Abstracts for Algebra, Geometry and Physics: a mathematical mosaic

We present a program which is designed to take combinatorial categorical data and produce geometric spaces
and classify universal algebraic operations. Specifically when applied to categories of graphs as founded by Borisov-Manin thought of as Feynman graphs, this program on one hand produces cubical and simplicial complexes corresponding to moduli spaces of Riemann surfaces and their Kontsevich-Penner compactifications.

Let $E_f: y^2=f(x)$ and $E_h: y^2=h(x)$ be elliptic curves over a field
$K$ of characteristic zero that are defined by cubic polynomials
$f(x)$ and $h(x)$ with coefficients in $K$.

Suppose that one of the polynomials is irreducible and the other
reducible.
We prove that if $E_f$ and $E_h$ are isogenous over an algebraic closure
$\bar{K}$ of $K,$ then they both are isogenous
to the elliptic curve $y^2=x^3-1$ over $\bar{K}$.

Yu. I. Manin greatly contributed to several areas of algebraic
geometry, in particular to questions of rationality, and to quantum
cohomology. I will speak about an application of the latter to the
former: the very general cubic fourfold is not rational. The main
ingredient is a blow up formula for genus 0 Gromov-Witten invariants
(the proof is expected soon). Another output is a definition of
"correct" cohomology and GW invariants for singular varieties.