Geometry, Topology, and Physics Seminar, Fall 2017

Part of the NSF/UCSB ‘Research Training Group’ in Topology and Geometry

Organizers: Dave Morrison and Zhenghan Wang.
Meets 4:00 - 5:30 p.m. on selected Fridays in South Hall 6635.

This fall, we shall explore a collection of loosely-related topics (connections among which will be described on October 13): higher Chow groups, dilogarithms, lines on the Dwork pencil of quintic threefolds, division algebras over algebraic number fields, volumes of hyperbolic 3-manifolds, and mirror symmetry for open strings.

October 13

### Higher Chow groups, van Geemen lines, and mirror symmetry for open strings

Abstract: We shall describe a puzzle which has arisen in the study of mirror symmetry for open strings: certain calculated numbers, which naïvely would be expected to be integers, are in fact irrational algebraic numbers. The numbers in question are related to lines on the so-called Dwork pencil of quintic threefolds, which include a positive-dimensional family of lines, the "van Geemen lines." An ongoing quest of mine, in collaboration with Hans Jockers and Johannes Walcher, is to understand open mirror symmetry in this case. As one step in our quest, we were led to consider higher Chow groups, and have successfully made a computation in that context (which will be the topic of a later lecture). The structure of that computation is reminiscent of the way in which hyperbolic three-manifolds are built from elementary pieces, so we are hoping to predict which hyperbolic three-manifold should serve as the mirror of the van Geemen family on the basis of this computation. There are important connections between hyperbolic three-manifolds and algebraic number theory which may lead to an explanation of the original computation, including an a priori explanation of which number number field the computation takes values in.

October 20

October 27

### The dilogarithm function

Abstract: This is the first of an occasional series of lectures explaining more of the details of the work on higher Chow groups, mirror symmetry, and hyperbolic 3-manifolds which I outlined two weeks ago. The dilogarithm function has played an interesting role in various branches of mathematics over the past 25 years or so. It finds uses in number theory, in algebraic K-theory, in the study of hyperbolic manifolds, and in conformal field theory. The basic properties of Euler's dilogarithm function and its modern variants (e.g. the Bloch-Wigner function) will be surveyed. along with applications.

December 8

### Higher Chow groups and the Abel-Jacobi map

Abstract: The original Abel-Jacobi map from a complex algebraic curve to its Jacobian was generalized by Griffiths to the following construction, also called the Abel-Jacobi map. The pth intermediate Jacobian of an algebraic variety $X$ is the complex torus $(H^{2p+1,0}(X) + H^{2p,1}(X) + ... + H^{p+1,p}(X))^*/H_{2p+1}(X,\mathbb{Z}),$ where the integer $(2p+1)$-cycles define linear maps on de Rham cohomology via integration.

Consider an algebraic cycle $Z$ on $X$ of (complex) dimension $p$ which is homologous to zero. Then there is a $(2p+1)$-chain $\Gamma$ whose boundary is $Z$. We can integrate $(2p+1)$-forms over $\Gamma$ and get answers which are well-defined up to integrals over $(2p+1)$-cycles. Thus, $Z$ determines a well-defined point in the pth intermediate Jacobian. The image only depends on the rational equivalence class of $Z$, i.e., on the corresponding element of the Chow group. This is Griffiths' version of the Abel-Jacobi map.

In this lecture, following a paper of Kerr, Lewis, and Müller-Stach, we will present a refinement of the Abel-Jacobi map which is relevant when $X$ has a natural mixed Hodge structure rather than a pure Hodge structure. It involves some higher Chow groups which were originally defined by Bloch.

This is an ingredient in my work in progress (with Jockers and Walcher) on higher Chow groups, mirror symmetry, and hyperbolic 3-manifolds.