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► Forbidden Posets: Small Posets on Small Lattices, Georgia Sanders, Laura Prince, Michael Pilson

Advisor:  Shanise Walker
Abstract:  Induced saturation for posets was introduced by Ferrara et al. in 2017. An induced family $\mathcal{F}$ in $\mathcal{B}_n$ is induced-$\mathcal{P}$-saturated if it does not contain an induced copy of $\mathcal{P}$, but every proper induced superset of $\mathcal{F}$ contains an induced copy of $\mathcal{P}$. Ferrara et al. established on the induced saturation number for various small posets and a logarithmic lower bound was proved for the saturation number of posets from a rich infinite family.

We proved lower bounds on size of an induced saturated family for the $k$-diamond poset $\mathcal{D}_k$, the fork with $k$ tines $\mathcal{V}_k$, and its dual $\Lambda_k$ when the family contains a maximal chain. For posets $\mathcal{V}_3$, and $\Lambda_3$, we give an exact result when the family lies in the $3$-dimensional Boolean lattices. We give an exact bound for the poset $2\mathcal{C}_2$ of two incomparable pairs in the 3-dimensional lattice.

► Markoff triples modulo p, Alette Wells, Vernon Naidu, Claire Dunn

Advisor:  Elisa Bellah
Abstract:  The Markoff equation $X_1^2+X_2^2+X_3^2 = 3X_1X_2X_3$ is well-known to have infinitely many integer solutions called Markoff triples, which can be arranged into a tree via the action of the Vieta group. Analogously, the modulo $p$ solutions to the Markoff equation can be arranged into a graph via the action of the Vieta group. It can be shown that this graph is connected if and only if any modulo $p$ solution to the Markoff equation has an integer lift.

In 2016, Bourgain, Gamburd, and Sarnak proved that a component of this graph called the cage is always connected by corresponding the orbit lengths of Markoff triples modulo $p$ to the order of certain matrices in $\text{GL}_2(\mathbb{Z}/p\mathbb{Z})$. In our research, we explore how close the special point $(1,1,1)$ is to the cage and how many Markoff triples are in the cage. Since $(1, 1, 1)$ remains fixed under reduction modulo any prime, the connectedness of $(1, 1, 1)$ to the cage would guarantee that every modulo $p$ point in the cage has an integer lift, and thus if there are ``many" points in the cage we can guarantee many integer lifts.

Our main result generalizes ideas from Vince in 1978 to show that $(1,1,1)$ is connected to the cage for special families of primes. A result of this theorem is that for Mersenne primes greater than $5$ where $p$ is congruent to $\pm 2$ modulo $5$, $(1,1,1)$ is in the cage. We also explore the probability that a point is in the cage and provide a lower bound using heuristic methods, as well as providing a promising method using the inclusion-exclusion principal to improve the lower bound.

To reach our main results, we experimented using various coded programs to gather data, which led to several further observations and conjectures. Namely, through numerical experimentation we provide a conjecture for the density of primes where $(1, 1, 1)$ is in the cage.