Craig McGhee

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Contents

Early Life

My name is Craig McGhee. I am a Computer Science and applied Mathematics Major. I picked up the math major pretty late but I’ve loved every second of it. This is my last semester at NKU and I am excited to go out into the world to use what I have learned. I grew up in a military family and moved around the country. I had many different teachers with different teaching styles. Math was always one of my best subjects, but I never really enjoyed it. I am not sure why, but if I had to guess it had to do with the different styles of teaching. Math also felt abstract to me. It felt like I had to learn it because I had to learn it. My senior year in high school changed that thinking. I had a calculus teacher who was able to make math feel like a something integral to the world. After I graduated, I had an appreciation for math, but I still did not really want to devote my time to studying it further. When I went to college, I first started studying biology but after a while I decided that it was not for me. I the switched my major to Computer Science and started to gain a deeper appreciation for math. Computer science to me, feels like what engineering is to physics. As I started to take more math classes, I fell in love with it. I decided to double major in Computer Science and Applied Mathematics so I could could explore math further. In my time as a math major, I have had very interesting classes and have seen many different sides of math. I was able to explore things like game theory, linear algebra, and differential equations.

People Who Inspired Me

There have been many people who inspired me to gain a greater appreciation in math, but the most influential person was my high school calculus teacher, Mr. Parrott. He had a special way of teaching. Before he showed the formula, he would spend nearly the entire class period deriving it from a simple starting place, like a graph. This gave me a feeling of tangency to math. If I forgot a formula on a test or homework, I would remember the staring place of the derivation and be able to work the formula back out. I have had other teachers give derivations for formulas, but there was something about the way Mr. Parrott explained it that made it feel more realistic. I am not sure exactly what he did differently but whatever it was, it worked. His class made math feel like a science. Like each formula was something that existed in the universe already and we were just discovering it for the first time.


After I started taking an interest in math, I found out about Alan Turing. I think the life he led was incredibly brave and the impact of his work is only comparable to Sir Isaac Newton. He helped to break the enigma cipher, take down the Nazis, and he invented the ideas behind the computer. He even solved a main part of the German naval indicator system, essentially for fun. Turing wrote papers about mathematical biology and predicted oscillating chemical reactions a decade before they were first observed. The law passed in the UK that retroactively pardons men who were convicted for crimes of homosexuality is informally called the Turing award. His most famous idea is the Turing test. This is a test that is designed to test how effectively an artificial intelligence can imitate a human. With so much knowledge in the topic of AI and computers, he was able to predict the rise and acceptance in the 2000’s. "I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted," (Aaron Couch, 2014). Alan Turing left an unbelievable legacy. His advancements in mathematics were so a head of his time that we are finally starting to appreciate them.

Lessons in Education From Math

One of the lessons that I learned from mathematics was to approach problems from a different angle. Whenever I am stuck on a problem, I often find the best thing to do is to start over and try solving it in a different way. Sometimes it works and sometimes it does not, but I always learn something new about the problem when I do this. This is something that I often take into my programming for computer science and especially take into my real life. Another lesson I have learned from math is critical thinking. The ability to look at a situation, take in all the critical information, and be able to make a decision on where to go from there. This is probably one of the best life skills I learned in college and in mathematics.


How Math at NKU Has Helped Me

Math at NKU has helped me immeasurably. It has helped me gain insight and knowledge into every program I have written as a Computer Science student. Math also helped me to understand how each program works on a theoretical level. Without these additional math classes, I do not think my software engineering skills would be anywhere near as sharp. The mathematics program has also helped me to make connections with many great people. I got to meet people from all over the world with many different viewpoints. It really helped to expand my horizons. The teachers in the math program are also some of the best teachers on campus. They were always there to answer any questions I had. The knowledge and passion that they had for their subjects was incredible. Finally, math has helped me improve my resume and general business knowledge. There have been times where I have been talking to businessmen and the only thing that let me know what they were talking about was my skill in math.

Recent Experience

The best math classes I have taken in recent memory are partial differential equations and cryptology. Partial differential equations was one of the hardest classes I have taken but solving each problem felt so rewarding. Partial differential equations are multivariable differential equations, and there was a decent amount of physics that you were expected to know (which I did not). The problems were tough and when I first started the class, I felt like there was no way I could possibly do it. I kept my nose to the grindstone the whole semester and I eventually got it. When I did get it, was one of the biggest “a ha” moments of my life.

