Eduardfest

Title: Algebraic geometry of a desmic system of tetrahedra

Abstract: In projective geometry, a desmic system of tetrahedra is a set of three tetrahedra such that any two of them are perspective from any vertex of the third one.

In algebraic geometry, it appears as the set of singular members in the pencil of quartic surfaces birationally isomorphic to the Kummer surface of the self-product of an elliptic curve. In my talk, if time permits, I will discuss the relationship of desmic tetrahedra to  Fano 3-folds of degree 6 and cubic 4-folds with the maximal number of nodes,  cubic surfaces, the Weyl group $W(F_4)$, and linear representations of Vinberg's $\theta$-groups.

Title: Matroids and the integral Hodge conjecture

Abstract: Associated to any regular matroid of rank g on k elements, one can associate a multivariable semistable degeneration of principally polarized abelian g-folds over a k-dimensional base. I will discuss joint work with de Gaay Fortman and Schreieder, proving that a combinatorial invariant of the matroid obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. Corollaries include the failure of the integral Hodge conjecture for abelian varieties of dimension ≥ 4 and the stable irrationality of very general cubic threefolds.

Title: Homological perturbation theory for L-infinity algebras and higher holonomy

Abstract: Lie showed how to construct the Lie algebra of a Lie algebra by integrating ordinary differential equations. In modern language, the solution of the ODE associates to a path in the Lie algebra its holonomy in the Lie group. A path in the Lie algebra is the same thing as a connection on the interval  [0,1], and its holonomy may be characterized as the constant connection that is gauge equivalent to it.

In this talk, we show how to perform a similar construction on higher dimensional cubes, defining higher dimensional holonomy of connections for differential graded Lie algebras (and more generally L-infinity algebras). We use homological perturbation theory for L-infinity algebras to obtain explicit formulas (that, in principle, could be calculated by computer).

Title: Hyperelliptic Torelli groups and higher algebraic cycles associated to hyperelliptic curves

Abstract: In this talk I will discuss the problem of constructing abelian quotients of hyperelliptic Torelli groups and its consequences for the existence of higher Chow cycles in self products of hyperelliptic curves. This parallels the well known relationship between the classical Johnson homomorphism and algebraic cycles in powers of a smooth projective curve. This talk will focus mainly on the topological aspects of the story.

Title: Mapping class groups of del Pezzo manifolds

Abstract: A del Pezzo manifold is the underlying smooth manifold of a del Pezzo surface, and is one of $\mathbb{CP}^2 \# n\overline{\mathbb{CP^2}}$ for $0 \leq n leq 8$ or $\mathbb{CP}^1 \times \mathbb{CP}^1$. In this talk I will discuss various Nielsen realization problems for the topological mapping class groups of del Pezzo manifolds with connections to Coxeter groups, birational geometry of $\mathbb{CP}^2$, and Lefschetz fibrations. Many of these results are inspired by work of Farb--Looijenga on the mapping class group of the K3 manifold. This talk includes joint work with Tudur Lewis and Sidhanth Raman.