One aspect of mathematics that I found beautiful is the surprising connections between different concepts that are seemingly unrelated (if you feel the same then you probably have heard of Proofs from THE BOOK, which I strongly recommend). I am interested in presenting such topics to the general audience with only basic backgrounds in mathematics.
Ideally, these topics should be understandable by undergraduate students and may not be commonly included in a standard undergraduate math course. An example of such a talk is this.
Other examples of such talks include the following:
Abstract : The principle of inclusion-exclusion is a fundamental theorem about counting, the fundamental theorem of finite differences is the discrete analogue of the fundamental theorem of calculus, while the Möbius inversion is a theorem in elementary number theory. What do they have in common? In this talk, I will show how they can be viewed as the same theorem using "triangular matrices". (Basic knowledge in linear algebra is assumed; knowing the Möbius inversion formula may make the listener more motivated, although it is not required.)
Abstract : In a regular calculus course, it is standard to first introduce the concept of limits for real-variable functions, then gradually derive the theory of differentiation and integration. What if we are intersted in functions that are only defined for the integers, for which the usual concept of limits no longer exists? Is it impossible to do calculus with them? In this talk, I will show that one can define differentiation and integration (very naturally) so that many "calculus" results (such as the power rule, the product rule, the fundamental theorem of calculus, and integration by parts, etc.) still hold for integer-variable functions. (Knowledge in one-variable calculus would be more than enough.)
Abstract : In 1850, Thomas P. Kirkman posted the following question: “fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast”. This is now known as Kirkman’s schoolgirl problem and is considered a (combinatorial) block design problem. Many block design problems have natural connections with graphs — a graph is a set of nodes connected by edges in some way (such as a city network). In this talk, we will give concrete examples of block design problems and the graphs behind them, and outline some ideas on how linear algebra may be used to solve related graph problems.