About the research meeting
Conference schedule
|
Wednesday | Thursday
|
Friday |
| 8:30-9:00 | coffee and bagels | coffee and bagels | coffee and bagels |
| 9:00-10:00 | Bates | Maclagan |
Sidman |
| 10:00-10:30 | break | break | break |
| 10:30-11:30 | Leykin | Yu |
Smith |
| 11:30-1:00 | lunch | lunch | lunch |
| 1:00-2:00 | Erman | Gray |
Swinarski |
| 2:00-3:00 | Sam | Anderson |
Eisenbud |
| 3:00-3:30 | coffee | coffee | coffee |
| 3:30-4:30 | Raicu | He |
| 6:00 | | Dinner | |
All talks will be in Altgeld 314.
Dinner will be at the Big Grove Tavern.
Titles and abstracts:
Lara Anderson - "The geometry of heterotic/F-theory duality"
[abstract]
I will provide an overview of compactification in string theory and the
way in which the structure of the physical theory is intrinsically
intertwined with questions in differential and algebraic geometry. In
particular, I will highlight heterotic string theory and F-theory and
describe an analysis of a broad class of dual heterotic and F-theory
compactification that give to four-dimensional supergravity theories. This
study will link moduli spaces and deformation problems of higher rank
sheaves over 3- (complex) dimensional Calabi-Yau threefolds to the
geometry of 4-dimensional Calabi-Yau manifolds.
Dan Bates - "Parameter homotopies for applications"
[abstract]
The standard methods of numerical algebraic geometry have been useful for approximating solutions of polynomial systems in a number of applications. For
problems involving parameters, there is a special technique that is particularly efficient - the parameter homotopy. In this talk, I will introduce this fairly
simple concept, briefly describe a new software package - Paramotopy - that simplifies such runs (joint with Dan Brake and Matt Niemerg), and describe two
applications, one in biochemistry (joint with Brake, Jeremy Gunawardena, Benjmin Gyori, and Chris Nam) and one in engineering (joint with Brake, Tony Maciejewski,
and Vakhtang Putkaradze).
David Eisenbud - "Tying Betti numbers together"
[abstract]
If $M$ is a graded module over $k[x_1, \dots x_n]$ (or a
module over a regular local ring) that is annihilated by a regular
sequence $f_1,\dots f_m$, then the homotopies for the $f_i$ make
$Tor(M,k)$ into a module over an exterior algebra on $m$ generators.
This action is often trivial, but if $M$ is actually a (sufficiently)
high syzygy module over $S/(f_1,\dots,f_m)$, then the action is highly
nontrivial. I will describe what I know about this, and pose some open
questions. This is joint work with Irena Peeva.
Daniel Erman - "Tate resolutions for products of projective spaces"
[abstract]
TBD
James Gray - "Computation of moduli spaces of holomorphic bundles in heterotic
string theory"
[abstract]
I will describe the importance of understanding the moduli
spaces of holomorphic bundles to the physics of heterotic string theory. I
will set up some examples as explicit algebraic varieties and will describe
how we analyze them using techniques of computational commutative algebra.
I will emphasize those properties of these varieties that we can compute,
as well as those that we would like to know about and as yet can not
obtain.
Yang-Hui He - " In Praise of Macaulay2: Some perspectives from Theoretical Physics"
[abstract]
We give an overview of some classes of problems in theoretical high
energy and mathematical physics wherein Macaulay2 and the methods of
computational algebraic geometry have been an indispensable tool.
Topics will include Calabi-Yau geometries in the context of string
compactifications, moduli space of gauge theories, bipartite graphs on Riemann surfaces and associated
toric geometry, as well as the interplay between operators, quivers and modularity.
Anton Leykin - "Numerical algorithms for detecting embedded components"
[abstract]
Given a finite set of polynomials one can describe the corresponding
(complex affine) variety using numerical data derived by methods of
numerical algebraic geometry. Can one extend numerical techniques to
deliver not only a set-theoretic but also a scheme-theoretic description?
On the quest to construct numerical recipes for schemes, we produce (and
implement in Macaulay2) algorithms to detect whether a variety presented
numerically corresponds to an associated component of a polynomial ideal.
