Conference Agenda

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Session Overview
Session
MS123, part 1: Asymptotic phenomena in algebra and statistics
Time:
Tuesday, 09/Jul/2019:
10:00am - 12:00pm

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

Presentations
10:00am - 12:00pm

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)

 

Strength and polynomial functors

Arthur Bik
University of Bern, Switzerland

Define the rank of an infinite-by-infinite matrix A as the supremum of the ranks of all its finite-size submatrices. This can be finite, which implies that the matrix A is the product of an infinite-by-n and an n-by-infinite matrix. Or else it is infinite. In this case, the set of matrices that can be obtained from A by a finite number of row and column operations is Zariski-dense in the space of all matrices.

Next, define the strength of a series f of fixed degree in infinitely many variables to be the minimal number of products of series of lower degree that sum up to f. Then f has a simpler description when its strength is finite. And when its strength is infinite, the set of series obtained from f by finitely many variable substitutions turns out to be dense in its ambient space.

This dichotomy exists in a much more general setting: an element of the inverse limit V of a polynomial functor either is in the image of a polynomial transformation from a simpler functor or has a dense orbit in V. This talk is about the complexity measure on V that associates to an element the minimal polynomial functor from which it arises. The results are part of joint work with Jan Draisma, Rob Eggermont and Andrew Snowden.

 

Asymptotics Proved by the Method of Cumulants

Hanna Döring
Universität Osnabrück, Germany

The method of cumulants is closely related to the method of moments both being a classical tool to prove central limit theorems. Having a good bound on the cumulants of a sequence of random variables, one can deduce precise asymptotics for the distribution function of the properly scaled random variables, and it implies large and moderate deviations as well as so-called mod-Phi-convergence. As an example, we will study dependency graphs.

 

FI-algebras: examples and counterexamples

Robert Krone
University of California at Davis, USA

We link the literature on algebras with an action of the infinite symmetric group to the literature on FI-algebras by identifying mutually adjoint functors in both directions. I will discuss some interesting examples and counterexamples of Noetherian and non-Noetherian FI-algebras and modules over them. Finitely generated modules over some Noetherian FI-algebras are Noetherian, and in this setting we can describe free resolutions and Betti numbers, but for other FI-algebras this fails. This is joint work with Jan Draisma and Alexei Krasilnikov.

 

Asymptotic behavior of chains of ideals with symmetry

Dinh Le Van
Universität Osnabrück, Germany

Chains of ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups arise in various contexts, including algebraic statistics and representation theory. In this talk, I will discuss the asymptotic behavior of some invariants of ideals in such chains, namely, the Krull dimension, the projective dimension, and the Castelnuovo-Mumford regularity. The Krull dimension is eventually a linear function whose slope can be described explicitly. We conjecture that the projective dimension and the Castelnuovo-Mumford regularity also grow eventually linearly, and provide linear bounds for these invariants. This is joint work with Uwe Nagel, Hop D. Nguyen, and Tim Römer.