Monte Carlo and the Coding Tale

By: Michelle Chen, Cameron Farrar, Laura O’Sullivan, and Cat Bassett

Introduction

Everything we do in life has a chance. That chance may come from picking the right card, picking a certain marble out of a bag or maybe deciding to give the first person who walks through a random door $100. Essentially,each chance has a certain trade-off of benefits. Often times we think about the chances as something will happen over the chance of something else taking place as we weigh possible outcomes. This is called risk analysis. One of the ways we can determine risk is we can use Monte Carlo simulations to replicate real life situations a large number of times in order to observe the long-term patterns without having the complications (cost, labor, materials, etc.) of manual repetition.

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Quantum Mechanics

By Izzy Cox, Divya Iyer, Wgoud Mansour, Ashleigh Sico, and Elizabeth Whetzel.

Quantum Mechanics is the physics of molecular and microscopic particles. However, it has applications in everyday life as well. If someone asked you if a human was a particle or a wave, what would you think? What about a ball? What about light? Not so easy now, is it? It turns out that all of those things, and in fact, everything around us, can be expressed in physics as both a particle and a wave. This might seem a little unbelievable, but for now, let’s start with the basics.

 

Classical Physics

Although Classical Physics sounds like a complicated idea, it’s the most simple branch of physics. It’s what you think of when someone says “physics”. Classical Physics lays the basic foundation to Quantum Physics with a few basic laws.

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RSA Encryption Cryptography

By Divya Aikat, Helena Harrison, Annie Qin, and Quinn Shanahan

The definition of cryptography is the art of writing and solving code. However, over the last two weeks, we learned so much more than just this textbook explanation. While working together within our team, we explored many different aspects behind cryptography. By building off our individual strengths, we prepared ourselves for higher level mathematics. The following is a synopsis of the progress we’ve made over the past two weeks.

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Scientific Computing: Recurrence Relations

By: Kathryn Benedict, Kate Allen, Sarai Ross, Rosy Nuam

Girls Talk Math is an all girls camp that introduces new topics that students would not normally see in their everyday math class at school. This camp also brings together many young women to better explore a field that is male dominated. During this camp we were able to research many important women that we able to make their own legacy while facing much adversity along the way. The camp wants to show not only the campers but also other women going into the field of math and science to not be afraid due to the gender difference, but instead use it as motivation to carry on doing what you love and making your own legacy along the way.

Our group consisted of four young women. Kathryn is a rising sophomore at Cedar Ridge High School. Kate is a rising sophomore at Carrboro High School. Sarai is a rising junior at Northern Vance High School. Rosy is a rising senior at East Chapel Hill High School.

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Elliptic Curve Cryptography

By Mukta Dharmapurikar, Anagha Jandhyala, Savanna Jones, and Ciara Renaud.

Have you ever wondered how your credit card number stays secure after shopping online? Every day millions of people’s personal information is entered online or stored in databases, where it seems like anyone could access it. However, a process called cryptography keeps theft from occurring.

Cryptography is the ancient art of keeping secret messages secure. Elliptic curve cryptography is one type of encryption that we spent the last two weeks learning about. It has some advantages over the more common cryptography method, known as RSA.

RSA relies on the difficulty of factoring very large prime numbers. Despite the current security, it’s feasible that one day a method could be invented that makes factoring large prime numbers realistic. In this blog post, we will be explaining the essential math behind how elliptic curves work and how they are used to encrypt messages.

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Mathematical Epidemiology

By Camilla Fratta, Ananya Jain, Sydney Mason, Gabby Matejowsky, and Nevaeh Pinkney*.

*Nevaeh helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

Mathematical Epidemiology explores the realm of mathematics applied to public health. It relies on modeling to use known information about certain scenarios regarding the spread of diseases and then uses it to predict future outcomes. By the end of the problem set, our group learned about the challenging process that comes with trying to predict population sizes in order to control the spreading of diseases. The equations that are faced in this branch of mathematics are at the heart of mathematical modeling.

Mathematical Models and Modeling

A mathematical model is an equation used to predict or model the most likely results to occur in a real-world situation.  We used these types of equations to model the spread of a disease in a population, tracking the flow of populations from susceptible to infected to recovered.  In real life scenarios, there are too many variables to fully account for, so we only were able to place a few in our equations. This made the models less accurate, but at the same time very useful to us in our problem set.  They gave us a good idea of how things worked in an actual epidemic and helped us to understand what mathematical modeling really is.

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The Art of Cryptography

By: Nia Beverly, Makayla McDaniel, Yuanyuan Matherly, and Tyler Deegan

Introduction

Cryptography is defined as the art of writing and solving codes. Upon first thought, many people picture codes as an antiquated war time communication technique. However, the field of cryptography is alive and well,  and it has become pervasive in our everyday lives. The world is becoming more and more connected through technology, and with this, there is a greater need to protect information. Encryption is probably the most widely used application of cryptography, and it is used to protect information by making it so only one person with a key can understand what is transmitted. In the following paragraphs we will walk through the steps to mathematically understanding one widely used type of encryption.

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RSA Encryption Cryptography

By Camille Clark, Layke Jones, Isabella Lane, Aza McFadden*, and Lizbeth Otero.

*Aza helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

Cryptography is a field of coding and decoding information. It relies on the framework of number theory. Therefore, it can be used to connect theories as well as teaching others the fundamental properties of integers. Relevant number theory topics are modular arithmetic, prime factorization, greatest common divisor, and theorems such as the Chinese Remainder Theorem and Euler’s Theorem. This blog post will focus on the first three topics.

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Classification of Surfaces

By Ayanna Blake, Lisa Oommen*, Myla Marve, Tamarr Moore, Caylah Vickers, and Lily Zeng.

*Lisa helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

The Girls Talk Math camp is about female high school students from different places who discuss mathematics, mathematicians, and theories. We were split up into groups and were assigned different math topics to learn. Our topic was classification of surfaces, which is listed under the umbrella topic of abstract geometry.

We thought the surfaces project was very interesting and cool to learn about, because it introduced us to college level math and allowed us to understand different parts of geometry. Along with gaining knowledge of surfaces, we also got to learn about other groups topics. Campers presented their topics on the last day and helped us to perceive the significance of the different subjects.

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