Kyle Hansen
PhD

UCSB Department of Mathematics
South Hall 6431K

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

Welcome

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

About Me

I used to be a graduate student in mathematics at UC Santa Barbara. I defended my thesis on 2025-05-07. I enjoy baking and long walks on the beach.

Here is a past CV.

Here is my portfolio for the Certificate in College and University Teaching (CCUT) program. Here is a description of the program.

Teaching

Past

Term	Course Name	Course Number	Institution	Role
2025 Spring	Math Lab	N/A	UCSB	Coordinator
2025 Winter	Methods of Analysis	MA-117	UCSB	TA
2024 Fall	Modern Euclidean and Non-Euclidean Geometry	MA-102A	UCSB	Grader
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.

Research, Talks, and Publications

Research

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 asked "Given a nilpotent Lie group G, what algebraic information is necessary/sufficient to guarantee that a closed manifold M is G-elliptic?" In some sense, this asks which manifolds M can be efficiently wrapped using the group G as your wrapping paper. Until recently, there has been little indication of how to solve this problem. Recent advances in related areas have opened up some clear paths forward in this investigation, where it seems likely that the Heisenberg group is a poor choice of wrapping paper for a flat torus.

Papers

Domesticated (i.e., peer-reviewed)

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

Feral (i.e., pre-prints & arXiv)

• Simple Tilings of Nilpotent Lie Groups (2024)
  • arXiv
  • PDF

Talks

• Volumes of Nullhomotopies in Nilpotent Spaces (PhD Defense, 2025/05/07)
  • PDF
• Mathematical Quasicrystals: The Taming of the Hullabaloo (Undergraduate Colloquium, 2025/02/05)
  • PDF
• Tiling Spaces Which are Fiber Bundles over Nilmanfiolds (41st Workshop in Geometric Topology, 2024/06/13)
  • PDF
• The Hull of a Manifold with Bounded Geometry (UCSB Topology Seminar, Advancement Talk 2022/05/24)
  • PDF
• An Introduction To Tiling Spaces (UCSB Graduate Topology Semniar, 2022/04/06)
  • PDF

Other Unpublished Works (definitely not peer-reviewed)

• Elliptic Curve Cryptography (2021)
  • PDF
• Much Ado About Nothing, or All For Naught? Investigating the Zeros of Complex Functions (Undergraduate Honors Thesis, 2019)
  • PDF

Contact

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 Representation Theory: A First Course. 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 you be? The Springer GTM Test