#### Presentation Title

What is the boundary of a finite subset of the pac-man universe?

#### Advisor Information

Patrick X. Rault

#### Presentation Type

Poster

#### Start Date

26-3-2021 12:00 AM

#### End Date

26-3-2021 12:00 AM

#### Abstract

Given a square matrix *A*, its numerical range is the image of a certain map *g _{A}* from the unit sphere to the complex plane. This numerical range is the convex hull of a “boundary generating curve”. It is useful in the theory of linear algebra in approximating eigenvalues, and has applications in quantum computing. In the modular situation of finite fields, which resembles the universe in the game pac-man where there is no concept of boundary, the aforementioned boundary generating curve has some very special properties. If

*A*is a 2-by-2 matrix, then

*g*is a two-to-one map (up to trivial multiples), except on the boundary generating curve, where it is a one-to-one map (up to trivial multiples). We will discuss the geometry of the image of

_{A}*g*(the numerical range of

_{A}*A*) using this boundary generating curve.

What is the boundary of a finite subset of the pac-man universe?

Given a square matrix *A*, its numerical range is the image of a certain map *g _{A}* from the unit sphere to the complex plane. This numerical range is the convex hull of a “boundary generating curve”. It is useful in the theory of linear algebra in approximating eigenvalues, and has applications in quantum computing. In the modular situation of finite fields, which resembles the universe in the game pac-man where there is no concept of boundary, the aforementioned boundary generating curve has some very special properties. If

*A*is a 2-by-2 matrix, then

*g*is a two-to-one map (up to trivial multiples), except on the boundary generating curve, where it is a one-to-one map (up to trivial multiples). We will discuss the geometry of the image of

_{A}*g*(the numerical range of

_{A}*A*) using this boundary generating curve.

## Additional Information (Optional)

Unavailable 12:30pm-5:00pm US Central due to work.