## Lower Division Courses

## M302 Introduction to Mathematics

**Prerequisite and degree relevance:** Texas Success Initiative (TSI) exemption or a TSI Mathematics Assessment score of 350 or higher. The placement 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 seeking knowledge of the nature of mathematics as well as training in mathematical thinking and problem-solving. Mathematics 302 and 303F may not both be counted. A student may not earn credit for Mathematics 302 after having received credit for any calculus course. May not be counted toward a degree in the College of Natural Sciences.

**Course Description:** This is a terminal course satisfying the University's general education 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:** An appropriate score on the mathematics placement exam. Mathematics 303D and 303F may not both be counted. A student may not earn credit for Mathematics 303D after having received credit for Mathematics 305G or any calculus course. May not be counted toward a degree in the College of Natural Sciences.

**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 non-technical 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:** An appropriate score on the mathematics placement exam. Mathematics 305G and any college-level trigonometry course may not both be counted. A student may not earn credit for Mathematics 305G after having received credit for any calculus course with a grade of at least C-. Mathematics 301, 305G, and equivalent courses may not be counted toward a degree in mathematics.

**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:** An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. Only one of the following may be counted: Mathematics 403K, 408C, 408K, 408N.

**Course description: **M408C is our standard first-year 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 theorem-provers. 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.

## M408D-AP 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 world-class 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 M 408C, M308L, M 408L, M 308S or M 408S. Only one of the following may be counted: Mathematics 403L, 408D, 408M (or 308M). Math majors are required to take both M408C and M408D (or either the equivalent sequence M408K, M408L, M408M; or the equivalent sequence M408N, M408S, M408M). Mathematics majors are required to make grades of C- or better in each of 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 first-year 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, its primary objective is not the production of theorem-provers. M408D contains a treatment of infinite series, and an introduction to vectors and vector calculus in 2-space and 3-space, 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:** An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. 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- or M408R with a grade of at least B. Only one of the following may be counted: Mathematics 403L, 408L (or 308L), 408S.

**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:** M408L or M408S with a grade of at least C-. Only one of the following may be counted: Mathematics 403L, 408D, 408M (or 308M).

**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 variables, partial derivatives, gradients, and multiple integrals.

## M408N Differential Calculus

**Prerequisite and degree relevance: **An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. Only one of the following may be counted: Mathematics 403K, 408C, 408K, 408N.

**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 Differential and Integral Calculus for Sciences

**Prerequisite and degree relevance: **An appropriate score on the mathematics placement exam or Mathematics 305G with a grade of at least B-. May not be counted by students with credit for Mathematics 408C, 408K, or 408N.

**Course description: **M408R** **is a 1-semester survey of calculus. As such, it covers more ground than the first semester of a 2-semester sequence, but with a very different emphasis. We will cover Chapters 1-6 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 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: **Mathematics 408C, 408K, or 408N with a grade of at least C-, or Mathematics 408R with a grade of at least B. Only one of the following may be counted: Mathematics 403L, 408L (or 308L), 408S.

**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 permission form must be submitted before registration. The forms are available in the department office (PMA 8.100) or in the Mathematics, Physics, and Astronomy Advising Center. 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 permission form must be submitted before registration. The forms are available in the department office (PMA 8.100) or in the Mathematics, Physics, and Astronomy Advising Center. 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.

## M210E Emerging Scholars Seminar

**Prerequisite and degree relevance:** Restricted to students in the Emerging Scholars Program. Three two-hour laboratory sessions for one semester. May be repeated for credit. Offered on a pass/fail basis only.

**Course Description:** Supplemental problem-solving 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 permission form must be submitted before registration. The forms are available in the department office (PMA 8.100) or in the Mathematics, Physics, and Astronomy Advising Center. 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.

## M310P Modern Mathematics: Plan II

**Prerequisite and degree relevance:** Restricted to Plan II students. May not be counted toward a degree in mathematics.

**Course description:** Significant developments in modern mathematics. Topics may include fractals, the fourth dimension statistics and society, and techniques for thinking about quantitative problems.

## 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:** Credit or registration for Mathematics 408C and enrollment in a teaching preparation program, or consent of instructor.

**Course description:** Students will engage in lab-based activities designed to strengthen 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 significant 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: **An appropriate score on the mathematics placement exam. M 316 is an elementary introduction to statistical methods for data analysis; knowledge of calculus is not assumed. Students with a background in calculus are advised to take M 362K plus either M 358K or M 378K instead. This course may not be counted toward the major requirement for the Bachelor of Arts with a major in mathematics or toward the Bachelor of Science in Mathematics. Students taking the course should have good basic algebra skills. 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:** One of the following with a grade of at least C-: Mathematics 301, 302, 303D, 305G, 316, Educational Psychology 371, Statistics and Data Sciences 302, 304, or 306. Restricted to students in a teacher preparation program. 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. 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.

M316K is intended for prospective elementary teachers and other students whose degree programs require it; it treats basic concepts of mathematical thought.

**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 Foundations of Geometry, Statistics, and Probability

**Prerequisite and degree relevance:** M316K with a grade of at least C. Restricted to students in a teacher preparation program. 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. 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.

**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 permission form must be submitted before registration. The forms may be obtained in the mathematics office,PMA 8.100, or in the Math, physics and Astronomy Advising office, PMA 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

## M325K-H Honors Discrete Mathematics

Taught in the Spring semester by Prof. Michael Starbird, the course covers topics in mathematics rarely seen in undergraduate courses, such as countability.

## M325K Discrete Mathematics

**Prerequisite:** Mathematics 408D, 408L, or 408S with a grade of at least C-. This is the first course that emphasizes understanding and creating proofs. Therefore, it provides a transition from the problem-solving 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 theorem-proving skills.

**Course Topics: **

- Fundamentals of logic
- Elementary Number theory
- Sequences and Mathematical Induction
- Set Theory

## M328K-H 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 PMA 4.101, during Fall registration.

## M326K Foundations of Number Systems

**Prerequisite and degree relevance:** M408D, M408L, or M408S with a grade of at least C-.Restricted to students in a teacher preparation program or who have the consent of the instructor.

**Course description: **Study of number-related 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 a pedagogical understanding of concepts and notation, and using both informal reasoning and proof. Three lecture hours a week for one semester.

