysis of the Common Core Movement and HB5 in Texas and their historical precedents. We will then turn our attention to policies relevant to college access and student success with particular attention to their consequences for populations traditionally underrepresented in higher education. In addition to analyzing primary documents, students will have an opportunity to interview influential policymakers as well as with former LBJ students who are now leading related policy initiatives.
There will be much reading and writing, group work, and field work. In addition, the practicum is set up to help support the career development of students with an interest in policy development, advocacy, and public service. The practicum typically draws secondyear MPAff and MBA students, thirdyear law students, and doctoral students in mathematics, mathematics education, journalism, and educational policy and leadership. Feel free to contact Professor Treisman if you have an interest in this course.