
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 looselyrelated 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 3manifolds, and mirror symmetry
for open strings.
Other Quarters: [
Fall, 2017;
Spring, 2017;
Wnter, 2017;
Fall, 2016;
Spring, 2016;
Winter, 2016;
Fall, 2015;
Spring, 2015;
Winter, 2014;
Fall, 2013;
Fall, 2012;
Fall, 2011;
Winter, 2011;
Spring, 2010;
Winter, 2010;
Fall, 2009;
Spring, 2009;
Winter, 2009;
Fall, 2008;
Spring, 2008;
Winter, 2008;
Fall, 2007;
Spring, 2007;
Winter, 2007;
Fall, 2006
]
October 13 
Dave Morrison (UCSB)
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 socalled Dwork pencil
of quintic threefolds,
which include a positivedimensional 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 threemanifolds are built from
elementary pieces, so we are hoping to predict which hyperbolic threemanifold
should serve as the mirror of the van Geemen family
on the basis of this computation.
There are important connections between hyperbolic threemanifolds 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.
Audio; Lecture notes.

October 20 
No meeting

October 27 
Dave Morrison (UCSB)
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 3manifolds 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 Ktheory, 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 BlochWigner function) will be surveyed. along with applications.
Audio; Lecture notes.

December 8 
Dave Morrison (UCSB)
Abstract:
The original AbelJacobi map from a complex algebraic curve to its Jacobian
was generalized by Griffiths to the following construction, also called the
AbelJacobi map. The p^{th} 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
welldefined up to integrals over $(2p+1)$cycles. Thus, $Z$ determines a
welldefined point in the p^{th} 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 AbelJacobi map.
In this lecture, following a paper of Kerr, Lewis, and MüllerStach,
we will present a refinement of the AbelJacobi 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 3manifolds.
Audio; Lecture notes v1; Lecture notes v2.


