I study algebraic topology, particularly homotopy theory, K-theory, and their applications to manifolds. My favorite problems involve understanding topological/geometric structure using the tools of homotopy theory. Currently I’m thinking about square *K*-theory, Floer homotopy theory, and some equivariant homotopy theory.

I like way Fields medalist Maryam Mirzakhani described mathematical research: it’s like “being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.”

**Research Papers
**

- The Spectrum of the Burnside Tamara Functor of a Cyclic Group (with Sam Ginnett). To appear in the
*Journal of Pure and Applied Algebra*. Also available on arXiv.

- We determine a family of prime Tambara ideals in the Burnside Tambara functor on a group
*G*. When*G*is cyclic, we show that this family comprises the entire prime ideal spectrum of the Burnside Tambara functor.

- The Tambara Structure of the Trace Ideal (with Sam Ginnett).
*Journal of Algebra*: Vol. 560 (2020). Also available on arXiv.

- We study the kernel of the Dress map as a morphism from the Burnside Tambara functor to the Grothendieck-Witt (Galois) Tambara Functor. In certain cases, we can explicitly determine the generators of this kernel as a Tambara ideal.

- Check out slides for my talks about this paper: a long version (~50 min) presented at the Reed College Student Colloquium (2020) and a short version (~15 min) presented at the Nebraska Conference for Undergraduate Women in Mathematics (2020).

- Sharp Sectional Curvature Bounds and a New Proof of the Spectral Theorem (with Corey Dunn).
*Involve, a Journal of Mathematics*: Vol. 13, No. 3 (2020). Also available on arXiv.

- We compute and bound the possible sectional curvature values for a canonical algebraic curvature tensor, and geometrically realize these results to produce a hypersurface with prescribed sectional curvatures at a point. We also give a relatively short proof of the spectral theorem for self-adjoint operators on a finite-dimensional real vector space.

*k*-Plane Constant Curvature Conditions.*Rose Hulman Undergraduate Journal of Mathematics*: Vol. 20, Iss. 2 (2019).

- Constant sectional curvature and constant vector curvature are two curvature invariants of an algebraic curvature tensor which take in
*2*-planes as input. We generalize these invariants to take*k*-planes as input and explore their structure. Just as in the*k=2*case, we show that a space with constant*k*-plane scalar curvature has a uniquely determined tensor and that a tensor can be recovered from its*k*-plane scalar curvature measurements.

- Check out slides for a talk (~20 min) which I presented at various undergraduate symposiums circa 2018.

**Theses**

- An Introduction to Symplectic Geometry for Lagrangian Floer Homology. Expository master’s thesis (2022) written as part of my Ph.D. qualifying exam, supervised by Prof. Jonathan Block.

- This thesis introduces symplectic geometry with an eye towards developing Floer homology for Lagrangian intersections. My secret motivation for producing this document is to provide a handy overview of symplectic geometry for homotopy theorists who are interested in learning more about Floer homotopy theory.

- Here are my notes from a talk (~50min) for the graduate geometry/topology seminar.

- Morse Theory and Flow Categories. Reed College undergraduate thesis (2020), advised by Prof. Kyle Ormsby.

- In an unpublished preprint from the 1990s, Cohen-Jones-Segal claim that the homotopy type of a manifold can be recovered from the classifying space of a flow category, which is formed from the data of a Morse function. The paper was never published due to gaps in the proofs, some of which have since been addressed. This thesis explore this story in more detail, starting with the basics of Morse theory, and attempts to fix one of the errors.

*Disclaimer:*there are some errors in the later sections that I am working to address.- Here’s a blog post about what went wrong. You can also check out a notes from a talk (~50min) I gave in the graduate geometry/topology seminar.

**Notes, Expository Papers, and Slides**

- Square
*K*-Theory and Manifold Invariants. Write-up for a talk at Talbot 2022: Scissors Congruence and Algebraic*K*-theory (Summer 2022). Check out the slides for my presentation (~50min). - A Bit About Infinite Loop Spaces. An expository overview of infinite loop space theory written for Math 619: Algebraic Topology I (Spring 2021) at UPenn, with Prof. Mona Merling. Check out the slides for my presentation (~25min).
- Freudenthal Suspension Theorem. Supplementary write-up to presentation for the Algebraic Topology Bridge Summer workshop (Summer 2020). Check out the slides for my presentation (~50 min).

Slides from other expository talks:

- Equivariant Bundle Theory and Classifying Spaces (~50m), presented at eCHT’s equivariant homotopy theory reading seminar (Fall 2021).
- Bousfield Localization (~50min), presented at UPenn’s chromatic homotopy theory seminar (Summer 2021).
- Topological K-theory (~50min), presented at Algebraic Topology Bridge Summer Workshop (Summer 2021).

**More Misc. Stuff
**

- Lens space: a CW story video I made for Math 810: Video Production for Mathematics seminar (Fall 2021) at UPenn, taught by Prof. Rob Ghrist.
- How to Write Mathematics Badly (transcript). A (somewhat incomplete) transcript of this public lecture given by Jean Paul Serre.

From undergraduate:

- I did some paintings of the Hopf fibration (inspired by this video by Niles Johnson) which I hung up in Reed’s math lounge.
- Configuration Spaces and Robots video, with Lucas Williams. We submitted the video to the Elevating Mathematics Video Competition and received an honorable mention. The video was inspired by this paper by Williams, completed under the supervision of Prof. Safia Chettih.
- Gauss’ Class Number Problems and the Determination of Imaginary Quadratic Fields with Class Number One. For Math 361: Number Theory (Spring 2019) at Reed College, with Prof. Jerry Shurman.
- On the Flipside: Refinements of Polytopal Subdivisions and Secondary Polytopes. For Math 341: Topics in Geometry, Polytopes (Fall 2018) at Reed College, with Prof. Angélica M. Osorno.
- A Case for Quotienting: Equivalence and Postmodal Mathematical Structuralism. For Phil 411: Advanced Topics in Metaphysics, Metaphysics of Science (Fall 2018) at Reed College, with Prof. Troy Cross.

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