Presentation topics, Mathematical Computation, Math 9171

Each student will give one presentation near the end of the semester. (See the timeline at the bottom of this page.) All students are expected to attend all presentations and to arrive on time.

Possible Topics: These are suggestions, but you can also propose other topics. Topics need to be discussed with me and approved. You should choose a topic that is not something you already know about. I can give more information about the topics and can suggest further references. You should also do some research about the topics.

Presentations should not just be a tutorial about some software. They should contain at least one part that is mathematically substantial. For example, it could be a proof of a theorem, or a description of a non-trivial algorithm for accomplishing a task (which might require a proof that it works, or a proof that it runs efficiently).

Many of these can be studied using Sage, but there are also other software packages that might be more appropriate. Or maybe your topic can be presented well without any software.

Number theory (software/algorithms).
Cryptography, e.g. RSA, lattice-based crypto, elliptic curve crypto, etc.
Integer factorization algorithms.
Grobner bases.
Combinatorics and/or graph theory (software/algorithms).
Linear programming, an important optimization technique.
Polytopes (software/algorithms).
SAT solvers, e.g. explaining the methods they use to efficiently solve many instances.
Numerical integration and differential equations (software/algorithms).
An introduction to another mathematical software package, such as Macaulay, Singular, Maxima, etc. (But you need to cover something mathematically substantial as well.)
Machine learning.
Topological data analysis (studying large data sets using algebraic topology).
Kenzo, a powerful tool for doing computations in algebraic topology. Available as part of Sage.
SageManifolds, or some other software for studying manifolds. (But it's tricky to explain this in one talk, and also to find something substantial to say.)
Knot theory and links (software/algorithms).
Quantum computing.
Functional programming, lazy evaluation, with Haskell as an example.
Prolog, a language for encoding problems using logic.
The unsolvability of the word problem for finitely presented groups, and/or other problems in mathematics: group isomorphism, Hilbert's tenth problem, homeomorphism of finite simplicial complexes, the Mortal matrix problem, etc.
Prove that a particular problem is NP-complete. I cover BUH and SAT in class.
Other complexity classes besides P and NP, and what is known about them.
A single algorithm due to Marcus Hutter that solves any well-defined problem in asymptotically optimal time, up to a factor of 5 and some lower order terms.

Duration: 40-45 minutes. The presentations are not long, so you will need to carefully select the appropriate amount of material to present. You should focus on the key ideas, with illustrative examples, motivation, necessary background, and history (e.g. attributions and years). The presentations can involve writing on the board, prepared slides, computer software, or some combination.

Grading: The presentations will be worth 40% of the overall mark in the course. They will be graded on:

Outline and organization: Well-organized; appropriate choice of topics and amount of material; good outline, handed in on time.
Knowledge of material. Be prepared to answer questions.
Clarity and style of presentation: speaking clearly, looking at audience, giving clear explanations, etc.
Blackboard/tablet use (if you use them): Use boards in order, don't erase what you've just written, don't stand in front of what you've written, legible, enough words written, etc.
Slides/software (if you use them): Well-designed slides, not too packed with content; designed to focus on the important material; not flipped by too fast.
Duration: End within the correct duration and go at an appropriate pace. You might want to build some flexibility into the end of your presentation so you can adjust on the fly. And take into account that there may be questions during your talk.

Note that knowledge of material is just a small part of the grade. The presentation itself is much more important. Because of this, you should practice the talk at least once or twice beforehand, with someone listening, and you should time how long it takes. This is extremely important. You should also address your presentation to your fellow students, not to me; students in the audience are strongly encouraged to ask questions during and after the talk.

I strongly recommend that you practice your talk at least once, to polish it and ensure that the timing is accurate.

Timeline:

Oct 6: start looking over the presentation topics.
In class on Oct 15 we'll discuss topics and dates. Bring at least two possible choices of topic.
Give me a brief outline (1 to 2 pages, in point form) ≥ 2 weeks ahead of your date.
Talks will take place roughly November 14 to December 8.

Course home page.