## M328K Introduction to Number Theory

**Prerequisite and degree relevance:** Mathematics 325K or 341 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 problem-solving 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 theorem-proving 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, Euler's j-function, 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**: Dirichlet's theorem, continued fractions, Pells equation, Liouville's theorem, algebraic and transcendental numbers, the transcendence of e and/or z.

## M329F Theory of Interest

**Prerequisite and degree relevance:** Mathematics 408D, 308L, 408L, or 408S with a grade of at least C-. This course covers the content for the SOA Exam FM. Topics include nominal and effective interest and discount rates, general accumulation functions and force of interest, yield rates, annuities including those with non-level payment patterns, amortization of loans, sinking funds, bonds, spot and forward rates, interest rate swaps, duration, and immunization.

## M329W Cooperative Mathematics

**Prerequisite and degree relevance:** Application through the College of Natural Sciences Career Design Center; Mathematics 408D, 408L, or 408S with a grade of at least C-; a grade of at least C- in two of the following courses: Mathematics 325K, 427J or 427K, 341, 362K, or 378K; 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 long-session 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: **Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 427J and 427K may not both be counted.

**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 an 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.

## M427K-H 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:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 427J and 427K may not both be counted.

**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 problem-oriented 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.

## M427L-AP Honors Advanced Calculus for Applications II

**Course description: **Rather than cover material of a standard calculus class, this course goes directly into an upper-division treatment of multivariable calculus, and covers this topic from a more advanced perspective. In addition, M427L-AP covers additional material, such as theorems of Stokes, Green, and Gauss, that are not found in general calculus. This material prepares students 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 upper-division math hours.

## M427L Advanced Calculus for Applications II

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 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:** Mathematics 408D, 408L, or 408S with a grade of at least C-; or upper-division standing and consent of instructor. These requirements are set to ensure 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 the understanding of the concepts and the ability to use the concepts in proving theorems. The course includes the study of axiom systems, transformational geometry, and an introduction to non-Euclidean 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.

## M339D Introduction to Financial Mathematics for Actuaries Syllabus

**Prerequisite and degree relevance:** Actuarial Foundations 329 or Mathematics 329F; and Mathematics 362K with a grade of at least C-. Moreover, the instructor advises that students will need a thorough understanding and operational knowledge of (at least) calculus, finite-stage-space probability, and the term structure of interest rates.

**Course description:** This course is intended to provide the mathematical foundations necessary to prepare for a portion of

(1) the joint SOA/CAS exam FM/2, as well as

(2) the SoA exam MFE and the financial economics portion of the CAS Exam 3.

Additionally, the course is aimed at building up the vocabulary and the techniques indispensable in the workplace at current financial and insurance institutions. This is not an exam-prep seminar. There is intellectual merit to the course beyond the ability to prepare for a professional exam.

The material exhibited includes elementary risk management, forward contracts, options, futures, swaps, the simple random walk, the binomial asset pricing model and its application to option pricing. The remainder of the Exam MFE/3F curriculum is exhibited in course M339W (also offered by the Department of Mathematics).

## M339J Probability Models with Actuarial Applications

**Prerequisite and degree relevance: **Mathematics 358K or 378K with a grade of at least C-.

**Course description:** Introductory actuarial models for life insurance, property insurance, and annuities. With Mathematics 349P, covers the syllabus for the professional actuarial exam on model construction.

Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.

## M339U Actuarial Contingent Payments I

** Prerequisite and degree relevance: **Mathematics 362K with a grade of at least C-; credit with a grade of at least C- or registration for Actuarial Foundations 329 or Mathematics 329F; and credit with a grade of at least C- or registration for Mathematics 340L or 341.

**Course description:** Intermediate actuarial models for life insurance, property insurance, and annuities.

Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.

## M339V Actuarial Contingent Payments II

** Prerequisite and degree relevance: **Actuarial Foundations 329 or Mathematics 329F, and Mathematics 339U with a grade of at least C- in each.

**Course description:** Advanced actuarial models for life insurance, property insurance, and annuities.

## M339W Financial Mathematics for Actuarial Applications

** Prerequisite and degree relevance: **Mathematics 339D with a grade of at least C-.

**Course description:** Pricing, stock price, and interest rate models for actuarial applications. Tools include lognormal distribution, Brownian motion, Black-Scholes, and delta hedging.

## M340L Matrices and Matrix Calculations

**Prerequisite and degree relevance:** Mathematics 408C, 408K, or 408N with a grade of at least C-. Only one of the following may count: Mathematics 340L, 341, Statistics and Data Sciences 329C, or Statistics and Scientific Computation 329C.

**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 applications-oriented students such as those in the natural and social sciences, engineering, and business. Topics might include matrix operations, systems of linear equations, introductory vector-space concepts (e.g., linear dependence and independence, basis, dimension), determinants, introductory concepts of eigensystems, introductory linear programming, and least square problems.

## M341-H 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 PMA 4.101, during Fall registration.

## M341 Linear Algebra and Matrix Theory

Formerly M311.

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Restricted to mathematics majors. Only one of the following may count: Mathematics 340L, 341, Statistics and Data Sciences 329C, or Statistics and Scientific Computation 329C. 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 systems 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: Cauchy-Schwarz inequality, orthonormal bases, the Gramm-Schmidt 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, Cramer's 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: Mathematics 325K or Philosophy 313K, Mathematics 328K, Mathematics 341. 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 Error-Correcting 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 error-correcting codes, including finite fields, error-correcting 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 second-semester course in symbolic logic: formal syntax and semantics, basic metatheory (soundness, completeness, compactness, and Löwenheim-Skolem theorems), and further topics in logic.

## M346 Applied Linear Algebra

**Prerequisite and degree relevance: **The prerequisite is M341 (or M311) or M340L with a grade of C- or better.

**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 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 object-oriented programming in an advanced language. Study and use of numerical methods for solutions of linear systems of equations, non-linear least-squares data fitting, numerical integration of multi-dimensional, non-linear equations and systems of initial value ordinary differential equations.

## M349P Actuarial Statistical Estimates I

**Prerequisite and degree relevance:** Mathematics 339J, and 341 or 340L, with a grade of at least C- in each. Please note that thorough knowledge of calculus, probability, and statistics will be assumed.

**Course description: **Statistical estimation procedures for random variables and related quantities in actuarial models. With Mathematics 339J, covers the syllabus for the professional actuarial exam on model construction.

Actuarial Course descriptions and guides in selecting actuarial courses are available at the Actuarial Course Descriptions.