Cryptology was one of the most enjoyable classes I have taken at NKU. Every problem felt like a puzzle. Sometimes you are given only a string of ciphertext and you must use every tool available to you to first understand the encryption method. After you have found out the encryption method, you then need to gather all possible information and break the cipher. As we got into more modern encryption methods, the class became less about breaking the cipher and more about understanding how they work. It was great to see how all of the encryption algorithms that I have heard about actually work. The class made me feel like I was a spy.


Mathematical Topics I Want to Study

These are some of the areas that I potentially want to study in grad school, or at the very least learn more about.


Chaos Theory

My roommate is not a math guy, but he is a big Jurassic Park fan. Whenever I start talking about math to him, he starts quoting Jeff Goldblum’s character and says I should start studying chaos theory. This has happened so many times that I started looking it up in my free time. Chaos theory has some interesting things in it, but a good deal of it goes over my head.

Chaos theory is a branch of mathematics that studies dynamic systems that appear to move in a random state, but the irregularities are actually governed by some underlying pattern. In chaos theory mathematicians study a system that has some initial conditions. Changing these initial conditions, even slightly, can cause the system to have a drastically different behavior. In other words chaos theory is the study of the butterfly affect.




Topology

Geometry is a field of math that I have a love hate relationship with. It always felt so different than any other math class I have taken. That said, I think that geometry offers some very interesting viewpoints on mathematics. I feel like this is even more true in topology. If I study topology, I would study some of the most interesting shapes and learn the mind-bending properties behind them.

Topology is the study of objects whose properties are preserved through deformations, twisting, and stretching. This leads to all sorts of interesting equivalencies, like a circle and an ellipse being the same. It includes things like mobius strips, fractals, and knots. It can even be extended to research computer networks.






Economics

For most of my life I was never interested in economics. It always seemed like some fougasse idea and I never really concerned myself with learning anything beyond what I needed to know. As I get older, however, I have started to take more of an interest in it. I think the reason for this is because I saw some of the equations my girlfriend, who is a business major, was using. It piqued my interest at first mainly because I wanted to know how the formulas were derived. Since then, I have started studying it more and more, trying to understand everything I can about it. This is a skill that I can use everyday, and the more I know about it the better.






Statistics

I have never been strong with statistics. I understood the formulas but getting to the point where I could use them was challenging. I think the only way to get better at that part would be to keep working on statistical problems. I have had to use statistics more times than I can count. I always feel like I am slow to recognize what formula should be used where, and when the formulas should be used. I want to study statistics more to overcome my shortcomings. Statistics is also used in plenty of other mathematical fields, so the more I learn about statistics the better. I also feel that having a good base in statistics will really help me in any mathematical career that I have.







Fluid Dynamics

My dad was a pilot, so I have always been pretty interested in all aspects of flying. Fluid dynamics is not just the study of how fluids behave, it also involves the study of aerodynamics. Learning how lift and drag affect different objects is something that is right up my ally. One of my dreams has always been to build my own ultra-light aircraft. Being able to understand the forces that effect the aircraft would be a huge help in its creation and development. Fluid dynamics more of a topic that I would study as a hobby. I would also love to have the opportunity to work with aircraft professionally.







Career Interest

The careers I am most interested in are in the field of software engineering or data science. During my time as a Computer Science student I had the chance to solve many interesting problems, which made me fall in love with the process of building software. Eventually I was able to get a job at a startup company where I am an Android Developer. This job gave me a greater appreciation for the software I was developing since it the software concepts become more tangible. It gave me insight into how software development works in a real world setting, and how to work with a team to overcome some of the challenges that arrose. The thing that I love most about this job is that at its core its just solving problems with math, which is something that both feels very familiar because I have been doing it my whole life, and very different as each problem is unique. The other career that I am interested in is data science. The idea of machine learning has always fascinated me. When I recieved the chance to take a class in it, I was ecstatic. This class was the heaviest mathematical programming class I have ever taken which made me love it even more. The best thing about this class though was the products that I was able to build. Even building simple AI products that were entirely based on statistics seemed incredible. As we moved forward and got into more complicated things, like genetic algorithms and neural networks. It seemed like I had reached the pinnacle of software. Getting the chance to design software like this for a living would be an incredible experience and an opportunity I would pass up.


