Conference Agenda

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Session Overview
Session
MS123, part 2: Asymptotic phenomena in algebra and statistics
Time:
Tuesday, 09/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F007
30 seats, 59m^2

Presentations
3:00pm - 5:00pm

Asymptotic phenomena in algebra and statistics

Chair(s): Rob H. Eggermont (Technical University Einhoven, Netherlands), Uwe Nagel (University of Kentucky, USA), Tim Römer (Universität Osnabrück, Germany)

Across several branches of mathematics, the following fundamental question arises: given a sequence of algebraic structures with maps between them, can the entire sequence be characterized by a finite segment? Here the maps are comprising symmetries of the objects as well as morphisms between them. An affirmative answer leads to a description of all structures by using finite data only. There is a growing body of work that establishes the desired finiteness result in varied contexts. Nevertheless, instances where stability is not well understood include:

  • In algebraic statistics, a typical object is a toric ideal arising from a statistical model, and the maps correspond to shrinking the state space of the variables. The question is whether these ideals stabilize as n approaches infinity.
  • In commutative algebra, a typical object is a free resolution of an ideal over a polynomial ring in n variables, and the maps are induced by injections of the rings. The question is whether the resolutions stabilize as n varies.
  • In representation stability, a typical object is a cohomology group of the configuration space of n labeled points in a manifold M, and the maps between the groups correspond to relabeling or forgetting points. The question is whether these groups stabilize.
  • In tensor decomposition, a typical object is the variety of n-way tensors of bounded border rank, and the maps correspond to actions of products of general linear groups acting on the tensor factors and contractions relating the varieties for different n. The question is whether equations or higher-order syzygies of them stabilize.

The aim of the minisymposium is to build bridges between the varied mathematicians and the different areas investigating stability phenomena. We propose a two half-day minisymposium with 8 speakers total. The proposed speakers have all expressed interest in speaking at the symposium.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Quantitative Properties of Ideals arising from Hierarchical Models

Aida Maraj
University of Kentucky, USA

We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented. One goal is to study the invariance properties of families of ideals arising from hierarchical models with varying parameters. We will present classes of examples where we have information about an equivariant Hilbert series. This is joint work with Uwe Nagel.

 

Bounding degrees of generators for sequences of ideals

Mateusz Michalek
Max-Planck-Institute MiS, Germany

By the celebrated Hilbert's basis theorem an ideal in a polynomial ring has a finite number of generators - in particular, there exists a bound on the degree of the generators. Varieties however often come to us in sequences and it may be highly nontrivial to establish a uniform degree bound. The questions one asks can have different flavour: one can ask for a set-, scheme- or ideal-theoretic description, an explicit or existential bound. We will report on several conjectures and theorems inspired by applied algebra, in particular, algebraic phylogenetics.

 

Asymptotic Phenomena in the homology groups of graph configuration spaces

Eric Ramos
University of Oregon, USA

A graph is a 1-dimensional compact, connected CW-complex. Given a graph G we define its n-fold configuration space UConf_n(G) to be the topological space of n distinct and unlabeled points on G. The study of the asymptotic behaviors of graph configuration spaces have taken two distinct paths in the literature. The first, and more classically flavored, involves fixing the graph G and increasing the number of points. In this case, Work of An, Drummond-Cole, and Knudsen, as well as independent work of the speaker, have shown that the homology groups can be equipped with the structure of finitely generated modules over a certain polynomial ring associated to the graph G. The second approach, appearing in work of Lutgehetmann, White, Proudfoot, and the speaker, involves fixing the number of points being configured and allowing the graph to vary in some regular way. In this case one once again recovers finite generation results for the homology groups, although describing what they are finitely generated over requires one to introduce concepts from the representation theory of categories. In this talk we will outline the state of the art with regards to both of these approaches, as well as how these two types of asymptotic stability can sometimes be related to one another.

 

Mirror spaces and stability in the homology of Vandermonde varieties

Cordian Riener
The University of Tromsø

The level sets of the first d Newton power sums in R^k for some d k have been called Vandermonde varieties by Arnold and Giventhal. These varieties have a natural action of the symmetric group, which induces an action on their cohomology groups. By using a formula of Solomon we can study the decomposition of the resulting S_k-module and generalise some of the results obtained by Arnold and Giventhal on the homology modules of such varieties. These results in particular also yield some insight into the representational stability in the homology modules. (Based on joint works with Saugata Basu.)