## M349R Applied Regression and Time Series

**Prerequisite and degree relevance: **Mathematics 339J or 339U, and 358K or 378K, with a grade of at least C- in each.

**Course description: **Introduction to simple and multiple linear regression and to elementary time-series models, including auto-regressive and moving-average models. Emphasizes fitting models to data, evaluating models, and interpreting results

## M358K Applied Statistics

**Prerequisite and degree relevance:** M362K with a 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 the 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; chi-squared 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.

## M360M Mathematics as Problem Solving

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C- and written consent of instructor.

**Course description:** Discussion of heuristics, strategies, and methods of problem-solving, and extensive practice in both group and individual problem-solving. Communicating mathematics, reasoning and connections amongst topics in mathematics are emphasized.

## M361 Functions of A Complex Variable

**Prerequisite and degree relevance:** Mathematics 427J, 427K, or 427L with a grade of at least C-.

**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 Cauchy's 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 the Undergraduate Mathematics Faculty Advisor 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 have received a grade of C- or better in Mathematics 365C may not take Mathematics 361K.

**Course description:** This is a rigorous treatment of the real number system, of real sequences, and of limits, continuity, derivatives, and integrals of real-valued functions of one real variable.

## M362K Probability I

**Prerequisite and degree relevance:** Mathematics 408D, 408L, or 408S with a grade of at least C-. Mathematics 362K and Statistics and Scientific Computation 321 may not both be counted.

**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 the Undergraduate Mathematics Faculty Advisor 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 361K before attempting 365C. Students planning to take Mathematics 365C and 373K concurrently should consult a mathematics adviser.

**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 real-valued 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 one-variable
- L
^{p}spaces, Hilbert spaces, linear operators

**Text: **Davidson & Donsig: **Real Analysis and Real Applications** Prentice-Hall ISBN 0-13-041647-9

## 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 Gauss-Bonnet Theorem.

**Textbook: ** Elementary Topics in Differential Geometry, John A. Thorpe, Springer--Verlag, New York, 1979. (ISBN 0-387-90357-7)

**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 n-dimensional 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:** Mathematics 361K or 365C 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:**1-1 correspondence, countability, and uncountability.**Definitions of topological space:**basis, sub-basis, 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 Science 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, two-point-boundary value problems, numerical approximation of partial differential equations, signal processing, optimization, and Monte Carlo methods.

## M371E Learning Assistant Experience in Mathematics

**Prerequisite and degree relevance: **Mathematics 408C, 408K, 408N, 408R, or equivalent, and consent of instructor.

**Course description: **Students assist instructors and TAs in mathematics courses. This is a hands-on experience in what it is like to teach and support students in the learning of mathematics in undergraduate courses. Students must attend classroom training and discussions and work in Calculus discussion sections or undergraduate classrooms where mathematics is being taught. One class hour and three hours of fieldwork in an undergraduate mathematics course a week for one semester.

## M175 Conference Course

**Prerequisite and degree relevance:** Vary with the topic, and are given in the course schedule. May not be counted toward a degree in mathematics. A signed permission form must be submitted before registration. The forms are available in the department office (PMA 8.100) or in the Mathematics, Physics, and Astronomy Advising Center.

**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.

## M275 Conference Course

**Prerequisite and degree relevance:** Vary with the topic, and are given in the course schedule. May not be counted toward a degree in mathematics. A signed permission form must be submitted before registration. The forms are available in the department office (PMA 8.100) or in the Mathematics, Physics, and Astronomy Advising Center.

**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.

## M372K PDE and Applications

**Prerequisite and degree relevance:** Mathematics 427J or 427K 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:** Either consent of the Undergraduate Mathematics Faculty Advisor 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:** Mathematics 373K with a grade of at least C-. 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 427J or 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 378Kwith 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.

## M374M Mathematical Modeling in Science and Engineering

**Prerequisite and degree relevance: **Mathematics 427J or 427K, and 340L or 341, 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 and physical sciences. Includes dimensional analysis and scaling, regular and singular perturbation methods, optimization and calculus of variations, and stability.

## M375 Conference Course

**Prerequisite and degree relevance:** Vary with the topic, and are given in the course schedule. May not be counted toward a degree in mathematics. A signed permission form must be submitted before registration. The forms are available in the department office (PMA 8.100) or in the Mathematics, Physics, and Astronomy Advising Center.

**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.

## M375C Conference Course

**Prerequisite and degree relevance:** Vary with the topic, and are given in the course schedule. A signed permission form must be submitted before registration. The forms are available in the department office (PMA 8.100) or in the Mathematics, Physics, and Astronomy Advising Center.

**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.

## M375T Functions and Modeling

- M375T Functions and Modeling runs concurrently 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 Introduction to Graduate Studies in Mathematics

- 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: **Mathematics 427J or 427K, and 340L or 341, 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 problems 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-. Same as Statistics and Data Sciences 378. Only one the following may be counted: Mathematics 378K, Statistics and Data Sciences 378, Statistics and Scientific Computation 378.

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 (p-values, 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, chi-squared, 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 graduate 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.

## M378N Generalized Linear Models

**Prerequisite and degree relevance: **Mathematics 378K with a grade of at least C- or consent of instructor. Mathematics 375T (Topic: Generalized Linear Models) and 378N may not both be counted.

**Course description:** Extensions to ordinary least-squares regression, including Poisson regression, the lasso, mixed models, and ridge regression.

## M379H Honors Tutorial Course

**Prerequisite and degree of relevance:** Admission to the Mathematics Honors Program; Mathematics 365C, 367K, 373K, or 374G with a grade of at least 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 perturbation methods, optimization and calculus of variations, and stability.

## Graduate Courses

**SPRING 2022 GRADUATE COURSES **

For information on preliminary course syllabi - please visit the **prelim courses syllabi**

## M 391C (Beckner) Fourier Analysis

Our lectures will survey the classical structure and techniques of Fourier Analysis – starting from the development of the Lebesgue integral and the framework of the Riemann-Lebesgue lemma, the Plancherel theorem, Young’s inequality, the Hausdorff-Young inequality, and the uncertainty principle. The drive for “best constants” for intrinsic inequalities that characterize smoothness and encode geometric information was initiated by Hardy & Littlewood. The essential role of symmetry emerges with the isoperimetric and Brunn-Minkowski inequalities, the Riesz-Sobolev rearrangement theorem and its application for Sobolev inequalities. Topics that will be discussed include: (1) distribution theory, the Schwartz class, spherical harmonics, and the Hecke-Bochner representation, (2) restriction, Bochner-Riesz means, and Strichartz inequalities, (3) Kunze-Stein phenomena on Lie groups, (4) Gaussian functions and log Sobolev

inequalities, (5) Riesz potentials, the fractional Laplacian, Stein-Weiss integrals, the Hardy-Littlewood-Sobolev inequality, and Moser- Trudinger inequalities, (6) embedding estimates on the sphere and hyperbolic space, and (7) analysis on the Heisenberg group using both conformal and symplectic geometry in moving forward beyond the Euclidean domain.

