Zero schemes of exact 1-forms have received more attention recently as moduli spaces associated to Calabi-Yau threefolds; they are called gradient schemes or critical schemes. In this talk I will introduce the notion of "multi-gradient schemes" as an obvious generalization and explain their classification in the codimension one and monomial cases, as well as how they naturally arise as certain moduli spaces associated to varieties with globally generated canonical bundles.

The classical zero-one law for first-order logic on random graphs says that for any first-order sentence F in the theory of graphs, the probability that the random graph G(n, p) satisfies F approaches either 0 or 1 as n grows. It is well known that this law fails to hold for properties involving parity phenomena (oddness/evenness): for certain properties, the probability that G(n, p) satisfies the property need not converge, and for others the limit may be strictly between 0 and 1.

In this talk, I will discuss the behavior of FO[parity], first order logic equipped with the parity quantifier, on random graphs. Our main result is a "modular convergence law" which precisely captures the behavior of FO[parity] properties on large random graphs.

I will give an overview of this result and its proof. Along the way, we will ask (and answer) some basic, natural questions about the distribution of subgraph counts *mod 2* in random graphs (what is the probability that G(n,p) has: an odd number of triangles? an even number of 4-cycles? an odd number of triangles and an even number of 4-cycles? etc.). Our approach is based on multivariate polynomials over finite fields, in particular, on a variation on the Gowers norm. The proof generalizes the original quantifier elimination approach to the zero-one law, and has analogies with the Razborov-Smolensky method from circuit complexity.

Joint work with Phokion Kolaitis.

Given a graph H, the size Ramsey number r_e(H,q) is the minimal number m for which there is a graph G with m edges such that every q-coloring of G contains a monochromatic copy of H. We study the size Ramsey number of the directed path of length n in oriented graphs, where no antiparallel edges are allowed. We give nearly tight bounds for every fixed number of colors. For the case of two colors we show that there are constants c_1,c_2 such that \frac{c_1 n^{2} \log n}{(\log\log n)^3} \leq r_e(P_n,2) \leq c_2 n^{2}(\log n)^2.

Joint work with Michael Krivelevich and Benny Sudakov.

The systolic inequality says that any Riemannian metric on an n-dimensional torus with volume 1 contains a non-contractible closed curve with length at most C(n) - a constant depending only on n. One remarkable feature of the inequality is it holds for such a wide class of metrics. It's much more general than an inequality that holds for all metrics obeying a certain curvature condition.

The systolic inequality is a difficult theorem, and each proof is guided by a metaphor that connects the systolic inequality to a different area of geometry or topology. In this talk, I will explain three metaphors. They connect the systolic inequality to minimal surface theory, topological dimension theory, and scalar curvature.