Abstracts for Seminar on Algebra, Geometry and Physics

My goal is  to explain how I use a sort of Schubert incidence geometry to
prove complex geometric properties, e.g., Kobayashi hyperbolicity, of
spaces of algebraic cycles in open orbits of real forms in flag manifolds
of their complexifications. These results can then be transferred back to
derive properties of the flag domains.  The main results will appear in the
American Journal some time in 2012 .
 

I will describe recent work with Goncalo Tabuada, where we consider analogs
of Grothendieck's standard conjectures C and D for a suitable category of
noncommutative numerical motives and we show that, assuming these conjectures,
one can make this category into a Tannakian category. The motivic Galois group
of this category surjects onto the kernel of the homomorphism from the motivic Galois
group of the category of (commutative) numerical motives to the multiplicative group,
determined by the inclusion of the subcategory of Tate motives.

 

A Calabi-Yau ($CY$) structure of dimension $d$ on a compact $A_\infty$ algebra $A$ is
a degree $d$ non-degenerate, cyclically invariant pairing on $A$. It is well known that the
complex of Hochschild chains of a compact $A_\infty$, $CY$ algebra has the structure of
a TQFT, namely it is an algebra over a PROP of chains in the moduli space
$\mathcal{M}_{g,\vec{m},\vec{n}}$ of Riemann surface of genus $g$ with $m$ incoming
and $n$ outgoing marked points with real tangent directions, where $m,n \geq 1$. It is

Dimer models are bipartite graphs on a torus,
originally introduced in the 60s as statistical mechanical models,
but recently attracts much attention as combinatorial objects
describing derived categories of coherent sheaves
on toric Calabi-Yau 3-folds. In the talk, I would like to give
a brief introduction to dimer models and their application
to homological mirror symmetry.