Textbook: Complex Analysis by Lars Ahlfors, 3rd edition
This course covers Sections 1.11.2, 2.12.3, 3.22.3, 4.14.6, 5.15.3, 5.5, and 6.1 of Ahlfors’ book. It contains the material necessary for passing the Qualifying Exam in Complex Analysis for PhD students at IUPUI.
Textbook: Linear Algebra by Howard Anton, 11th edition
This course covers Chapters 1-6. It uncludes the following topics: Systems of linear equations, matrices, vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and applications.
Textbook: Reading, Writing, and Proving by Daepp and Gorkin, 2nd edition
3 credits, Chapters 124 (some chapters will be omitted). The core material for this course is drawn from basic set theory with the ultimate goal of learning how to compare sizes of different infinite sets. The material provides you with good foundation for courses in abstract algebra and real analysis. In spirit, the course is designed to help you bridge the gap between elementary and advanced courses in mathematics, move from solving problems to rigorously proving mathematical statements.
Textbook: Discrete Mathematics (Elementary and Beyond) by Lovász, Pelikán, and Vesztergombi
3 credits, Chapters 113 and 15. The class includes the following topics: Basic Logic, Counting and Induction, Discrete Probability, Integer Algorithms, and Graph Theory.
Textbook: Differential Equations by Blanchard, Devaney, and Hall, 4th edition (primary text)
Diffy Qs by J. Lebl, version 6.0 (secondary text, open source, can be found here)
3 credits, Chapters 14 and 6 (some sections will be omitted). Many basic principles in physics, engineering, economics, and other fields are expressed mathematically in terms of equations involving a function of one variable and its derivatives. This goal of this course is to familiarize students with solution techniques for such equations and develop an intuition for solution properties. Applications of differential equations in mathematical models will be frequently discussed in the course. The class includes the following topics: first and secondorder differential equations; systems of firstorder linear differential equations; qualitative analysis of solutions for differential equations; Laplace transforms; numerical methods.
Textbook: Multivariable Calculus by James Stewart, 7th edition
This is the second course in a 4course sequence for Math, Science, and Engineering majors (MATH 16500 16600 17100 26100). This course is equivalent to IU MATH M216. Topics include transcendental functions, techniques of integration, indeterminant forms and improper integrals, basic differential equations, polar coordinates, sequences, infinite series, and power series.
Textbook: Calculus by James Stewart, 7th edition
This is the first course in a 4course sequence for Math, Science, and Engineering majors (MATH 16500 16600 17100 26100). This course is equivalent to PU MATH 16300 and IU MATH M215. Topics include plane analytic geometry and trigonometry, functions, limits, differentiation and applications, integration and applications, and the Fundamental Theorem of Calculus.
Textbook: Ordinary Differential Equations by M. Tenenbaum and H. Pollard;
Ordinary Differential Equations by T. MyintU (secondary)
This course covers:
• basic concepts: definition and solutions of ODEs; directional fields
• special types of the first order ODEs: ODEs with separable variables; ODEs with homogeneous coefficients; exact ODEs; integrating factors
• series method for the first order ODEs: Taylor series and convergence of series; linear ODEs; general first order ODEs
• existence theorem: Picard’s approximations; convergence of sequences of functions • existence theorem; continuity theorem
• higher order linear ODEs: basic facts; Laplace transform and Gamma function; linear ODEs with constant coefficient; reduction of order method; power series method at regular and singular regular points
• systems of ODEs: basic facts; linear systems; exponential of a matrix; Jordan canonical form of a matrix; linear systems with constant coefficients; linear systems with periodic coefficients; stability of autonomous systems
• physical problems leading to ODEs (independent study).
Textbook: Multivariable Calculus by James Stewart, 6th edition
This course continues the study of multivariable calculus and focuses on integration of functions of several variables. The course, which is the second in the sequence, covers Chapters 16 and 17 of Stewart.
Textbook: Calculus by Michael Spivak, 4th edition
This course is third in the series of three and tentatively covers Chapters 2030 of the book. The material includes application of the approximation by polynomials, infi- nite sequences and series, uniform convergence, basics of the complex function theory, and construction of real numbers.
Textbook: Calculus by Michael Spivak, 4th edition
This course is first in the series of three and tentatively covers Chapters 1119 of the book. The material includes application of the differentiation, inverse functions, trigonometric, exponential and logarithmic functions, and the theory of integration.
Textbook: Calculus by Michael Spivak, 4th edition
This course is first in the series of three and tentatively covers Chapters 111 of the book. The material includes basic properties of numbers, definition and properties of functions, limits, continuity, properties of continuous functions, and differentiation.
Textbook: Discrete and Combinatorial Mathematics by R.P. Grimaldi
This course continues the introduction to the subject of discrete mathematics. Topics include: graphs and trees; optimization and matching; groups, rings, and fields. The course, which is the third in the sequence, covers most of Chapters 1114 and parts of Chapters 16 and 17 of Grimaldi.
Textbook: Discrete and Combinatorial Mathematics by R.P. Grimaldi
This course continues the introduction to the subject of discrete mathematics. Topics include: abstract theory of functions and relations; finite state machines; counting via the principle of inclusion and exclusion; counting via generating functions; first and second order linear recurrence relations. The course, which is the second in the sequence, covers most of Chapters 510 of Grimaldi.
Textbook: Discrete and Combinatorial Mathematics by R.P. Grimaldi
This course introduces students to the subject of discrete mathematics. Topics include: fundamental principles of combinatorics; elementary logic; basic set theory; introduction to discrete probability; integer arithmetic. The course, which is the first in the sequence, covers most of the first four chapters of Grimaldi.