Discrete Mathematics and Computer Algorithm Theory: Calculus, Mathematical Analysis and Topology

Course 11: Discrete Mathematics and Computer Algorithm Theory: Calculus, Mathematical Analysis and Topology

I. Course Description

Topologies are flexible geometries. In geometry, an object is considered inflexible, but in topology, you can stretch and bend it without bonding or cutting it. Thus, from a topological perspective, the shape of the Earth is a sphere, even if it is not completely circular, and therefore, geometrically speaking, it is not a sphere. One of the central questions of topology and general relativity is what is the shape of the universe that we live in. Construction logic and more general construction mathematics process objects, such as numbers, which can be generated as the output of certain finite computer algorithms. Any computer algorithm can be represented by its binary code, so the set of all possible numbers that can be obtained in this way can be enumerated with positive integers. By definition, this means a set of all algorithms and therefore all construction numbers are countable. The set of all rational numbers are also countable but surprisingly the set of all irrational numbers is uncountable.

In this course, we will first introduce the concepts of sequence and series, and then learn some basic metric space and point set topologies. Next, we explore constructive mathematics, enumable sets and decidable sets, focusing on the well-known downtime problems and the existence of enumumable but undecidable sets and unscalable computer algorithms.

II. Professor Introduction

Vladimir Chernov – Tenured Professor at Dartmouth College

Vladimir Chernov Professor is currently deputy Dean of the School of Mathematics at Dartmouth College and a tenured professor of the School of Mathematics at Dartmouth College. He is also chairman of the Dartmouth Syer Prize Mathematics Competition Committee and a member of the Dartmouth Committee. In addition, Professor Vladimir Chernov is also a faculty advisor to the Undergraduate Mathematics Society at Dartmouth College, providing teaching guidance and support to other teachers.

He has also twice won the Simmons Foundation “Mathematicians Cooperation Award” award, which is awarded to three or four mathematicians who are at the peak of creation, leading the research field, creating a new direction, and being effective in training young scholars. He has published several important academic papers, such as “Minimizing the number of intersection points under virtual Homotopic”, “causality and high-dimensional space and time”.

III. Syllabus

1. Column and limit
2. Series and convergence
3. The Taylor series and the basis of the analysis
4. Continuity and complete space
5. Metric space and topological space
6. Homomorphic and countable sets
7. Structural mathematics and computability
8. Undecidability and downtime problems
9. Structural real numbers and functions
10. Research projects and applications