## cSplash 2017

A First Glance into the World of the AbstractIn this talk, I want to talk about plain, simple algebra. I'm not going to talk about polynomials or roots or anything like that. I'm going to get a little more abstract. This lecture will be an introduction into the subject of Modern Algebra. We will define what a group is, lay out the groundwork for some important Group theory results and then delve into specific examples of groups such as the cyclic group, the symmetric group, and the dihedral group. Applications of Mathematics to the Problems in InsuranceThe problems in determining insurance premiums for different kinds of insurance policies and the amount an insurance policy has to keep in reserve to stay in business involves the application of different branches of mathematics, namely algebra, calculus, probability and statistics. This talk will focus on these areas and will include an example in automobile theft insurance, which deals with the problem of how an insurance company manages to insure automobile owners against the risk of theft. Dark Matter!In the 1970’s, Vera Rubin and her collaborators discovered and established the “Dark Matter Phenomenon”. Despite four decades of intense effort, the nature of Dark Matter remains a major unsolved mystery of particle physics and astrophysics. This talk will highlight some particularly puzzling features and current theoretical, observational and experimental efforts to understand Dark Matter. The Dark Matter Phenomenon also presents a sociological puzzle: why wasn't a Nobel Prize awarded for its discovery? Data Analytics and VisualizationTechnological advancement has empowered humans to achieve the impossible. Whether it's about self-driving vehicles, predicting loan delinquency, forecasting demands. The list is endless. All this is a result of capturing and leveraging data in an intelligent manner. In this class, we'll talk about the different qualitative and quantitative techniques and processes used to derive insights from data and generate elegant visualizations to help us make strategic business decisions. Entropy factor, or why improbable events happen.Very often we hear about coincidences that seem bizarre or at least highly unlikely, such as unexpected encounters with long lost friends at remote locations. Isn't this ubiquity of improbable events odd? If an event is unlikely, doesn't this exactly mean that it happens rarely? I will try to explain what is going on with the help of basic models of probability theory. |
Exploring the Hidden with Maths--An Intro to Inverse ProblemsWhy do we know how the earth looks inside? How are we able to image the insides of our body without surgery? How do we find oil under the oceans? We know these things from the combination of mathematical models with indirect observations. For instance, from combining a model how earthquake waves travel through the earth with seismograms, we are able to learn about the inside of our planet. Or, from a model how hydrogen atoms react to a magnetic field, we can generate detailed images of the human brain. These are examples of inverse problems. Fibonacci and the Golden Ratio in FlowersDo you know of Vi Hart? Her YouTube videos on doodling, flowers, and the Fibonacci sequence are fun to watch! Try watching "doodling in math: spirals, fibonacci, and being a plant" at www.youtube.com/user/Vihart to see fascinating connections to plants. In our talk, we will pick some of what she mentions about the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... and phi (the golden ratio), examine them in depth, and provide a proof or two. How to Talk to Strangers: An Introduction to Public-Key CryptographyPublic-key cryptography allows people who have never met or coordinated before to securely communicate with one another. To motivate its discussion, we set the stage with some simpler methods of clandestine communication and show what public-key cryptography brings to the table. Introduction to Coding TheoryCoding Theory is the study of error-correcting codes. When a message is sent through a channel, noise can alter the message so that the data received is different from the data sent. This noise can be anything from human error or faulty transmission equipment to scratches on a CD. Error-correcting codes encodes the message in a way such that errors can be detected and fixed. Introduction to ProofsThe purpose of this talk is to introduce high school students to the two basic mathematical proofs needed for college-level mathematics: Proof by Contradiction and Proof by Induction. This foundational knowledge is necessary for every advanced mathematics class, but is not always formally taught. In the end, I will also introduce Cantor's Diagonalization Proof on the different sizes of infinity to show different types of elegant proofs mathematicians have constructed in the past. |