Mathematics

The Degrees of Varieties

Jason McCullough (Iowa State University)

Abstract: Let $K$ be an algebraically closed field and let $V$ be a projective variety (reduced, irreducible, nondegenerate) embedded in $\mathbb{P}^n_K$. There are two notions of degree associated to $V$: (1) the degrees of the generators of the defining homogeneous ideal of $V$, and (2) the number of points of intersection between $V$ and a linear space of complementary dimension. In the plane these two notions agree, but in higher dimension there was a only the conjectured inequality $\mathrm{maxdeg}(P) \leq \deg(V(P))$ for a prime ideal $P$. Recent joint work with Irena Peeva produced counterexamples to this conjecture and the Eisenbud-Goto conjecture. However, using the recent solution to Stillman's Conjecture by Ananyan and Hochster, we show that there is a bound on the defining equaltions of any variety purely in terms of its degree. This is joint work with Caviglia, Chardin, Peeva, and Varbaro.

The Erdős–Gallai theorem for cycles in hypergraphs

Ruth Luo (UIUC Math)

Abstract: The Erdős–Gallai theorem states that if a graph $G$ on $n$ vertices has no cycle of length $k$ or longer, then $e(G) \leq (k-1)(n-1)/2$. We present a hypergraph analogue of this theorem. A berge-cycle of length $\ell$ in an $r$-uniform hypergraph is a set of $\ell$ hyperedges $\{e_1, ..., e_\ell\}$ and $\ell$ vertices $\{v_1, ..., v_\ell\}$ such that hyperedge $e_i$ contains the vertices $v_i$ and $v_{i+1}$. We show that for $r \geq k+1$, if $H$ is an $r$-uniform hypergraph on $n$ vertices with no berge-cycle of length $k$ or longer, then $|H| \leq (k-1)(n-1)/r$. This is joint work with Alexandr Kostochka.

Department Awards Ceremony