Scott Neville

We subsantially resolve the Asymptotic sign coherence conjecture of Gekhtman and Nakanishi, which states that if you take a quiver with frozen vertices and mutate long enough, then it will eventually satisfy the sign coherence phenomena (which states that each mutable vertex is either a source or a sink with respect to all of the frozen vertices). We do this by introducing a new class of quivers, called BROG. We also resolve the asymptotic sign coherence conjecture for mutation abundant (always at least 2 arrows between the mutable vertices) 3-vertex quivers.

There are at least two ways to describe a variety associated to a cluster algebra. One can either take the union over all seeds, selecting nonzero elements for each variable (for each seed, this gives a cluster tori. Call the union the cluster manifold), or one can directly look at maps from all cluster variables simultaneously which satisfy all of the cluster relations (the cluster variety). These two choices are subtly different. Call points in the variety but not the manifold 'deep'. It was conjectured that the deep points of any locally acyclic cluster algebra would all have a simple witness of their depth; a stabilizer in the cluster dilation group. We show this conjecture holds for tree quivers (without multiplicities), but fails for key quivers (this is the literal name of a class of acyclic quivers, see the related paper by Ervin).

A cyclically ordered quiver is a quiver endowed with an additional structure of a cyclic ordering of its vertices - one might imagine drawing the vertices of the quiver on the unit circle. This structure, which naturally arises in many important applications, gives rise to new powerful mutation (semi) invariants. The invariants we find are truly mutation invariants when the cyclic ordering is 'totally proper,' we describe several families of quivers with totally proper cyclic orderings.
For quivers coming from divides, the most natural cyclic ordering leads to the Alexander polynomial of the associated knot.
I resolved an important conjecture from this paper in the follow-up. I showed that all acyclic quivers are totally proper, and thus we have new necessary conditions for a quiver to be mutation equivalent to an acyclic one. This relies heavily on a construction by Ahmet Seven.
Additional follow-up, in preperation with Hugh Thomas, extended cyclically ordered quivers to skew-symmetrizable matrices. Preprint in preperation.
With Tucker Ervin, I developed python code for many mutation invariants, including several from this project.

This project studies mutation cycles in quivers. We prove the existence of arbitrarily long cycles on n vertices for every fixed n ≥ 4 and construct large parameter families exhibiting strong rigidity properties.
Follow-up work with Tucker Ervin showed how this construction is part of a huge family of long mutation cycles, which can be formed by arbitrary triangular extensions of quivers with reddening sequences. This publication includes a large zoo of new, strange examples.

In the directed polymer model, we have two different kinds of randomness being played against each other. First, we have the environment, consisting of random weights on the points in a lattice, which we resample independently after each time step. Second, a random walk, which has some preferred distribution of next 'steps' on the lattice. The model produces a distribution on the possible resulting walks from the origin, where the walk is biased by the random environment based on a parameter (the temperature).
In dimension 1, the directed polymer model is in the celebrated KPZ universality class, and for all positive temperatures, a typical polymer path shows non-Brownian KPZ scaling behavior. In dimensions 3 or larger, it is a classical fact that the polymer has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can induce the phase transition in the directed polymer model.

Neural networks often outperform intentionally designed programs for many 'real world' tasks. A natural question is if we can use a trained neural network to find a more classical algorithm in reasonable time. One theory was that neural networks work by extracting and removing invariants of the data. This begs a computational question: what are some transformations of the input data which do not change a neural networks output. We show this is generally infeasible, in the sense that determining if a neural network is invariant to some permutation of the inputs is NP-hard (even under loose definitions of invariant).

The 1983 Putnam competition included a problem which asked about the number of partitions with only 2^k parts (binary partitions) with each part used at most 3 (=2^2-1) times. We generalize this to m-ary partitions (so parts m^k) with each part used at most m^j -1 times. Generating functions are very up to this task, and reveal a cute symmetry. One can interchange the roles of k and j. We give a bijection to explain this symmetry, based on a classical transposition argument for partitions with bounded sizes vs bounded number of parts. (Color the boxes and read backwards!)
Story: Timothy Flowers gave a short talk on the problem and generating function results at an AMS mathfest. I thought the problem fun, and found a bijection to explain their results over lunch.