Classically, it is known that every algebraic variety is birational to a hypersurface in some projective or affine space. Using generic linear projections, Doherty proved that over the complex numbers, this hypersurface can be taken to have at worst semi-log canonical singularities in dimensions up to five. This extends classically known results for curves and surfaces. We present a positive-characteristic analogue of Doherty's theorem, by showing that in the same dimensions, generic projection hypersurfaces in positive characteristic are F-pure. This proves cases of a conjecture of Bombieri, Andreotti, and Holm. To show our result, we study F-injective singularities, which are the analogue of Du Bois singularities in positive characteristic, and their behavior under flat morphisms. This work is joint with Rankeya Datta.

Given a graph $H$, let $\hat{R}(H)$ be the minimum $m$ such that there exists a graph $G$ with $m$ edges such that in every $2$-coloring of the edges $G$, there is a monochromatic copy of $H$. To prove that $\hat{R}(P_n)>m$, one must show that every graph on $m$ edges can be $2$-colored such that every monochromatic path has order less than $n$. We show that $\hat{R}{P_n} > (3.75 - o(1))n$ thereby improving the previous best-known lower bound of $(2.5 - o(1))n$ due to Dudek and Pralat. We also discuss some results concerning the $r$-color version of the problem. This is joint work with Louis DeBiasio.

Cluster varieties are blow up of toric varieties. They come in pairs (A,X), with A and X built from dual tori. Compactifications of A, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the A and the X cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists a positive polytope for the type A cluster varieties which give us a hint to the Batyrev-Borisov construction.