### GSSE Mathematics Course Summer 2024

The first week of the course will consist in a formal introduction

to proof writing. We will partially use the book Introduction to

Abstract Mathematics by C. Plaut. We will cover the following

topics: Logic, Proof Writing, Integers and Induction, Primes and

Divisibility, Fields.

In the second week of the course we will cover Linear Algebra

concepts including Vectors, Matrices, Invertibility, and

Determinants. We will discuss algebraic, geometric, and physical

meanings for these concepts.

In the third week we will present applications to Coding Theory,

Computer Graphics, Search Engines, and Hadamard Matrices.

This part of the course will rely on class notes, without the use of

a book.

In the last week we will investigate research-level questions on

Hadamard Matrices. This will allow us to formulate and prove

conjectures on topics from Number Theory and Linear Algebra.

We will also work on student presentations.

#### 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 - 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 - Check if a set with two given binary operations is a ring or a

field - 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, prime, prime-factorization
- 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)

**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.