# The topology of projective codes and the distribution of zeros of odd maps

@inproceedings{Adams2021TheTO, title={The topology of projective codes and the distribution of zeros of odd maps}, author={Henry Adams and Johnathan Bush and Florian Frick}, year={2021} }

. We show that the size of codes in projective space controls structural results for zeros of odd maps from spheres to Euclidean space. In fact, this relation is given through the topology of the space of probability measures on the sphere whose supports have diameter bounded by some speciﬁc parameter. Our main result is a generalization of the Borsuk–Ulam theorem, and we derive four consequences of it: (i) We give a new proof of a result of Simonyi and Tardos on topological lower bounds for…

## 6 Citations

### The Persistent Topology of Optimal Transport Based Metric Thickenings

- Mathematics
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A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X . Thickenings have found use in applications of topology to data analysis, where one may…

### Vietoris thickenings and complexes have isomorphic homotopy groups

- MathematicsJournal of Applied and Computational Topology
- 2022

. We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let U be a cover of a separable metric space X by open sets with a uniform…

### On Vietoris-Rips complexes of hypercube graphs

- MathematicsJ. Appl. Comput. Topol.
- 2022

We describe the homotopy types of Vietoris–Rips complexes of hypercube graphs at small scale parameters. In more detail, let Qn be the vertex set of the hypercube graph with 2n vertices, equipped…

### On Borsuk-Ulam theorems and convex sets

- Mathematics
- 2021

. The Intermediate Value Theorem is used to give an elementary proof of a Borsuk-Ulam theorem of Adams, Bush and Frick [1] that, if f : S 1 → R 2 k +1 is a continuous function on the unit circle S 1…

### Statement of Recent Work : Henry Adams

- Mathematics
- 2022

4 C ec hA nd V ie to ri sR ip s. nb Large sets of high-dimensional data are common in most branches of science, and their shapes reflect important patterns within. The goal of topological data…

### Research Statement: Bridging applied and quantitative topology

- Mathematics
- 2022

Henry Adams, Colorado State University Large sets of high-dimensional data are common in most branches of science, and their shapes reflect important patterns within. The goal of topological data…

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