An Inverse to the Delay Move on k-Graphs

Presentation Type

Poster Presentation

Category

STEM (science, technology, engineering, mathematics)

Abstract/Artist Statement

The goal of this research was the try and define a new move on k-graphs, particularly one that would serve as an inverse to the previously defied move called “delay”. C* algebras are a mathematical object in the realm of analysis. There are many ways to think of C* algebras; the arguably most approachable way is to think of them as matrices with infinite dimensions. C* algebras have applications to group representation theory and quantum physics. However, they are very difficult to construct. One type of C* algebra is a graph C* algebra and these are easier to construct. Not only that, but there are also connections between the graph’s structure and the graph C* algebra’s structure. Therefore, graph C* algebras became a point of interests among C* algebraists. A central research question was asked: when do two graphs give rise to the same graph C* algebra? This question has been answered using a set of “moves” on the graphs and their inverses that preserve the graph structure. However, not all C* algebras are graph C* algebras, which brings us to our research. Kumjain and Pask generalized the theory of graph C* algebras to k-graphs, which can be thought of as graphs in multiple dimensions. The new natural, analogous question then becomes: when do two k-graphs give rise to the same C* algebra? This question is still open. Currently, C* algebraists have been trying to find “moves” on k-graphs that preserve the k-graph structure, but not all the moves and their inverses have been discovered. One move that has been discovered and proven to work is called “delay”. However, a good definition for the inverse of delay has not been offered (there is one working definition but it is extremely restrictive, only working in specific settings and not in general). Our collective research the previous two semesters and summer have been to try and find a better inverse for delay. The first step was taken by Davis in the spring of 2021 as part of his masters project. Davis offered a new move, called “fast forward”, which serves as a one-sided inverse to delay. That is, if one fast forwards in a k-graph and then delays, the result is the original k-graph. Davis finished the proof of this fact in the fall of 2021. Meanwhile, Lippert expanded on this idea when he noticed that the concept of fast forward could be generalized even further. During the summer of 2021 and the fall of 2021, Lippert defined a new move called “neighborhood reduction”, of which fast forward is a special case. Davis and Lippert gave a two-part talk on fast forward and neighborhood reduction at the end of the fall 2021 semester.

Mentor Name

Elizabeth Gillaspy

Personal Statement

The goal of this research was the try and define a new move on k-graphs, particularly one that would serve as an inverse to the previously defied move called “delay”. C* algebras are a mathematical object in the realm of analysis. There are many ways to think of C* algebras; the arguably most approachable way is to think of them as matrices with infinite dimensions. C* algebras have applications to group representation theory and quantum physics. However, they are very difficult to construct. One type of C* algebra is a graph C* algebra and these are easier to construct. Not only that, but there are also connections between the graph’s structure and the graph C* algebra’s structure. Therefore, graph C* algebras became a point of interests among C* algebraists. A central research question was asked: when do two graphs give rise to the same graph C* algebra? This question has been answered using a set of “moves” on the graphs and their inverses that preserve the graph structure. However, not all C* algebras are graph C* algebras, which brings us to our research. Kumjain and Pask generalized the theory of graph C* algebras to k-graphs, which can be thought of as graphs in multiple dimensions. The new natural, analogous question then becomes: when do two k-graphs give rise to the same C* algebra? This question is still open. Currently, C* algebraists have been trying to find “moves” on k-graphs that preserve the k-graph structure, but not all the moves and their inverses have been discovered. One move that has been discovered and proven to work is called “delay”. However, a good definition for the inverse of delay has not been offered (there is one working definition but it is extremely restrictive, only working in specific settings and not in general). Our collective research the previous two semesters and summer have been to try and find a better inverse for delay. The first step was taken by Davis in the spring of 2021 as part of his masters project. Davis offered a new move, called “fast forward”, which serves as a one-sided inverse to delay. That is, if one fast forwards in a k-graph and then delays, the result is the original k-graph. Davis finished the proof of this fact in the fall of 2021. Meanwhile, Lippert expanded on this idea when he noticed that the concept of fast forward could be generalized even further. During the summer of 2021 and the fall of 2021, Lippert defined a new move called “neighborhood reduction”, of which fast forward is a special case. Davis and Lippert gave a two-part talk on fast forward and neighborhood reduction at the end of the fall 2021 semester.

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Mar 4th, 5:00 PM Mar 4th, 6:00 PM

An Inverse to the Delay Move on k-Graphs

UC North Ballroom

The goal of this research was the try and define a new move on k-graphs, particularly one that would serve as an inverse to the previously defied move called “delay”. C* algebras are a mathematical object in the realm of analysis. There are many ways to think of C* algebras; the arguably most approachable way is to think of them as matrices with infinite dimensions. C* algebras have applications to group representation theory and quantum physics. However, they are very difficult to construct. One type of C* algebra is a graph C* algebra and these are easier to construct. Not only that, but there are also connections between the graph’s structure and the graph C* algebra’s structure. Therefore, graph C* algebras became a point of interests among C* algebraists. A central research question was asked: when do two graphs give rise to the same graph C* algebra? This question has been answered using a set of “moves” on the graphs and their inverses that preserve the graph structure. However, not all C* algebras are graph C* algebras, which brings us to our research. Kumjain and Pask generalized the theory of graph C* algebras to k-graphs, which can be thought of as graphs in multiple dimensions. The new natural, analogous question then becomes: when do two k-graphs give rise to the same C* algebra? This question is still open. Currently, C* algebraists have been trying to find “moves” on k-graphs that preserve the k-graph structure, but not all the moves and their inverses have been discovered. One move that has been discovered and proven to work is called “delay”. However, a good definition for the inverse of delay has not been offered (there is one working definition but it is extremely restrictive, only working in specific settings and not in general). Our collective research the previous two semesters and summer have been to try and find a better inverse for delay. The first step was taken by Davis in the spring of 2021 as part of his masters project. Davis offered a new move, called “fast forward”, which serves as a one-sided inverse to delay. That is, if one fast forwards in a k-graph and then delays, the result is the original k-graph. Davis finished the proof of this fact in the fall of 2021. Meanwhile, Lippert expanded on this idea when he noticed that the concept of fast forward could be generalized even further. During the summer of 2021 and the fall of 2021, Lippert defined a new move called “neighborhood reduction”, of which fast forward is a special case. Davis and Lippert gave a two-part talk on fast forward and neighborhood reduction at the end of the fall 2021 semester.