by Camille Clark, Layke Jones, Aekta Kallepalli, Maya Mukerjee, and Caroline Zhou.

Euler Characteristics

Euler characteristics are defined by the equation V- E + F = 2 where V = number of vertices, E = number of edges or nodes, and F = number of faces. Sometimes though, the equation V – E + F = 2 does not work for all situations because the solution can give various outcomes due to the dimensions and simplicity of the object. If 2 objects are topologically the same, they will have the same Euler characteristics. For all simple polygons, the Euler characteristics equal one. Figures with holes don’t follow these conventions as the holes in these figures add additional faces and edges not proportional to the formulas for simple figures.

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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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by Erin Gottschalk, Simon Johnson, Meghan, Elizabeth Nguyen, and Brooke Rogers*.

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

What We Did:

Knot theory has many different applications in math including algebra and geometry, and (outside of math) physics. We learned that we can use algebraic techniques to describe knots. When trying to understand knot theory we learned that it is very helpful to work in a group and read the definitions out loud. Us working together was key in understanding knot theory.

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By Miranda Copenhaver, Nancy Hindman*, Efiotu Jagun, and Gloria Su.

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

Number systems are how we represent numbers like 1, 32, and 75. We use the base ten (decimal) system for our numbers most of the time. It’s called base ten because it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. But what if I told you that 1001101 and 4D both mean seventy-seven? Crazy, right? There are countless number systems, but today we will be focusing on two: hexadecimal (base sixteen) and binary (base two)!

As we’ve said before, the binary system is base two; it only uses 0 and 1. Since only 1 or 0 can be used, the placement of each digit is important. Computers use binary to store and transfer information. It is used in communication (Morse code, braille) and everything electronic like computers, lights, calculators, MP3s, MIDI, JPEG, etc. 

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by Nya Batson, Monique Dacanay, Emily Gao, and Staci Tranquille.

Hello! And welcome to the realm of quantum mechanics!  First off, what in the world is quantum mechanics? Let’s start with a brief introduction.

What is Quantum Mechanics?

Quantum mechanics is one of the most important branches of physics. It focuses on the laws of nature at three different levels: molecular, atomic, and subatomic. Quantum mechanics has a variety of important concepts; the following are some that we learned through our problem set: Planck’s law, the photoelectric effect, and wave-particle duality.  A crucial element of quantum mechanics is understanding that everything has characteristics of both waves and particles. We will touch on this and many other topics later on.

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