## M 391C (Lafleche) Semiclassical Dynamics

This course will present some of the mathematical tools to describe the links between quantum and classical theories. Semiclassical analysis aims to understand asymptotic expansions in terms of a small parameter often corresponding to the Planck constant. Such expansions however usually require a certain regularity that must be proved for dynamical models.

The main topics of the lectures will be

- the basis of the underlying physical theories

- the Wigner and Husimi transforms, the Weyl and Wick quantizations

- operator theory, Schatten spaces, trace and semiclassical inequalities

- the quantum Wasserstein distances and Sobolev spaces

- the large particle approximations and the limit from the Hartree to the Vlasov equation

## M 392C (Siebert) Algebraic Geometry

This is a second course in algebraic geometry, assuming some knowledge of scheme theory as contained e.g. in Chapter 2 of Hartshorne's classic book on the subject. We first introduce cohomological methods and then, as an application of the learned machinery, study the moduli space of stable curves as an algebraic stack. Further topics will be added if time permits.

## M 392C (Stecker) Discrete Subgroups of Lie Groups

A Lie group G, such as the group of invertible nxn matrices, carries both the structure of a group and that of a manifold, in particular it comes with a topology. So we can ask which subgroups of G are discrete in this topology. These subgroups play an important role in geometry. For example, (finitely generated torsion-free) discrete subgroups of the group PSL(2,R) are exactly the holonomy groups of hyperbolic surfaces, and so understanding them is essentially equivalent to understanding the geometry of hyperbolic surfaces.

A relatively new field often called "Higher Teichmüller Theory" aims to study discrete subgroups of higher rank semisimple Lie groups, for example PSL(n,R), by similar geometric methods. This is often done by focusing on classes of discrete subgroups with particularly nice properties. Examples are Hitchin representations, maximal representations, divisible convex sets, or the more general Anosov representations.

The plan for this course is to give an introduction into these tools and explore some of their fascinating geometric and dynamical properties. We will start with the discrete subgroups of rank one groups like PSL(2,R) and some of the hyperbolic geometry contained in them, and then build up to the definition of Anosov representations into PSL(n,R). After that,

further topics could for example be: the characterization of Anosov representations via singular values, other Lie groups, Hitchin representations, maximal representations, positivity, convex projective manifolds, the limit cone of Zariski dense groups.

Prerequisites: it would be good to know basic differential topology (smooth manifolds, Lie groups), differential geometry (the hyperbolic

## M 392C (Bajaj) Geometric Foundations Data Science/Predictive Machine Learning

Discuss foundational mathematical, statistical and computational theory of data sciences. Explore how this data driven predictive machine learning theory is applied to stochastic dynamical systems, optimal control and multi-player games.

## M 392C (Chen) The Gromov norm and bounded cohomology

The aim of the course is to introduce the relatively new and fast-growing field of study on Gromov's norm and bounded cohomology, their variants, and most importantly applications to more classical topics (mostly in geometry, topology and dynamics). The hope is that the course will provide useful tools and points of views for students studying (hyperbolic) geometry, (low-dimensional) topology and dynamics.

Mostow's rigidity asserts that hyperbolic structure is unique (if exists) on closed manifolds of dimension at least 3. In particular, the hyperbolic volume is (surprisingly) a topological invariant (in all dimensions). As a wonderful explanation of this fact, Gromov introduced the simplicial volume that measures the topological complexity of the fundamental class and showed that it is proportional to the hyperbolic volume if the manifold is hyperbolic.

The simplicial volume is a special case of Gromov's (simplicial) norm, which equips each homology group (of a given space with R coefficients) a semi-norm that measures the complexity of each homology class. For the second homology group of a 3-manifold, this turns out to be proportional to the Thurston norm, which reveals how the manifold fibers over the circle.

Bounded cohomology is the dual theory of simplicial norm (similar to the L^\infty-L^1 duality). The two theories complement each other and have numerous applications to geometry, topology and dynamics (see below), especially in the understanding of groups that arise naturally in these fields. We will also discuss variants of the simplicial norm and their corresponding dual theory.

We will cover a subset of the following topics: Simplicial volume and Gromov's proportionality, a proof of Mostow's rigidity; Thurston's norm and fibrations of 3-manifolds, relation between Thurston's norm and Gromov's norm; bounded cohomology, quasimorphisms, applications to transformation groups; examples and constructions of quasimorphisms on free groups, hyperbolic groups, and mapping class groups; groups acting on the circle, Poincare's rotation number, and the bounded Euler class, relation to left- and circular-orderability; stable commutator length, applications to the existence of surface (and free) subgroups of certain groups acting on trees and/or random groups; Burger-Monod's vanishing theorem for bounded cohomology of higher rank irreducible lattices and its application to rigidity of such lattices; the spectrum of stable commutator length and the spectrum of simplicial volume, spectral gap properties.

Prerequisites: The students are expected to be familiar with basic notions from algebraic topology (fundamental groups, free groups, Euler characteristic, covering spaces). Having some ideas about hyperbolic geometry would be helpful for certain topics of the course, but it is not required. Otherwise, prerequisites will be kept to a minimum.

## M 392C (Perutz) A Second Course in Algebraic Topology

This course will build on the foundation provided by the Algebraic Topology prelim course, and will cover some of the central ideas of the subject, concerning homotopy theory and cohomology. This is material that is widely used in differential and algebraic geometry, geometric topology, and algebra, as well as by specialists in algebraic topology. The following is an aspirational list of topics, of which I hope to cover several:

**Homological algebra: **Examples of derived functors: Tor, Ext, group (co)homology

**Singular homology: **Review of singular and cellular homology, Eilenberg-Steenrod axioms. Homology of products. Cohomology and universal coefficients. Simplicial spaces; construction of classifying spaces for topological groups.

**Cup products and duality: ** Cross, cup and cap products. Manifolds, orientation and Poincaré duality. Submanifolds and transverse intersections.

