It may be only a minor thing in the space of examples that you seem to be considering, but I have had a lot of success with my practice of requiring students in my graduate courses to write a substantial term paper on an original topic.
The aim is for them to undertake a simulacrum of the research experience. I definitely do not want them to just give me an account of some difficult topic on which they read elsewhere. Rather, we try to find a suitable original but manageable topic, which they will have to figure out themselves, and then write up their results in the form of a paper.
I insist that these term papers give the appearance of a standard research article, with proper title, abstract, grant or support acknowledgement, proper introduction, definitions, statement of main results and proof, with references and so on. Furthermore, I insist that the students use TeX, which I insist they learn on their own if they do not yet know it.
The most difficult part for the instructor is to find suitable topics. One rich source of topics is to take a standard topic that is well-treated elsewhere, but then make a small change in the set-up, giving the student having the task to work out how things behave in this slightly revised setting. For example, in a computability theory class, there is a standard definition of the busy beaver function, with many results known, but one can insist on a slightly different model of Turing machine (such as one-way infinite tape instead of two, or change the halt rule, or have extra symbols or extra tape), where the standard calculations are no longer relevant, but many of the ideas will have a new analogue in this new setting. But also there are usually many suitable topics if one just thinks with curiosity about some of the main ideas in the course and some relevant examples.
I always insist that the topics be pre-approved by me in advance, because I want to avoid the situation of a student just writing up something difficult they read, but rather have them really do real mathematical research on their own. Often, I meet with each student several times and we make some discoveries together, which they then work out more completely for their paper.
After students submit their final draft (I do not call it a first draft, since I want them to do several drafts on their own before showing me anything, and I don't want to look at anything that they regard as a "first draft"), then I give comments in the style of a referee report, and they make final revisions before submitting the "publication" version, which I sometimes gather into a Kinko's style bound issue Proceedings of Graduate Set Theory, Fall 2014 or whatever, and distribute to them and to the department.
Finally, on the last lecture of the course, we usually have student talks of them making presentations on their work. For example, see the student talks given for my course on infinitary computability last fall.
I think it works quite well, and gives the students some real experience of what it is like to do mathematical research. In a few exceptional cases, the terms papers have subsequently turned into actual journal publications, when the students got some strong enough and interesting enough results, and that has been really special.
The workflow for me is to assign normal problem sets in the early part of the course, and then start suggesting topics, with the students coming to me and we discuss possibilities. Then, as the work on the paper ramps up, the problem sets taper off, until they are submitted, with additional problem sets at the end of the course, except when they are making their revisions.
(And I never accept papers after the end of the course.)
Rudolf K. Luneburg, author of Mathematical Theory of Optics (1944), also published the book Mathematical analysis of binocular vision (Princeton UP, 1947) in which he argued that the "psychological space of binocular vision" carries a hyperbolic metric. From the Bull. AMS review:
Some of the topics which are treated in considerable detail are the horopter problem (geodesic lines), the alley problem, and rigid transformations of the hyperbolic visual space. (...) Using the hyperbolic metric, the author calculates the shape of distorted rooms which are congruent to rectangular rooms, that is, rooms with distorted walls and windows which appear (under fixed conditions) to be identical to rectangular rooms with rectangular windows.
In 1959, R. Penrose and independently J. Terrell showed that we won't perceive rapidly moving objects as Lorentz contracted but as rotated. Form V. Weisskopf's Physics Today review:
James Terrell (...) does away with an old prejudice held by practically all of us. We all believed that, according to special relativity, an object in motion appears to be contracted in the direction of motion by a factor $[1-(v/c)^2]^{1/2}$. A passenger in a fast space ship, looking out of the window, so it seemed to us, would see spherical objects contracted to ellipsoids. This is definitely not so according to Terrell's considerations, which for the special case of a sphere were also carried out by R. Penrose. The reason is quite simple. When we see or photograph an object, we record light quanta emitted by the object when they arrive simultaneously at the retina or at the photographic film. This implies that these light quanta have not been emitted simultaneously by all points of the object. (...) In special relativity, this distortion has the remarkable effect of canceling the Lorentz contraction so that objects appear undistorted but only rotated. This is exactly true only for objects which subtend a small solid angle.
Best Answer
David, I'd suggest to use physics or biology as targets.
I mean, try to build a bridge between your research area and the applications... which ultimately could turn into a technological or daily application.
Examples:
Differential geometry -> General relativity (gravitation) -> Fine corrections in GPS devices.
Lineal Algebra -> Quantum mechanics -> Transistors -> Computer and cell phones
Chaos theory -> Critical points -> improvement of Weather predictions
Dynamical systems -> (Population modelling ...)
Path integrals -> Financial market
And so on...
Good luck with the seminar... and enjoy it!!!
P.D.: Include graphics, short videos or simulations, cartoons... I'd also suggest you to watch the film Freakonomics, could help you.