Kyle Hansen

PhD Candidate

UCSB Department of Mathematics

South Hall 6431K

kylehansen(at)math(dot)ucsb(dot)edu

PhD Candidate

UCSB Department of Mathematics

South Hall 6431K

kylehansen(at)math(dot)ucsb(dot)edu

Click on one of the links to the right to learn more about me!

I'm a 5th year graduate student in mathematics at UC Santa Barbara. I enjoy baking and long walks on the beach.

Here is my current CV.

I am currently on summer break, and will return to TA for Transition to Higher Mathematics (MA-8) in Summer Session B 2024 at UCSB.

Please see the official class webpages for more details there.

Term | Course Name | Course Number | Institution | Role |
---|---|---|---|---|

2024 Summer |
Transition to Higher Mathematics | MA-8 | UCSB | TA |

2024 Spring |
"Abstract" Linear Algebra | MA-108A | UCSB | TA |

2024 Winter |
Numerical Methods | MA-104B | UCSB | TA |

2023 Fall |
"Abstract" Linear Algebra | MA-108A | UCSB | TA |

2023 Fall |
Discrete Mathematics | MA/CS-015 | Westmont College | Instructor of Record |

2023 Summer |
Transition to Higher Mathematics | MA-8 | UCSB | TA |

2023 Spring |
Discrete Mathematics/Foundations of Computer Science | CMPSC-040 | UCSB | Instructor of Record |

2022 Fall |
Discrete Mathematics | MA/CS-015 | Westmont College | Instructor of Record |

2022 Fall |
Linear Algebra with Applications | MA-4A | UCSB | TA |

2022 Summer |
Linear Algebra with Applications | MA-4A | UCSB | TA |

2022 Winter* |
"Abstract" Linear Algebra | MA-108A | UCSB | TA |

2021 Fall |
Discrete Mathematics | MA/CS-015 | Westmont College | Instructor of Record |

2021 Fall |
Methods of Analysis | MA-117 | UCSB | TA |

2021 Summer* |
Methods of Analysis | MA-117 | UCSB | TA |

2021 Spring* |
Differential Equations | MA-4B | UCSB | TA |

2021 Winter* |
Transition to Higher Mathematics | MA-8 | UCSB | TA |

2020 Fall* |
Linear Algebra with Applications | MA-4A | UCSB | TA |

2020 Summer* |
Transition to Higher Mathematics | MA-8 | UCSB | TA |

2020 Spring* |
Transition to Higher Mathematics | MA-8 | UCSB | TA |

2020 Winter |
Calculus for Social Sciences I | MA-34A | UCSB | TA |

2019 Fall |
Calculus I | MA-3A | UCSB | TA |

* This term was significantly altered by the COVID-19 pandemic.

If you want to study the complexity of a topological space, you might study the algebraic invariants of this space. Often, however, topological spaces such as manifolds come equipped with some geometric structure as well. How does this algebraic topological complexity translate into geometric complexity? This broad question motivates the field of so-called "quantitative topology".

For example, if you have a nullhomotopic map (a priori, algebraic information) into your space, what is the most efficient way to perform this nullhomotopy (geometric information)? My advisor Fedya Manin has asked (and answered) similar questions in the context of simpy connected target spaces, and some of my current work is an extension of this to nilpotent targets.

I also study tiling spaces, where I am interested in extending some existing theory to the relevant nilpotent setting. For example, I extend work about the topology of Euclidean tiling spaces to tiling spaces of certain nilpotent Lie groups.

I have also studied elliptic manifolds, where I askied "Given a nilpotent Lie group G, what algebraic information is necessary/sufficient to guarantee that a closed manifold M is G-elliptic?", until I realized just how hard this question is.

- Clues About Bluffing in Clue: Is Conventional Wisdom Wise? (2019)

- Tiling Spaces Which are Fiber Bundles over Nilmanfiolds (41st Workshop in Geometric Topology, 6/13/2024)
- The Hull of a Manifold with Bounded Geometry (UCSB Topology Seminar, Advancement Talk 5/24/2022)
- An Introduction To Tiling Spaces (UCSB Graduate Topology Semniar, 4/6/2022)

You may reach me at the email address in the above banner.

I completed the 40th Annual Santa Barbara Triathlon with a time of about 3:45. You can view some photos of that below. I've posted them here, and not on any of my social media accounts, so that might tell you something about me.

If I were a Springer-Verlag Graduate Text in Mathematics, I would be William Fulton and Joe Harris's My primary goal is to introduce the beginner to the finite-dimensional representations of Lie groups and Lie algebras. Intended to serve non-specialists, my concentration is on examples. The general theory is developed sparingly, and then mainly as a useful and unifying language to describe phenomena already encountered in concrete cases. I begin with a brief tour through representation theory of finite groups, with emphasis determined by what is useful for Lie groups; in particular, the symmetric groups are treated in some detail. My focus then turns to Lie groups and Lie algebras and finally to my heart: working out the finite dimensional representations of the classical groups and exploring the related geometry. The goal of my last portion is to make a bridge between the example-oriented approach of the earlier parts and the general theory. Which Springer GTM would |

"The moving power of mathematical invention is not reasoning, but imagination." ~Augustus DeMorgan