(Joint work with Robert Krone)
Diane Maclagan - "Computational algebra for tropical geometry"
[abstract]
In this talk I will discuss two variants of Groebner bases
that are relevant in tropical geometry. The first is for ideals in a
polynomial ring where the coefficient field has a valuation. In this
case the term order is modified to take the term order into account.
There is an algorithm to compute this (joint with Andrew Chan) that
has been implemented in Macaulay 2. The second variant concerns
ideals in the semiring of tropical polynomials, where multiplication
is addition and addition is minimum. I will discuss the first
variant, and outline some of the computational challenges of the
second.
Claudiu Raicu - "The syzygies of some thickenings of determinantal varieties"
[abstract]
The space of m x n matrices admits a natural action of the group GL_m x GL_n via row and column operations on the matrix entries. The invariant closed subsets are the determinantal varieties defined by (reduced) ideals of minors of the generic m x n matrix. The minimal free resolutions for these ideals are well-understood by work of Lascoux and others. There are however many more invariant ideals which are non-reduced, and whose syzygies are quite mysterious. These ideals correspond to nilpotent structures on the determinantal varieties, and they have been completely classified by De Concini, Eisenbud and Procesi. In my talk I will recall the classical description of syzygies of determinantal varieties, and explain how this can be extended to a large collection of their thickenings. Joint work with Jerzy Weyman.
Steven Sam - "Moduli spaces of Coble hypersurfaces"
[abstract]
I will explain two related projects where Macaulay2 has been
used. The first is joint with Laurent Gruson and concerns the study of
Coble cubics, which is closely tied to the study of abelian surfaces.
The moduli space can be constructed explicitly via invariant theory. I
will explain how Macaulay2 was useful for getting refined information
about the construction and for the study of degenerations. The second is
joint with Qingchun Ren, Gus Schrader, and Bernd Sturmfels, and concerns
Coble quartics, which are related to abelian 3-folds. An embedding of
the moduli space was described by Coble and also by Dolgachev-Ortland,
and we use Macaulay2 to calculate the defining ideal and prove basic
properties, like that it is Gorenstein.
Jessica Sidman - "Rigidity Theory: An algebraic perspective"
[abstract]
The central question in rigidity theory is to determine whether a structure
built out of rigid parts assembled according to certain geometric
constraints is rigid or flexible. I will give a brief introduction to
rigidity theory and the role that algebra and combinatorics can play in
analyzing structures. Specifically, I will discuss the role of
combinatorics in characterizing structures that are generically rigid and
the role of algebra in finding "special positions" which admit internal
motions. I will also discuss particular issues that arise in body-and-cad
rigidity theory, which was designed to model systems coming from constraint
based computer aided design software.
This is joint work with James Farre, Helena Kleinschmidt, Audrey St. John,
Stephanie Stark, and Louis Theran.
Gregory G. Smith - "Cones of Hilbert functions"
[abstract]
In this talk, we examine the closed convex hull of various collections of
Hilbert functions. Working over a standard graded polynomial ring with
modules
that are generated in degree zero, we describe the supporting hyperplanes and
extreme rays for the cones generated by the Hilbert functions of all modules,
all modules with bounded a-invariant, and all modules with bounded
Castelnuovo-Mumford regularity.
David Swinarski - "State polytopes and geometric invariant theory"
[abstract]
A major open question in algebraic geometry is to investigate the birational geometry of the moduli space of stable genus $g$ algebraic curves
$\overline{\mathcal{M}}_g$. I will survey work by Bayer, Morrison, Hassett, Hyeon, Lee, Swinarski, Alper, Fedorchuk, Smyth, and Deopurkar on how calculations of
state polytopes of ideals can be used to prove geometric invariant theory stability, which in turn may help us to construct and analyze new birational models of
$\overline{\mathcal{M}}_g$.
Josephine Yu - "Stable Intersection for Tropical Varieties"
[abstract]
Tropical cycles are pure dimensional rational fans that satisfy a
balancing condition. They form a commutative ring with a product called
stable intersection, which is closely related to the intersection
product on toric varieties and has been studied by many people. We give
several characterizations of stable intersection and establish their
fundamental properties using elementary means. We show that the stable
intersection of two tropical varieties is the tropicalization of the
intersection of the classical varieties after a generic rescaling. We
will discuss relations to McMullen's polytope algebra and answer some
questions regarding connectivities of tropical varieties.