**Homotopy theory: **Homotopy groups; fiber bundles and fibrations. Cofibrations. Weak equivalence and Whitehead’s theorems. The homotopy exact sequence of a fibration. The Hurewicz theorem. Eilenberg-MacLane spaces.

**The Serre spectral sequence: **The spectral sequence of a filtered complex. The Serre spectral sequence; examples; transgression. Proof of the Hurewicz theorem.

**Localization: **Serre classes** **of abelian groups; homotopy and homology theory modulo a Serre class; applications.

** Prerequisite:** Algebraic Topology at the level of the prelim: fundamental groups, covering spaces, basics of homology theory (e.g. Mayer-Vietoris). You should also know the basics of rings and modules, as in the Algebra I prelim - for instance, the tensor product of modules.

** Texts: **I plan to draw on Haynes Miller’s MIT lecture notes in Algebraic Topology (https://math.mit.edu/~hrm/papers/lectures-905-906.pdf), as well texts including those of Allen Hatcher (http://pi.math.cornell.edu/~hatcher/AT/ATpage.html) and Peter May (https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf).

## M 393C (Ward) Foundational Techniques of Machine Learning and Data Sciences

This course will be a mathematically rigorous introduction to topics from linear algebra, high-dimensional probability, optimization, statistics, which are foundational tools for data science, or the science of making predictions from structured data. A secondary aim of the course is to become comfortable with experimenting and exploring data science problems through programming.

## M 393C (Vasseur) Partial Differential Equations II

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 to the questions of well-posedness, stability, and regularity.

## M 394C (Zitkovic) Volatility Modeling

Volatility is a local measure of variability of the price of a financial asset. It plays a central role in modern finance, not only because it is the main ingredient in the celebrated Black-Scholes option-pricing formula. One of its most enticing aspects is that it is as interesting to mathematicians and statisticians as it is to financial practitioners. As the markets, and our understanding of them, evolve and as our statistical prowess grows, the models we use to describe volatility become more and more sophisticated.

The goal of this course is to give an overview of various models of volatility, together with their most important mathematical aspects. In addition, these models provide a perfect excuse to talk about various classes of stochastic processes (Gaussian processes, affine

diffusions or rough processes).

While the main focus will remain on the underlying mathematics, some time will be spent on statistical properties of these models and their fit to data.

Thorough familiarity with measure-theoretic probability and basic concepts of stochastic analysis (at the level of M385C/M385D "Theory of Probability I/II", respectively) will be assumed. No prior knowledge of finance or statistics will be required.

**FALL 2021 GRADUATE COURSES**

## M 380C (Raskin) Algebra

The prelim sequence is M380C and M380D

**Description**

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, *p*-groups, direct products and sums, semi-direct products, permutation groups, simple groups, finite Abelian groups; infinite groups, including normal and composition series, solvable and nilpotent groups, Jordan-Holder theorem, free groups.

**References:** Goldhaber Ehrlich, Ch. I except 14; Hungerford, Ch. I, II; Rotman, Ch. I-VI, 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, Wedderburn-Artin 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 (Caffarelli) Real Analysis

**Description**

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.

**Measure Theory and the Lebesgue Integral**

Basic properties of Lebesgue measure and the Lebesgue integral on**R**(see [5], Ch. 1-4) and general measure and integration theory in an abstract measure space (see [5], Ch. 11-12; and especially [6], Ch. 1-2). L^{n}spaces (see [6], Ch. 3); convergence almost everywhere, in norm and measure; approximation in L^{p}-norm and L^{p}-L^{p}duality; integration in product spaces (see [6], Ch. 8) and convolution on^{q}**R**; and the concept of a Banach space, Hilbert space, dual space and the Riesz representation theorem.^{n}**Holomorphic Functions and Contour Integration**

Basic properties of analytic functions of one complex variable (see [1], Ch. 4-5; [2], Ch. 4-7; [4], Ch. 4-8; or [6], Ch. 10-12 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 singularities, entire and meromorphic functions, Laurent series, infinite products, Weierstrass factorization, conformal mapping, Riemann mapping theorem, analytic continuation, "little" Picard theorem.**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.**Specific Important Theorems**

Students should be familiar with Monotone and Dominated Convergence theorems, Fatou's lemma, Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Fubini-Tonelli 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,* McGraw-Hill, New York, 1979.

2. J.B. Conway, *Functions of One complex Variable,* second edition, Springer-Verlag, 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, Springer-Verlag, New York, 1995.

5. H.L. Royden, *Real Analysis,* Macmillan, New York, 1988.

6. W. Rudin, *Real and Complex Analysis,* third edition, McGraw-Hill, New York, 1987.

7. R. Wheeden and A. Zygmund, *Measure and Integral,* Marcel Dekker, New York, 1977.

## M 382C (Gordon) Algebraic Topology

**Description**

**MANIFOLDS AND CELL COMPLEXES:**

Identification (quotient) spaces and maps;

Topological n-manifolds including surfaces, S^n, RP^n, CP^n;

CW decompositions, including these examples.

**FUNDAMENTAL GROUP AND COVERING SPACES:**

Fundamental group, the first example S^1;

Functoriality and homotopy-type invariance;

Retraction and deformation retraction;

Van Kampen's Theorem;

Covering spaces and lifting properties;

Covering transformations;

The covering spaces <--> subgroups of pi_1 correspondence;

Regular covers;

Further examples including RP^n and lens spaces.

**SINGULAR HOMOLOGY:**

Definitions, functoriality, homotopy-type invariance;

Relative homology, excision, the Eilenberg-Steenrod axioms;

Mayer-Vietoris and examples, including S^n, CP^n;

Homology of cell complexes;

Further examples, including RP^n;

Local homology and orientations, degree of a map : S^n \to S^n;

Brouwer fixed point theorem, Jordan separation theorem;

Euler characteristic, Lefschetz fixed point theorem.

**PRINCIPAL TEXT:**

Hatcher, Algebraic Topology (available for free download)

The syllabus is approximately Chapters 0-2 of Hatcher.

**OLDER REFERENCES** (all 35+ years old but still useful):

Armstrong, Basic Topology, Springer

Greenberg, Lectures on Algebraic Topology

Massey, Algebraic Topology, an Introduction

Munkres, Elements of Algebraic Topology

## M 383C (Arbogast) Methods of Applied Mathematics

**Description**

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 eight areas. The first four are discussed in M383C and will be covered in the first part of the Prelim examination:

**Banach spaces**: Normed linear spaces, convexity, and examples; convergence, completeness, and Banach spaces; continuity, open sets, and closed sets; bounded linear transformations; Hahn-Banach Extension Theorem and its applications; the Baire Theorem and uniform boundedness; Open Mapping and Closed Graph Theorems; linear functionals, dual and reflexive spaces, and weak convergence.

