### Hadamard Matrices, Secret Codes, Concert Halls and Quantum Teleportation Schemes

The first half of the course will consist in a formal introduction to proof writing. We will use the book *Introduction to Abstract Mathematics *by C. Plaut, and we will cover the following topics: Logic and Set Theory, the Real Numbers, the Integers and Induction.

In the second half of the course we will discuss some advanced algebra and number theory concepts involving matrices, primes, and modular arithmetic. We will also present applications to subjects such as Coding Theory, Computer Graphics, Fibonacci Patterns, and Hadamard Matrices. This part of the course will rely on class notes, without the use of a book.

#### Student Learning Outcomes

Upon successful completion of this course, a student will be able to:

- Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments
- Perform set operations on finite and infinite collections of sets and be familiar with properties of set operations
- Write solutions to problems and proofs of theorems that meet rigorous standards based on content, organization and coherence, argument and support, and style and mechanics
- Describe the real line as a complete, ordered field
- Check if a set with two given binary operations is a ring or a Give examples and non-examples of rings and fields (including examples from matrix theory and from modular arithmetic)
- Construct direct and indirect proofs and proofs by induction and determine the appropriateness of each type in a particular setting
- Master the concepts of divisibility, congruence, greatest common divisor, prime, prime- factorization, and quadratic reciprocity
- Master matrix algebra concepts such as matrices, vectors, and linear transformations
- Be familiar with the definition, properties and current state of classification of Hadamard matrices
- Formulate and prove conjectures about numerical patterns in number theory and matrix algebra

#### Course Requirements/Examinations

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

- Completion of weekly homework (60% of the grade)
- Class participation (20% of the grade)
- Team-based preparation of a poster/presentation (20% of the grade)

#### 2021 Mathematics Research

Click on the link below to view research from 2021 GSSE Mathematics scholars.

2021 GSSE Online Research Symposium

**Credit Hours: 3**

#### Course Instructors

#### Dr. Remus Nicoara, Course Director

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 a 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.