Why is Topology MA342 relevant to Maths students?Topology can be fun. It is also a major branch of mathematics, as demonstrated by the number of Fields Medals awarded to topologists such as Atiyah, Donaldson, Freedman, Jones, Milnor, Mumford, Novikov, Perelman, Quillen, Serre, Smale, Thom, Thurston, Voevodsky ... . The module MA342 tries to give students a taste of this vast subject.
If you'd ike to do a final-year project in topology then please get in touch.
Why is Topology MA342 relevant to Computer Science students?In the last decade or so, topologists have been trying to harness the power of modern computers to apply topological ideas to problems in science and engineering. The aim is to use the deformation invariant notions of topology to provide qualitative answers to problems; see, for instance, details of the research network on Applied Computational Algebraic Topology . The module MA342 tries to hint at these applications through a discussion of Euler characteristics of digital images and Euler integration in sensor networks.
If you'd ike to do a final-year project in computational topology then please get in touch
Why is Topology MA342 relevant to Financial Maths & Economics Students?Fixed point theorems play an important role in theoretical economics; see, for instance, the textbook Fixed point theorems with applications to economics. The module MA342 provides the outline of a proof of Brouwer's fixed point theorem and an explanation of how Brouwer's theorem can be used to prove the existence of Nash equilibria. This latter notion is due to the mathematician John Nash who was awarded the Nobel Prize for Economics for his work in this area.
If you'd ike to do a final-year project on applications of fixed pont theory in economics then please get in touch.
Why is Topology MA342 relevant to Mathematics & Education Students?Much of school mathematics focuses on procedural tasks: teach children the procedures for calculating answers to problems and then test their ability to do mathematics by asking them a range of problems to which the procedures can be applied. The core Maths modules in the Mathematics & Education BA programme also tend to focus to a large extent on procedural mathematics: evaluate a multiple integral; evaluate a complex integral, calculate the inverse of a matrix; determine a probability using Bayes' Rule; decipher an encrypted message by first using Euclid's algorithm to solve a system of equations; use differentiation to calculate the maximum/minimum value of some quantity; ... .
Project Maths has been introduced into schools with the noble aim of complementing childrens' procedural knowledge of mathematics with a strong conceptual knowledge. One difficulty facing teachers of Project Maths is: how can a child's conceptual knowledge of a topic be developed, and how can it be reliably assessed?
The MA342 module is primarily concerned with developing students' conceptual knowledge of a particular area of mathematics. Even though topology, per se, is unlikely to enter into the Project Maths curriculum in the near future, the module should give students some ideas for developing and assessing conceptual mathematics.
lectures will be based on the first five chapters of the
text: Basic Topology by M.A.
Armstrong, Undergraduate Texts in Mathematics, Springer-Verlag.
The book emphasizes the geometric motivations for topology and I
recommend that you take a look at it. There are some copies in
the library. The text:
Topology and Groupoids by R. Brown is also a great book which
covers similar material and should be consulted too. It is available online here.
(Another good topology book is Undergraduate Topology - A Working Textbook by Aisling McCluskey and Brian McMaster, Oxford University Press . This text places more emphasis on the importance of topology in analysis. It is available here and I think there are a few copies in the library.)
Assessment: The end of semester exam will count for 70% of the assessment. The continuous assessment will count for 30% of the assessment.
Continuous Assessment: The continuous assessment will consist of three equally weighted in-class tests based on the homework/tutorial problems. Each test will consist of questions taken verbatim from the homework/tutorial sheet. Many questions on the end of semester exam will also be taken, almost verbatim, from the homework/tutorial sheet too.
Homework sheet will be available here.
Lectures take place at 12 pm Monday in the McMunn Theater and 12pm Wednesday in AC215. The lecturer is Graham Ellis.
Tutorials take place at 2pm Wednesday in AC201 and Wednesdays at 6pm Wednesday in AC202. The tutor is Adib Makroon.
Student feedback on this module will be posted here.
Previous years: The material covered in the module this year and last year is a bit different to previous years. For that reason the exam questions will be very closely based on the homework/tutorial sheet.