**Hilbert spaces:** Inner products, basic geometry, orthogonality, bases, projections, and examples; the Riesz Representation Theorem; Bessel’s inequality and the Parseval Theorem; maximal orthonormal sets and the Riesz-Fischer Theorem; weak convergence.

**Spectral theory**: Resolvent and spectrum; compact operators; spectral theory for compact, self-adjoint, and normal operators; the Fredholm alternative; the Ascoli-Arzelà Theorem; Sturm-Liouville Theory.

**Distributions**: Seminorms and locally convex spaces; test functions and distributions; operations with distributions; approximations to the identity; applications to linear differential operators.

#### The following four areas are discussed in M383D and will be covered in the second part of the Prelim examination:

**The Fourier transform**: The Schwartz space and tempered distributions; the Fourier transform on L-2 and tempered distributions; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s.

**Sobolev spaces**: Definitions and basic properties; extensions theorems; the Sobolev Embedding Theorem; compactness and the Rellich-Kondrachov Theorem; fractional order spaces and trace theorems.

**Variational Boundary Value Problems (BVP)**: Weak solutions to elliptic BVP’s; variational forms and satisfaction of Dirichlet and Neumann boundary conditions; Closed Range Theorem; Lax-Milgram Theorem; Galerkin methods; Green’s functions.

**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 and the Euler-Lagrange equation; problems with constraints and Lagrange multipliers.

**References**

R. A. Adams, Sobolev Spaces, Academic Press, 1975.

T. Arbogast and J. L. Bona, Functional Analysis for the Applied Mathematician, 2020.

A. Avez, Introduction to Functional Analysis, Banach Spaces, and Differential Calculus, Wiley, 1986.

C. Carath'eodory, Calculus of Variations and Partial Differential Equations of the First Order, 2nd English Edition, Chelsea, 1982.

Ph. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013.

L. Debnath and P. Mikusi'nski, Introduction to Hilbert Spaces with Applications, Academic Press, 1990.

I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall, 1963.

E. Kreyszig, Introductory Functional Analysis with Applications,1978.

J.T. Oden and L.F. Demkowicz, Applied Functional Analysis, CRC Press, 1996.

M. Reed and B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.

W. Rudin, Functional Analysis, McGraw-Hill, 1991.

W. Rudin, Real and Complex Analysis,3rd Edition, McGraw-Hill, 1987.

K. Yosida, Functional Analysis, Springer-Verlag, 1980.

## M 383E (Martinsson) Numerical Analysis: Linear Algebra

**Description**

Matrix computations form the core of much of scientific computing, and are omnipresent in applications such as statistics, data mining and machine learning, economics, and many more. This first year graduate course focuses on some of the fundamental computations that occur in these applications. Specific topics include direct and iterative methods for solving linear systems, standard factorizations of matrices (LU, QR, SVD), and techniques for solving least squares problems. We will also learn about basic principles of numerical computations, including perturbation theory and condition numbers, effects of roundoff error on algorithms and analysis of the speed of algorithms.

Pre-requisites for this course are a solid knowledge of undergraduate linear algebra, some familiarity with numerical analysis, and prior experience with writing mathematical proofs.

## M 384C Mathematical Statistics I

**Description**

The two semesters of this course (M 384C and M 384D) are designed to provide a solid theoretical foundation in mathematical statistics.

During the TWO-SEMESTER course, the statistical topics include the properties of a random sample, principles of data reduction (sufficiency principle, likelihood principle, and the invariance principle), and theoretical results relevant to point estimation, interval estimation, hypothesis testing with some work on asymptotic results.

During the first semester, M384C, students are expected to use their knowledge of an undergraduate upper-level probability course and extend those ideas in enough depth to support the theory of statistics, including some work in hierarchical models to support working with Bayesian statistics in the second semester. Students are expected to be able to apply basic statistical techniques of estimation and hypothesis testing and also to derive some of those techniques using methods typically covered in an undergraduate upper-level mathematical statistics course. A brief review of some of those topics is included. Probability methods are used to derive the usual sampling distributions (min, max, the t and F distributions, the Central Limit Theorem, etc.) Methods of data reduction are also discussed, particularly through sufficient statistics. This includes the five chapters of the text and part of the sixth chapter as well as some additional material on estimation and hypothesis testing.

**Prerequisite:**

M362K, Probability, and M378K, Introduction to Mathematical Statistics, or the equivalent. Course descriptions of 362K and 378K are available on the web and more information about equivalencies is available from http://www.ma.utexas.edu/users/parker/384/prereq/

**Textbook: ***Statistical Inference* by George Casella and Roger L. Berger, second edition

**Consent of Instructor Required: **Yes.

## M 385C (Neeman) Theory of Probability

**Prerequisites:**

- Real Analysis (M365C or equivalent),
- Linear Algebra (M341 or equivalent),
- Probability (M362K or equivalent).
- R. Durrett, Probability: theory and examples, fifth ed., Duxbury Press, Belmont, CA, 1996. (required)
- D. Williams, Probability with martingales, Cambridge University Press, Cambridge, 1991. (recommended)

**Literature**

5th edition of Durrett's book. A free pdf of this book is available at Durrett's website (https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf)

**Syllabus:**** **(Note: all references are to Durrett's book)

**Description**

**Foundations of Probability:**

- Random variables (Sections 1.1, 1.2, 1.3): probability spaces, σ-algebras, measurability, continuity of probabilities, product spaces, random variables, distribution functions, Lebesgue-Stieltjes measures (without proof), random vectors, generation, a.s.-convergence
- Expected value (Section 1.4, 1.5, 1.6): abstract Lebesgue integration (without proofs), inequalities (Jensen, Cauchy-Schwarz, Chebyshev, Markov, Hölder, Minkowski), limit theorems (Fatou's lemma, monotone convergence and dominated convergence theorems), change-of-variables formula,
- Dependence (Section 2.1): independence, pairwise independence, Dynkin's - theorem, convolution of measures, Fubini's theorem, Kolmogorov's extension theorem (without proof)

**Classical Theorems:**

- Weak laws of large numbers (Sections 2.2, 2.3, 2.4): the L
^{2}-weak law of large numbers, triangular arrays, Borel-Cantelli lemmas, modes of convergence, inequalities (Markov, Chebyshev, Jensen, Hölder), the weak law of large numbers - Central limit theorems (Sections 3.1, 3.2, 3.3, 3.4): 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

** **

**Discrete-Time Martingale Theory:**

- Conditional expectation (Sections 4.1.1, 4.1.2): Radon-Nikodym theorem (without proof), conditional expectation, filtrations, predictability and adaptivity
- Martingales (Sections 4.2, 4.4, 4.5, 4.6, 4.7): martingale transforms, the optional sampling theorem, the upcrossing inequality, Doob's decomposition, Doob's inequality, Lp -convergence, maximum inequalities, L
^{2}-theory, uniform integrability, backwards martingales and the strong law of large numbers.

