Mini-courses
Harm Derksen (Northeastern University, USA)
Michael Lesnick (State University of New York, SUNY, USA)
Research Talks
Ben Blum-Smith (John Hopkins University, USA)
Peter Bubenik (University of Florida, USA)
Abigail Hickok (Columbia University, USA)
Harold Nieuwboer (University of Copenhagen, Denmark)
Rafael Oliveira (University of Waterloo, Canada)
Jose Perea (Northeastern University, USA)
Francesca Tombari (Max Planck Institute, Germany)
Mini-courses:
Harm Derksen (Northeastern University, USA)
Title: Invariant Theory, Quiver Representations and Complexity
Abstract: The focus of these lectures is on Geometric Invariant Theory, quivers, and complexity. We start with an introduction to representations of quivers and quivers with relations. Some topics include indecomposable representations, finite/tame/wild type quivers and some homological algebra. In geometric invariant theory one uses polynomial invariants to construct quotients of varieties with a group action. Using Geometric Invariant Theory (GIT) we can construct moduli spaces of representations of a fixed quiver and dimension vector. We will discuss some combinatorial aspects of GIT quotients of quiver representations, constructive aspects, and connections to algebraic complexity.
Michael Lesnick (State University of New York, SUNY, USA)
Title: Representation Theory and Topological Data Analysis
Abstract: Persistent homology, the flagship tool of topological data analysis (TDA), analyzes the shape of a data set by constructing a filtered topological space and applying homology with field coefficients. The isomorphism type of the resulting diagram of vector spaces is fully described by a simple invariant called a barcode.
Often, a single filtered space does not adequately capture the structure of interest in the data. One is then led to consider multiparameter persistence, which associates to the data a space equipped with a multiparameter filtration. In contrast to the 1-parameter case, the homology of this object has wild representation type. Hence, there is no fully satisfactory definition of a barcode in the multiparameter setting. This makes extending the 1-parameter persistence theory and methodology to multiple parameters an interesting and challenging problem.
In the last several years, multiparameter persistence has become one of the most active areas of research within TDA, with exciting progress on several fronts. These talks will introduce this subject and survey some of this progress. We will aim to understand the difficulties created by the “wildness” of multiparameter persistence, the ideas that have been developed to circumvent these difficulties, and the challenges that remain. Among other topics, we will cover stability theory, signed barcodes, and the computational and practical aspects.
For more information and references, click here!
Research Talks:
Ben Blum-Smith (John Hopkins University, USA)
Title: Degree bounds for rational invariants
Abstract: Degree bounds have a long history in invariant theory. The Noether bound on the degrees of algebra generators for a ring of invariants is over a century old, and there is a vast literature sharpening and generalizing it. In the last two decades, there has also been an active program on degree bounds for invariants which are able to distinguish orbits as well as algebra generators can (known as separating invariants).
In this talk I make the case that generators for the field of rational invariants represent an exciting avenue for research on degree bounds as well. I present new lower and upper bounds. It will transpire that even the case of G=Z/pZ, uninteresting from the point of view of generating and separating invariants, has a story to tell for rational invariants. I argue that this domain of inquiry is both interesting in itself and well-motivated by applications.
This talk is based on joint work with Afonso Bandeira, Joe Kileel, Jonathan Niles-Weed, Amelia Perry, Alexander Wein, Thays Garcia, Rawin Hidalgo, Consuelo Rodriguez, Alexander Kirillov Jr., Sylvan Crane, Karla Guzman, Alexis Menenses, and Maxine Song-Hurewitz.
Peter Bubenik (University of Florida, USA)
Title: The derived category of persistence modules and exact weights and path metrics for triangulated categories
Abstract: In topological data analysis we consider diagrams of simplicial complexes. Taking simplicial chain complexes with coefficients in a field, we obtain diagrams of chain complexes of vector spaces. I will discuss how from the category of these diagrams we obtain the derived category of persistence modules, and how this is a triangulated category. I will define exact weights on a triangulated category, give some of their properties, and show how they can be used to define path metrics and Wasserstein distances.
This is joint work with Jose Velez Marulanda.
Abigail Hickok (Columbia University, USA)
Title: Persistence Diagram Bundles: A generalization of vineyards for multiparameter topological data analysis
Abstract: Topological data analysis (TDA) is a way to understand the “shape” of a data set by using algebraic topology. The primary tool of TDA is persistent homology, which tracks the connected components, holes, and higher-dimensional homology classes as they emerge and disappear at increasing scale. I’ll start by surveying methods for analyzing how the topology of a data set changes as multiple parameters vary, which is a very active area of research. In particular, I’ll discuss a construction called “persistence diagram bundles” that I introduced, as well as its relation to other objects in TDA such as “vineyards”, “multiparameter persistent homology”, and the “persistent homology transform”.
Harold Nieuwboer (University of Copenhagen, Denmark)
Title: Moment polytopes in action
Abstract: Moment polytopes are a classical object associated with actions of reductive groups on algebraic varieties appearing in geometric invariant theory, and capture. They also have strong connections to for example Horn's problem on spectra of sums of Hermitian matrices, asymptotic (non-)vanishing of Kronecker coefficients, and (asymptotic) properties of tensors and quantum states.
I will survey some recent developments on practical determination of properties of moment polytopes, in particular rigorous algorithms for determining (semi)stability, moment polytope membership, and certain orbit problems.
Rafael Oliveira (University of Waterloo, Canada)
Title: The null-cone problem in invariant theory and its connections to computer science and other areas
Abstract: In this talk we will explore the null-cone problem in invariant theory from the algebraic and from the optimization perspectives. Combining both of these lenses, we will see how the null-cone problem is intrinsically connected to important problems in computer science (for instance the rational identity testing problem and perfect matching in bipartite graphs), as well as problems in other areas of science, such as functional analysis (Brascamp-Lieb inequalities), the one-body quantum marginal problem, and statistics (sample complexity of Matrix and Tensor Normal Models).
Jose Perea (Northeastern University, USA)
Title: DREiMac: Dimensionality Reduction with Eilenberg-Maclane Coordinates
Abstract: Dimensionality reduction is the machine learning problem of taking a data set whose elements are described with potentially many features (e.g., the pixels in an image), and computing representations which are as economical as possible (i.e., with few coordinates). In this talk, I will present a framework to leverage the topological structure of data (measured via persistent cohomology) and construct low dimensional coordinates in classifying spaces consistent with the underlying data topology.
Francesca Tombari (Max Planck Institute, Germany)
Title: Koszul complexes and tame functors indexed by realisations
Abstract: In this talk, we discuss tame functors indexed by posets and their minimal free resolutions. These functors are generalisations of persistence modules in TDA. We show classes of (infinite) indexing posets of functors for which the homologies of Koszul complexes give the Betti numbers of the functors. We show these results for functors indexed by upper semilattices and realisations, a new class of poset we introduce. Tameness is also a fundamental condition in this context. Tame functors assume different values only at a finite number of indices and are constant otherwise. This allows us to reduce to the finite case, where Koszul complexes are available.