Mathematics

Algebraic topology is concerned with the construction of algebraic invariants associated to topological spaces which serve to distinguish between them. This course focuses on the concept of the fundamental group of a topological space, and discusses its relation to other important notions in topology such as homotopy, covering space, etc.  

Topics include homotopy of paths, covering spaces, the fundamental group of the circle, retractions and fixed points, the Borsuk-Ulam theorem, deformation retracts and homotopy type, the Jordan curve theorem, imbedding graphs in the plane, the winding number of a simple closed curve, the Cauchy integral formula, the Seifert-van Kampen theorem, the fundamental group of a wedge of circles, adjoining a two-cell, the fundamental group of the torus and the dunce cap, the classification theorem, equivalence of covering spaces, and existence of covering spaces. 

Prerequisite: Topology 1 

This is an independent research course with research arranged between the student and faculty member. The specific research topics vary each term and are described on a special project form for each student. A substantial paper is required. The number of units varies with the student’s project, contact hours, and method of assessment, as defined on the student’s special study project form.

This course is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, it develops the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. 

This course is a first introduction to algebraic topology, the area of mathematics in which algebra is used to study topological spaces. It defines the fundamental group and singular homology and studies their basic properties and applications. The course introduces foundational competencies in algebraic topology. Important concepts include homotopy, homotopy equivalence, fundamental group, covering space, chain complex, and homology. Prerequisites include knowledge about general topology and abelian groups, as obtained through courses such as Topology and Algebra 2, and Advanced Vector Spaces.
 

Analyze time series, to explain temporal components such as trend and seasonality. Identify the appropriate model for a time series and according to this make future data predictions.

This course examines mathematical language and techniques to unravel many seemingly unrelated problems. The course content addresses five major pillars of discrete mathematics: set theory, number theory, proofs and logic, combinatorics, and graph theory. 

This course is an introduction to computability theory and Gödel's incompleteness theorems. The first half of the course focuses on computability theory, and includes Recursive and primitive recursive functions; Turing machines and computable functions; basic results in computability theory including Kleene's Normal Form Theorem, the s-m-n Theorem, Kleene's Recursion Theorem, Recursively enumerable sets, the halting problem and decision problems in general; as well as hierarchy theory, relative computability, and Turing degrees. The second part of the course focuses on Gödel's first incompleteness theorem, and includes Axiom systems for number theory, representable relations and functions, arithmetization of syntax, the Fixed-Point Lemma, and Gödel's first incompleteness theorem, as well as Gödel's second incompleteness theorem.

Mathematics underpins virtually everything that we take for granted in our daily lives, and it is sometimes referred to as the “Queen of Science” due to its demands of logical rigor and cold calculations. However, despite its intimidating veneer, mathematics is the culmination of millennia of human endeavor. The purpose of this course is to give an accessible overview of some of the key developments in mathematics, covering the period from the time of the ancients, up to the early modern period. The course also provides an opportunity to apply historical mathematical methods to solve problems. While covering the well-known Greek, Chinese, Islamic, and European mathematicians, the course also addresses Japanese mathematics during the Edo period.　
 

While the course includes written assignments, to properly understand and follow the thinking of the mathematicians, the course covers problem solving using historical mathematical methods. While a background in high-school level math is useful, an enthusiasm for critical thinking and problem solving could replace that prerequisite, since the mathematical concepts will be introduced as they were historically considered.