Mathematics

The course discusses integration and differentiation in a mathematically rigorous manner. It also discusses the series of functions and their convergence.

The course covers the basic model theory and proof theory of 1st order languages, the Gödel Completeness Theorem and the Godel Incompleteness Theorems characterizing the non-provability of the consistency of a formal system within that system. These theorems are the foundations of 20th century logic.

This course looks at problems that are associated with discrete rather than continuous situations. So the nature of the problems is quite distinct from those that are considered in a calculus paper because the important underlying set is the integers rather than the sets of real or complex numbers. The curriculum includes a selection from the following topics: combinatorics, counting techniques, logic, graph theory, set theory, relations, number theory. There will be an emphasis on both proof techniques and practical algorithms.

This course introduces and develops some of the basic ideas in the areas of Combinatorics and Number Theory. Topics include: mathematical induction, permutations and combinations, counting arguments, modular arithmetic, Euclidean algorithm, Fermat's and Euler's theorems, fundamental theorem of arithmetic, systems of linear congruences, and the Chinese remainder theorem. Students learn to recognize, read, and use standard mathematical symbols and notation. Students learn to ask pertinent questions, to decide which questions are relevant, answerable, and so on. Students gain an understanding of the reasoning behind any methods or procedures they use and are able to demonstrate that understanding. Students also learn to produce examples themselves, in order to illustrate a definition, show a method, or test boundaries of an idea.

This course provides a mathematical background essential for understanding the theory behind various machine learning techniques. After this course, students are able to design and execute machine learning algorithms such as regression analysis, classification with support vector machine, feedforward neural network, principal component analysis, k-means clustering, etc. without relying on pre-programmed packages. Topics of this course include (1) linear algebra (basic and advanced), (2) probability and information theory, (3) analytical geometry, (4) calculus (basic and advanced), (5) optimization, (6) machine learning applications: distance-based classifiers, Naive Bayes classifier, linear & logistic regression, neural network, SVM, PCA, k-means clustering, etc.

Probability theory, the mathematical description of chance, is a subject in its own right but also the bedrock on which statistics and data science are built. We are surrounded by important questions involving chance but our intuition on the subject is often wrong. This course gives an understanding of the subject that help students understand issues where chance plays a central role as well as preparing them for further study.