In mathematics, a matrix is a rectangular table of numbers arranged in rows and columns. Matrices can be added and multiplied, just like numbers. However, matrices can also behave very differently from numbers. For instance, matrices do not commute when multiplied, meaning that A times B may not equal B times A. In the first part of the course, we will study matrices and their applications to computer graphics, the ranking of web pages in Google, and quantum mechanics.

We will then focus on a special class of matrices called Hadamard matrices. These are matrices having n rows and n columns and all entries equal to 1 or -1, such that each two distinct rows agree on exactly half positions (and disagree on the other half). It is easy to draw them by representing the 1 entries as white squares, and the -1 entries as a black squares. Here are some examples of Hadamard matrices:

The definition of a Hadamard matrix is deceptively simple, but it is not known for which n’s such matrices exist. The famous Hadamard conjecture asserts that for every integer n divisible by 4 there exists a Hadamard matrix of size n, but nobody has managed to prove this 150-year old conjecture.

We will investigate different ways to construct Hadamard matrices. Then, we will focus on their applications to coding theory, acoustics, and quantum teleportation.

The criteria for assigning grades for the course are the following:

- completion of weekly homework
- class participation, including board work
- preparation of a poster/presentation (team-based)

Dr. Remus Nicoara earned his Ph.D. in Mathematics from UCLA, and his Bachelor's Degree from the University of Bucharest, Romania. He is currently an Associate Professor of Mathematics and Director of the Math Honors Program at the University of Tennessee. His main research interest lies in von Neumann algebras, which are algebras of operators that model quantum mechanical systems. Outside of work, Remus likes to hike, bike and garden while thinking about math. He enjoys meditation, Sci-Fi books, and Hanayama puzzles. He is also an avid gamer and he currently teaches a class about video games and math, called Math Effect.

Adam is a proud graduate of UT in mathematics and computer science and is an upcoming graduate student in mathematics at Purdue University. This is Adam’s third year working as a teaching assistant for the mathematics Governor’s School program, and he is excited to assist the incoming Governor's School mathematics students as they learn about linear algebra, quantum teleportation, and goats! In addition to studying mathematics, Adam enjoys outdoor activities, contra dancing, and practicing handstands.

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