Mathematical Spotlight

The mathematical topic I choose to focus on was bitcoin and how it works. I chose to focus on this topic because of three reasons. The first reason is that in the past couple of years it has expanded rapidly. I feel it is important to have a proper understanding of bitcoin, especially for those who want to invest in it. The second reason is the math, cryptography, and computer science that allow for it to work. Bitcoin is mind-bendingly complicated, but at the same time provides a simple and elegant solution to the problems that it faces. The third and final reason is that I think that the idea of a decentralized currency is, at the very least, an interesting thought experiment to see how real-world money and economies behave.

Bitcoin, at its heart, is a decentralized currency system meaning that there is no central bank or government backing it. The bitcoin network is a ledger of all transactions that everyone has access to. If someone wants to make a transaction, then they send a signal out to everyone on the network saying that they are making transaction and sign it with a digital signature.







Eventually, once enough transactions have occurred, the transactions get grouped together into a structure called a block. A block is the heart of the bitcoin network. It consists of a group of transactions, some metadata, and a hash. The hash is the answer to a difficult math problem that is designed to take a computer 10 minutes to solve the problem. Computers that find answers to these math equations are called miners. Once a miner has found an answer to the equation other miners check the block to make sure that the answer is valid and all transactions in the block are legitimate. If the miners agree that the block and its contents are valid it gets uploaded to the block chain (which is just all previous blocks strung together) and the miner that found the right answer receives some amount of bitcoins for their work.








This graph shows the reward for successfully mining a block. As more bitcoins are mined, fewer new bitcoins enter circulation. Currently for every successful block mined the miner receives 6.25 bitcoins. If you want more information on bitcoin and how it operates, I have a paper that explains it here: Media:bitcoin.pdf.

Sources

Hong, Euny. “How Does Bitcoin Mining Work?” Investopedia, Investopedia, 20 Apr. 2021, www.investopedia.com/tech/how-does-bitcoin-mining-work/.

ComodoSSLStore. What Is Digital Signature and How It Works?, 23 Nov. 2020, comodosslstore.com/blog/what-is-digital-signature-how-does-it-work.html.

“Bitcoin Halving Reduces Mining Rewards for Third Time in Brief History.” Blockchain News, 11 May 2020, blockchain.news/news/bitcoin-halving-reduces-mining-rewards-for-third-time-in-brief-history.

“Chaos Theory | Brilliant Math & Science Wiki.” Brilliant, brilliant.org/wiki/chaos-theory.

Bang, Starts With. “Chaos Theory, The Butterfly Effect, And The Computer Glitch That Started It All.” Forbes, 14 Feb. 2018, www.forbes.com/sites/startswithabang/2018/02/13/chaos-theory-the-butterfly-effect-and-the-computer-glitch-that-started-it-all/?sh=d9fe5a269f6c.

“Topology -- from Wolfram MathWorld.” Mathworld.Wolfram, 29 Apr. 2021, mathworld.wolfram.com/Topology.html.

Darling, David. “Topology.” Daviddarling, 2016, www.daviddarling.info/encyclopedia/T/topology.html.

“During Economic Highs and Lows, the Arts Are Key Segment of U.S. Economy.” National Endowment for the Arts, 17 Mar. 2020, www.arts.gov/about/news/2020/during-economic-highs-and-lows-arts-are-key-segment-us-economy.

edX. “Probability and Statistics in Data Science Using Python.” EdX, 2021, www.edx.org/course/probability-and-statistics-in-data-science-using-p.

“Aerolite 103: Light Aircraft DB & Sales.” Pilotmix Light Aircraft DB & Sales, www.pilotmix.com/aerolite-103.

Wikipedia contributors. “Alan Turing.” Wikipedia, 28 Apr. 2021, en.wikipedia.org/wiki/Alan_Turing.

Couch, Aaron. “Alan Turing's 5 Most Powerful Quotes.” The Hollywood Reporter, 30 Nov. 2014, www.hollywoodreporter.com/news/alan-turings-5-powerful-quotes-752669.

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