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Hülya Argüz and Pierrick Bousseau
MATHEMATICS
UGA
Description:  In elementary classical algebra, we have the commutativity'' of products, which means that when we multiply two variables, the order we multiply them does not change the result. For instance, if we have two variables x and y, then the product xy is the same as the product yx. However, for more advanced mathematical objects, there are algebras which are not commutative, and in particular, in which the order we multiply variables matters. For example, if A and B are two matrices, AB is not equal to BA in general. Such algebras are called non-commutative algebras.
Many non-commutative algebras appear as deformation" of classical commutative algebras, obtained by adding an additional variable which keeps track of deformations. For instance, an algebra where two variables x and y satisfy xy=qyx is commutative if q=1 but non-commutative if q is not equal to 1. Here, the additional variable q'' is often referred to as a quantum variable'' and appears frequently in the study of algebras that arise in quantum physics, which roughly describes particles at the atomic level. The process of producing a non-commutative algebra from a commutative algebra is called quantization", as it relates to the passage from classical physics to quantum physics. The goal of this project is to construct explicit non-commutative algebras as quantization of commutative algebras of great interest in geometry, influenced by quantum physics.
Activities: Students will spend 2 weeks engaged in collaborative activities such as the following:
• Lectures and discussions: Students will learn the basic tools for calculating non-commutative algebras. They are expected to participate in lectures, which will take roughly 5 hours each week. The schedule will be flexible and determined together with the students to avoid overlaps with their other classes
• Research reports and presentations: Students are expected to work together and prepare a report at the end of the two weeks on the outcomes of the research.
Prerequisite:  Students are expected to have some experience with linear algebra
Stipend: Each student will receive \$1400 stipend. We also have some funding to support travel and accommodation costs.
Application: Interested students should send the following documents to Hulya.Arguz@uga.edu :
• Unofficial transcript
• A personal statement (no more than one page):
• Tell us about yourself: What topics and subjects are you interested in? What courses have you most enjoyed?
• Tell us why you are interested in participating in this REU, and how this experience would help you achieve your future goals.