Research Experience for Undergraduates
SUMMER 2024: Non-commutative algebra from geometry
Leaders: Dr Hülya Argüz and Dr Pierrick Bousseau
Dates: August 12, 2024 to August 23, 2024
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.
Application Information: https://www.math.uga.edu/events/content/2024/research-experience-undergraduates
Currently, all positions for this REU are filled. Please check back on the news/events link of the UGA Maths Department's website for future REU's.Thanks for your interest.
Objectives:
Objective I: Understand what is a non-commutative algebra and what does it mean for a non-commutative algebra to be a deformation quantization of a commutative algebra.
Objective II: Understanding combinatorial gadgets called ``scattering diagrams'' and construct commutative algebras using them.
Objective III: Understanding ``quantum scattering diagrams'' and construct non-commutative algebras using them.
Coming up: REU student presentations
• October 1st 2024, at 4 pm, Boyd 304: Ava Kuhlman
• October 15th 2024, at 4 pm, Boyd 30: Sara Logsdon
This REU is supported by the NSF grant DMS 2302116 New Bridges to Gromov-Witten Theory.