## M 387C (Engquist) Numerical Analysis: Algebra & Approximations

**Description**

This is the first part of the Prelim sequence for Numerical Analysis, and it covers development and analysis of numerical algorithms for algebra and approximation. The second part covers differential equations. Below is an outline of topics for M387C.

Numerical solution of linear and nonlinear systems of equations including direct and iterative methods for linear problems, fixed point iteration and Newton type techniques for nonlinear systems. Eigenvalue and singular value problems.

Optimization algorithms: search techniques, gradient and Hessian based methods and constrained optimization techniques including Kuhn-Tucker theory.

Interpolation and approximation theory and algorithms including splines, orthogonal polynomials, FFT and wavelets.

Numerical integration and differentiation including Monte Carlo methods.

## M 392C (Ben-Zvi) Algebraic Geometry

**Description**

This will be a first course in modern algebraic geometry, largely following the textbook by Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry. Some familiarity with basics of category theory and commutative algebra recommended.

## M 391C (Radziwill) Introduction to Analytic Number Theory

**Description**

This course will be an introduction to analytic number theory. We will focus on multiplicative and additive aspects. As far as multiplicative number theory is concerned we will cover the prime number theorem, the Bombieri-Vinogradov theorem, properties of the Riemann zeta-function and L-functions, sieve theory and the method of bilinear forms. We will also cover some of the main tools of additive number theory: namely the circle method and methods for bounding exponential sums and see how these tools are applied in practice, for instance to proving Birch's theorem or studying rational points lying close to curves. While we will cover the basics I will also emphasize the modern directions of the field; e.g how the circle method can be used to count points on varieties (e.g Birch's theorem, Manin's conjectures) and how the knowledge of the analytic properties of L-functions is relevant to anybody interested in automorphic forms or the broader Langlands program.

The anticipated start date is September 14. Please contact Prof. Radziwill at

## M 392C (Freed) Differential Geometry

**Description**

Differential geometry is the application of calculus to geometry on smooth manifolds. Felix Klein's Erlangen program defines geometry in terms of symmetry, and in the first part of the course we delve into its manifestation in smooth geometry. So we begin with basics about Lie groups and move on to the geometry of connections on principal bundles. We focus in particular on the bundle of frames and geometric structures on manifolds.

Armed with this general theory, we can move in many directions. Possible topics include Chern-Weil theory of characteristic classes; topics in Riemannian geometry, symplectic geometry, and spin geometry; differential equations on manifolds; curvature and topology. Students' interest will influence the particular topics covered.

**Prerequisites:** Familiarity with smooth manifolds and calculus on smooth manifolds at least at the level of the prelim class.

## M 392C (Kileel) Geometric Methods in Data Science

**Description**

This is a graduate topics course on geometric methods in data science. Data sets in applications often have interesting geometry. For example, individual data points might consist of images or volumes. Alternatively, the totality of the data may be well-approximated by a low-dimensional space. This course surveys computational tools that exploit geometric structure in data, as well as some of the underlying mathematics.

The syllabus will adapt to the interests of course participants, but we plan to survey some of the following topics:

- principal component analysis (including streaming PCA),
- tensor decomposition and latent variable estimation (higher-order PCA),
- clustering methods (k-means and mixture of Gaussians),
- non-linear dimensionality reduction (manifold learning),
- numerical optimization (semi-definite programming and possibly semi-algebraic optimization or manifold optimization).

Time permitting, we might look at applications of group theory in data science.

We will be reading excerpts from important papers and monographs. Students will present some fraction of the lectures (with coaching from the instructor), write up lecture notes, and submit a final project with a written report. For the project, students may choose between applying methods to real data sets or writing a synopsis of a theoretical paper. For real data sets, possible sources include signal processing, microscopy or computer vision applications, among others.

The course's main prerequisites are linear algebra, basic probability, and mathematical maturity. Programming familiarity (or willingness to learn) will help with certain projects. A few elements of differential and algebraic geometry will be developed along the way.

## M 392C (Bowen) Hyperbolic Geometry and Teichmüller Theory

**Description**

The aim of this course is to give students a working knowledge of hyperbolic geometry and Teichmuller spaces. Topics may include: Hausdorff dimension of limit sets, Patterson-Sullivan theory, geodesic flows, quasi-conformal maps, quadratic differentials, flat surfaces, measured laminations, Weil-Peterson symplectic form, pseudo-Anosov maps, mapping class groups, Teichmuller flows, quasi-Fuchsian surfaces, hyperbolic 3-manifolds that fiber of the circle, Surface Subgroup Theorem, Agol’s virtual fibering Theorem.

**Prerequisites:** Students are expected to be familiar with measure theory, L^p spaces and smooth manifolds. Otherwise, prerequisites will be kept to a minimum.

