AMZI JEFFS

I am an NSF Postdoctoral Associate at Carnegie Mellon University, working in combinatorics and discrete geometry with Florian Frick. In 2021 I completed my PhD at the University of Washington under Isabella Novik. My most recent work focuses on classifying convex neural codes and their embedding dimensions.
PUBLICATIONS (click titles below for pdfs)

- Morphisms, Minors, and Minimal Obstructions to Convexity of Neural Codes

PhD Thesis. 2021. - Order-Forcing in Neural Codes

with Caitlin Lienkaemper and Nora Youngs. 2020. - Non-Monotonicity of Closed Convexity in Neural Codes

with Brianna Gambacini, Sam Macdonald, and Anne Shiu. Under review, 2020. - Embedding Dimension Phenomena in Intersection Complete Codes

Under review, 2019. - Sunflowers of Convex Open Sets

Published in*Advances in Applied Mathematics*, Vol 111, 2019. - Convex Union Representability and Convex Codes

with Isabella Novik. Published in*International Mathematics Research Notices*, 2019. - Morphisms of Neural Codes

Published in*SIAM Journal on Applied Algebra and Geometry*, Vol 4, 99-122, 2020. - Neural Ideal Preserving Homomorphisms

with Mohamed Omar and Nora Youngs. Published in*Journal of Pure and Applied Algebra*, Vol 222, 3470-3482, 2018. - Sparse Neural Codes and Convexity

with Mohamed Omar, Natchanon Suaysom, Aleina Wachtel, and Nora Youngs. Published in*Involve*, Vol 12, 737-754, 2019. - Characterizing the Cryptographic Properties of Reactive 2-Party Functionalities

with Mike Rosulek. Published in*Theory of Cryptography Conference*, 2013.

RESEARCH (return to top)

I am interested in discrete geometry and combinatorics, particularly applications of geometric techniques to combinatorial problems. My undergrade and graduate studies have focused on convex neural codes. I plan to pursue further research in this field while exploring related topics such as matroids and convex geometry.

**Convex neural codes:**Given a collection of

*n*convex open sets in Euclidean space, one can form a combinatorial code (a subset of the Boolean lattice on

*n*elements) that describes which of the sets have nonempty intersection, which ones cover others, and so forth. The reverse problem is more difficult: given a code, can you find a collection of convex open sets that corresponds to it? If so, we call the code convex. The problem of classifying convex codes is motivated by a problem in neuroscience, and was introduced to the math community by Curto et al in 2013. Since then it has generated a sizeable body of work and motivated the development of new perspectives and tools in combinatorics, topology, and discrete geometry.

During my graduate work I have approached this question from a combinatorial and geometric direction. In 2018 I introduced a notion of "minors" for codes, in analogy to graph minors, with the goal of isolating "minimal obstructions" to convexity in codes. Kunin, Lienkaemper, and Rosen recently used this framework to illustrate connections between the theory of convex codes and the theory of oriented matroids.

I have also formulated new discrete geometry theorems that help explain why certain codes are not convex. These theorems describe constraints on "sunflowers" of convex sets and generalizations thereof. A good summary of this work can be found in my talk "Convex Sunflower Theorems and Neural Codes" from June 2020.

ANNOTATED FIGURES (return to top)

Below are figures from several of my papers with accompanying commentary. I chose these either because they illustrate a particularly interesting idea, or just for their visual appeal. In general I greatly enjoy the concrete and visual side of discrete geometry, and if you ever find yourself in need of a mathematical illustration I would be happy to chat!

The figure on the left illustrates the positions of several codes in the poset of all codes ordered by "minors." The property of being a convex code is closed under taking minors, so convex codes form a down-set in this partial order. The boundary of this down-set is indicated by the thin wavy line, and determining the codes that lie on this boundary (i.e. the "forbidden minors of convexity" or "minimally non-convex" codes) is a question of some interest.

The code C is the first example of a locally good non-convex code, described by Lienkaemper, Shiu, and Woodstock. It turns out the code C is not minimally non-convex, but moving downwards by a couple of covering relations in the partial order gives us a minimally non-convex code. The codes D and E help us recognize this. This figure appears in my first grad school paper [4] listed above.

The code C is the first example of a locally good non-convex code, described by Lienkaemper, Shiu, and Woodstock. It turns out the code C is not minimally non-convex, but moving downwards by a couple of covering relations in the partial order gives us a minimally non-convex code. The codes D and E help us recognize this. This figure appears in my first grad school paper [4] listed above.

The figure above shows that a closed convex realization of a code can be turned into an open realization by adding a small open ball to each set, as long as the code is a simplicial complex. To my knowledge, this result was fairly well known for several years, but the first time it was put in writing was in the paper [7] listed above.

This figure comes from [5], and shows that a "convex union representation" of a simplicial complex can be extended in a consistent way to a convex union representation of a full simplex. We used this construction to show that the Alexander dual of any convex union representable complex is collapsible.

Both figures above concern "sunflowers" of convex open sets. I first started working with these arrangements of convex sets in [6], but both figures above relate more closely to the content of [7]. Understanding these arrangements, in particular the smallest dimension in which one can achieve them, is a key question in understanding the convexity of codes. It turns out that the minimum dimension can be nicely understood for the (infinite) family partially illustrated in the top figure, but that the family beginning in the bottom figure is much more mysterious. All of the illustrations above are done in the smallest dimension possible.

The first figure on the left illustrates a "flexible" sunflower of convex open sets, a choice of one point from each set in the flexible sunflower, and the convex hull of these points. Constraints on sunflowers of convex open sets can be described by a nice Helly-style theorem (see [6]), and I was able to generalize this theorem to flexible sunflowers in [7]. In particular, the fact that the grey convex hull overlaps with the square where all sets in the sunflower meet is a general phenomenon.

The sunflower phenomena mentioned above can be described and interpreted using codes and minors thereof. The figure on the left illustrates a family of codes related to sunflowers which has a particularly nice structure: it sits in the layer of codes that have convex realizations in

OTHER ACTIVITIES (return to top)
The sunflower phenomena mentioned above can be described and interpreted using codes and minors thereof. The figure on the left illustrates a family of codes related to sunflowers which has a particularly nice structure: it sits in the layer of codes that have convex realizations in

*m*-dimensional space, and it has a unique minimum minor. Both figures on the left come from [7].During my graduate studies I was an active organizer and Joint Council member with UAW4121, the union of academic workers at the University of Washington. I am proud to have taken action alongside fellow grad students and workers to win gains like our fully-subsidized transit passes, strong protections from harassment, trans-inclusive healthcare measures, and improved mental health care. I am also active in grassroots political organizing, for example volunteering with the Tax Amazon movement, which won hundreds of millions of dollars in funding for affordable housing in Seattle. More recreationally, I enjoy hiking, cooking, and parkour.