General
Teachers
- Kalle Kytölä
Teaching assistants:
- David Adame-Carrillo (Head TA)
- Kai Hippi (TA)
- Teemu Tasanen (TA)
Practicalities
- The lectures will be in the form of Zoom-meetings on Mondays 10-12 and Wednesdays 14-16. See the Lectures tab for zoom links.
- The exercise sessions will be in the form of Zoom-meetings. There are five groups, each with 2h/week exercise session.
- The course discussion forum is a Zulip chat hosted by Aalto CS. It will be used for announcemnts and discussions about the course topics, lectures, exercises, practicalities, etc., and some quizzes and polls will be held there. You can and should login with your Aalto account. The data is hosted at Aalto University servers, but the policy is to not post any sensitive information in this discussion forum; see the general rules of zulip.cs.aalto.fi.
Description of the course
This course is about topological properties of spaces, including the familiar n-dimensional Euclidean space \(\mathbb{R}^n\) but also for example function spaces arising in applications. Topological concepts include, e.g., limits of sequences and continuity of functions, familiar already from basic courses. Somewhat more advanced topological notions treated in this course are connectedness ("the space can not be torn apart to separate pieces"), compactness ("the space has no infinite directions to escape towards"), and completeness ("sequences that look convergent actually have limits").
A central theme of the course is a (modest) increase in the level of abstraction, particularly for the purposes of mathematical analysis. An increased level of abstraction brings about the following benefits, in particular. The desired results are derived with more general (less restrictive) assumptions, so that a similar result does not have to be proven for each particular application case separately. Also as the reasoning can only resort to the minimal amount of assumptions, it conveys the essence of the result in a clearer manner. The increase of the level of abstraction requires careful logical reasoning, to avoid being mislead by intuitive ideas or particularities of specific concrete cases. Intuition is, of course, a key to coming up with the correct general ideas in the first place, but it eventually has to be turned into rigorous logical arguments. Counterexamples are a useful tool of abstract reasoning, as they can explicitly point out in which ways intuition could be misleading.
The topic of the course has applications in various fields of mathematics. For example the convergence of sequences and series generalized to function spaces is necessary for the theory of Fourier series and Fourier transforms, which in turn is a key tool in signal processing, among other application areas. In statistics, on the other hand, one often needs limits of random variables and probability distributions, necessitating working in spaces of random variables or probability measures. Geometry, with its own applications, is a close cousin of topology to start with. And for example theories of particle physics feature continuous symmetries, and topological properties largely dictate to what extent the analysis of infinitesimally small symmetry transformations yields results about all symmetry transformations. Although we do not elaborate on the application areas, the topological notions studied in the course underlie almost all of mathematics. (To see this, you may want to check how many MSc level courses assume MS-C1540/MS-C1541 as a prerequisite).
Course contents and learning objectives
Key topics of the course include:
- real numbers, metric, norm, inner product, open and closed sets, continuous mappings, sequences and limits, compactness, completeness, connectedness.
Upon successful completion of this course, you. . .
- . . . recognize what structure different spaces are equipped with (e.g. inner product, norm, metric, topology, vector space structure, ...);
- . . . know the definitions and fundamental properties of continuous functions, convergent sequences, compact sets, completeness, and connectedness, required in advanced studies in mathematics;
- . . . can perform rigorous logical reasoning (specifically with concepts of metric space topology);
- . . . can apply abstract and general results (about metric spaces in particular) to concrete and specific cases(including the Euclidean space \(\mathbb{R}^n\) and various function spaces).
Prerequisites
The course starts from mathematical foundations, and therefore logically relies on minimal prerequisites. A broader prior familiarity with mathematics is of course helpful for putting the abstract results in context and for appreciating the achieved generalizations.
A sufficient prior knowledge is a good command of:
MS-A01XX Differential and integral calculus 1
Completing the course
It is possible to take the course based on either exercises and course exam (KT), or final exam only (T0 or T1). The grade is based on the maximum of the following two options:
- exercises 50% + course exam 50%
- final exam 100%.
The final exam (T0) held simultaneously with the course exam
has mostly overlapping problems, so the student can attempt both completion options simultaneously, and the one leading to better grade will be considered.
The exam arrangements for KT and T0 on 25.2.2021 are detailed in the Exam section.
- KT: February 25, 2021
- T0: February 25, 2021
- T01: April 16, 2021
- T02: May 31, 2021