**References**

**Marden, Albert **Hyperbolic manifolds. An introduction in 2 and 3 dimensions. *Cambridge University Press, Cambridge,* 2016. xviii+515 pp. ISBN: 978-1-107-11674-0

**Casson, Andrew J.; Bleiler, Steven A.** Automorphisms of surfaces after Nielsen and Thurston. **London Mathematical Society Student Texts, 9.** *Cambridge University Press, Cambridge,* 1988. iv+105 pp. ISBN: 0-521-34203-1

**Farb, Benson; Margalit, Dan** A primer on mapping class groups. **Princeton Mathematical Series, 49. ***Princeton University Press, Princeton, NJ,* 2012. xiv+472 pp. ISBN: 978-0-691-14794-9

**Gardiner, Frederick P.; Lakic, Nikola** Quasiconformal Teichmüller theory. **Mathematical Surveys and Monographs,** **76.**** ***American Mathematical Society, Providence, RI,* 2000. xx+372 pp. ISBN: 0-8218-1983-6

**Matheus Silva Santos, Carlos** Dynamical aspects of Teichmüller theory.

*SL*(2,ℝ)-action on moduli spaces of flat surfaces. **Atlantis Studies in Dynamical Systems, 7.** *Atlantis Press, [Paris]; Springer, Cham,* 2018. xiv+122 pp. ISBN: 978-3-319-92158-7; 978-3-319-92159-4

**Kapovich, Michael **Hyperbolic manifolds and discrete groups. Reprint of the 2001 edition. **Modern Birkhäuser Classics.** *Birkhäuser Boston, Ltd., Boston, MA,* 2009. xxviii+467 pp. ISBN: 978-0-8176-4912-8

## M 392C (Allcock) Lie Groups

**Description**

**Material:**** **We will cover the basic theory of Lie Groups and Lie algebras, from the Lie correspondence through the classification theorem (over the complex numbers) and highest weight modules. If there is time, then I would also like to talk about Kazhdan's property (T), and/or the structure theory of real Lie groups and algebras. This structure theory tells you how to think about real Lie groups, but we will not be able to cover their classification. This theory is a prerequisite to understanding infinite-dimensional representations of Lie groups, but we will not be able to cover any of that, either.

**Prerequisites:** I will assume you are comfortable with the material from our first year graduate courses in topology and algebra. From topology, you absolutely need fluency with the fundamental group, covering spaces, and the language of differentiable manifolds. We will use deRham coholomogy a little bit, but you could get by with just a high-level understanding of it. From algebra you need fluency with group theory and multilinear algebra (including bilinear and Hermitian forms, and tensor products). No Galois theory will be needed, only a tiny bit of commutative algebra, and nothing with a ground field other than the real or complex numbers. From analysis we need only a little: enough to understand statements about Haar measure, and maybe what a Banach space is.

**Textbook:** notes prepared by me.

## M 392C (Gualdani) PDE I

The topic of the graduate course will be on Nonlinear partial diﬀerential Equations, with emphases on elliptic and parabolic equations.

**Second order Elliptic equations**

- Existence of weak solutions
- Interior and exterior regularity
- Maximum principle
- Harnack’s inequality
- Cacciopoli’s inequality and applications
- Campanato’s spaces
- Schauder’s theory
- Hölder regularity estimates
- Calderon Zygmund decomposition

**Second order Parabolic equations**

- Existence of weak solutions
- Regularity
- Maximum principle
- Schauder’s estimates

**Applications to kinetic equations**

- Landau equation
- Landau-Fermi-Dirac equation

**Bibliography**

L.E. Evans, *Partial Diﬀerential equations*

E. H. Lieb and M. Loss, *Analysis*

G. M. Lieberman, *Second order parabolic differential equations** *

O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva, *Linear and quasilinear equations of parabolic type** *

*D. *Gilbarg, N.S. Trudinger, *Elliptic Partial Differential Equations of Second Order*

## M 393C (Maggi) Minimal Surfaces from the viewpoint of Geometric Measure Theory

**Abstract:** The course addresses the study of minimal surfaces from the viewpoint of Geometric Measure Theory. A basic goal is creating a theory of non-smooth surfaces which is flexible enough to contain limits of sequences of minimal surfaces under natural geometric bounds. Another direction is using the compactness theorems so obtained to apply variational methods to the study of minimal surfaces, for example, in proving the existence of minimal surfaces satisfying certain sets of constraints. Finally, we shall address the regularity problem for the generalized minimal surfaces created in the process.

## M 393C (Tsai) Mathematics in Deep Learning

**Description**

**We will discuss some key mathematical ingredients in the components of a typical deep learning algorithm.**The components include (but not limited to): approximation theory of neural networks, including new theories connecting data and initialization and over-parameterization, theory of optimal control, numerical optimization, and optimal transport.

A survey of the prominent deep learning models will be provided.

This course should be regarded as a numerical analysis course.

The course will be conducted with a mixture of regular and student-led lectures, and discussion.

Participants of this course are expected to present/lecture on relevant concepts from suggested reading assignments.

**Prerequisites**

**For undergraduate students who want to enroll in this class, please talk to the instructor. **The following qualification is highly recommended:

M341 or 340L with a grade of at least B.

M348, M368K with a grade of at least B.

M365C with a grade of at least B.

## M 394C (Zariphopoulou) Stochastic Processes I

**Description**

The course offers a comprehensive study of Ito-diffusion processes, stochastic calculus and stochastic integration. It covers stochastic differential equations and their connection to classical analysis. It also exposes the students to optimal stochastic control of diffusion processes, the Hamilton-Jacobi-Bellman equation (classical and viscosity solutions), singular stochastic control and linear filtering. Applications, mainly from mathematical finance, inventory theory, decision analysis and insurance will be presented. If time permits, the course will offer a brief overview of multi-scale problems in stochastic analysis.

*Topics*

- Review of fundamental concepts in probability
- Martingales and filtrations
- Brownian Motion
- Stochastic Integration
- Stochastic Calculus
- Stochastic Differential Equations
- Feynman-Kac formula and connection with linear PDE
- Introduction to Optimal Stochastic Control
- Introduction to Singular Stochastic Control
- Filtering
- Applications

* *

*Background*

- Knowledge of Probability and fundamental concepts of Stochastic Processes.
- Measure Theory and Real Analysis are highly recommended.
- The level of the class will be mostly similar to the one in references 1-3 below.

*Suggested readings besides the class notes*

*“An introduction to Stochastic Differential Equations”, L. C. Evans.**“Stochastic Differential Equations”,*B. Oksendal (6^{th}edition)*“Brownian Motion and Stochastic Calculus”,*I. Karatzas and S. Shreve*“The theory of Stochastic Processes, I”,*I. Gihman and A. Skorokhod*“The theory of Stochastic Processes, II,*I. Gihman and A. Skorokhod*“Controlled Markov Processes and Viscosity Solutions”,*W.H. Fleming and M. Soner

*Some organizational issues about the course:*

- Class notes will be distributed via email. Other readings will be suggested and distributed later on.

- Homework will be assigned every week with strong expectation to be completed (and returned to me) within a week.

- Two take-home exams will be given.

- The grade will consist of the take-home exams (2x20%) and the HW assignments (60%).