In this course, we study elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation. Then we study Fourier and harmonic analysis, emphasizing applications of Fourier analysis. We will see some applications in combinatorics / number theory, like the Gauss circle problem, but mostly focus on applications in PDE, like the Calderon-Zygmund inequality for the Laplacian, and the Strichartz inequality for the Schrodinger equation. In the last part of the course, we study solutions to the linear and the non-linear Schrodinger equation. All through the course, we work on the craft of proving estimates.

This course describes discrete mathematics, which involves processes that consist of sequences of individual steps (as compared to calculus, which describes processes that change in a continuous manner). The principal topics presented in this course are logic and proof, induction and recursion, discrete probability, and finite state machines. Upon successful completion of this course, the student will be able to: Create compound statements, expressed in mathematical symbols or in English, to determine the truth or falseness of compound statements and to use the rules of inference to prove a conclusion statement from hypothesis statements by applying the rules of propositional and predicate calculus logic; Prove mathematical statements involving numbers by applying various proof methods, which are based on the rules of inference from logic; Prove the validity of sequences and series and the correctness or repeated processes by applying mathematical induction; Define and identify the terms, rules, and properties of set theory and use these as tools to support problem solving and reasoning in applications of logic, functions, number theory, sequences, counting, probability, trees and graphs, and automata; Calculate probabilities and apply counting rules; Solve recursive problems by applying knowledge of recursive sequences; Create graphs and trees to represent and help prove or disprove statements, make decisions or select from alternative choices to calculate probabilities, to document derivation steps, or to solve problems; Construct and analyze finite state automata, formal languages, and regular expressions. (Computer Science 202)

The Elementary Math Education course was developed through the Ohio Department of Higher Education OER Innovation Grant. This work was completed and the course was posted in October 2019. Team LeadBradford Findell                                Ohio State UniversityContent ContributorsVictor Ferdinand                               Ohio State UniversityHea-Jin Lee                                      Ohio State University LimaJenny Sheldon                                  Ohio State UniversityBart Snapp                                       Ohio State UniversityRajeev Swami                                  Central State UniversityRon Zielker                                       Ohio Dominican UniversityLibrarianCarolyn Sanders                               Central State UniversityReview TeamAlice Taylor                                       University of Rio Grande

A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.

This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions.