August 1
(Monday) ～ August 5 (Friday), 2016
ABSTRACT
| August, 1 (Mon) | August, 2 (Tue) | August, 3 (Wed) | August, 4 (Thu) | August, 5 (Fri) |
| 9:30--10:30 | Uwe Nagel (University of Kentucky) "Equivariant Hilbert Series" |
| Ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid of strictly increasing functions arise in various contexts. We study such an ideal using an ascending chain of invariant ideals. We establish that the associated equivariant Hilbert series is a rational function in two variables. This is used to prove that the Krull dimensions and multiplicities of ideals in such an invariant filtration grow eventually linearly and exponentially, respectively. Furthermore, we determine the terms that dominate this growth. This may also be viewed as a method for assigning new asymptotic invariants to a homogenous ideal in a noetherian polynomial ring. | |
| 10:45--11:45 | David Cook II （Eastern Illinois University）
"Smooth toric varieties satisfying many Laplace equations" |
| Mezzetti, MirLo-Roig, and Ottaviani showed that in some cases the failure of the weak Lefschetz property for a graded ideal can be used to produce a variety satisfying a Laplace equation. The distance from maximal rank for the Lefschetz map is the Lefschetz defect. In joint work with Uwe Nagel, we combinatorially construct ideals having asymptotically optimal Lefschetz defects. In turn, we thus have smooth toric varieties satisfying many Laplace equations. | |
| 13:15--14:15 | Andreas Paffenholz （Technische Universität Darmstadt）
"Structure and Classifications of Lattice Polytopes" |
|
Lattice polytopes, i.e. polytopes whose vertices are contained in the
integer lattice, are an important subclass of polytopes with
applications in number theory, optimization, and algebraic geometry,
among others. Lattice polytopes naturally correspond to toric
varieties, and many properties of the polytope or variety are
reflected on the other side. Various subclasses of lattice polytopes are of particular importance in applications. Fano polytopes, i.e. polytopes with only one interior lattice point, correspond to Fano toric varieties. They are reflexive if the variety is also Gorenstein. Smooth polytopes are those of nonsingular varieties. Knowledge of the structure of such particular classes are an important step to understand more general classes. To approach these polytopes we can study their combinatorial or geometric structure, and we can try to computationally classify subfamilies to deduce or test conjectures. In my talk I will report on currently known structural properties and known classifications for lattice polytopes. I will mainly focus on Fano polytopes and some subfamilies, but also discuss some other classes. |
|
| 14:30--15:30 | Thomas Prince （Imperial College London）
"Complete intersection constructions for certain toric degenerations of toric Fano varieties" |
| Smoothing toric Fano varieties is expected to provide a rich source of as yet unknown Fano varieties. Rather than attempt a systematic study of the deformations of a toric Fano variety, we restrict attention to deformations coming from toric complete intersection models. Our main result is the construction of a complete intersection model from any choice of extra data called a 'scaffolding' on a Fano polytope via a process called Laurent Inversion (studied jointly with Coates and Kasprzyk), inspired by ideas in Mirror Symmetry. We show how 527 new Fano 4-folds found (by a different method) in joint work with Coates, Kasprzyk appear as examples of this technique, and how this method might lead to the discovery of further Fano 4-folds. We review the situation for 3-folds and state some partial results toward combinatorially detecting when a scaffolding produces a smooth variety. | |
| 15:45--16:45 | Fu Liu (University of California, Davis)
"Ehrhart positivity and BV-α-positivity for generalized permutohedra" |
| The Ehrhart polynomial
$i(P,m)$ of an integral polytope P counts the number
of lattice points in dilations of P. We say a polytope has Ehrhart
positivity if all the coefficients in its Ehrhart polynomial are positive.
There are few families of polytopes that are known to have Ehrhart
positivity. De Loera et al conjectured that matroid polytopes have Ehrhat
positivity. In our work, we consider generalized permutohedra, which
contain matroid polytopes, and conjecture they all have Ehrhart
positivity.
We study this conjecture by using the existence of McMullen formula: $|P \cap Z^n| = \sum_{F} \alpha(F, P) nvol(F)$, where the sum is over all faces of P, and $\alpha(F, P)$ are rational numbers depending only on the feasible cone of P at F. The function $\alpha(F, P)$ is not uniquely determined and different constructions have been discovered. We explore a particular one given by Berline and Vergne, which we call BV-$\alpha$. We conjecture that BV-$\alpha$s arising from generalized permutohedra are all positive, which easily imply our first conjecture. We show our conjectures hold for dimension up to 6, and for faces of codimension up to 3, as well as give two equivalent statements to the second conjecture in terms of mixed valuations and Todd class, respectively. This is joint work with Federico Castillo. |