Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS143, part 1: Algebraic geometry in topological data analysis
Time:
Tuesday, 09/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F006
30 seats, 57m^2

Presentations
10:00am - 12:00pm

Algebraic geometry in topological data analysis

Chair(s): Nina Otter (UCLA, United States of America)

In the last 20 years methods from topology, the mathematical area that studies “shapes", have proven successful in studying data that is complex, and whose underlying shape is not known a priori. This practice has become known as topological data analysis (TDA). As additional methods from topology still find their application in the study of complex structure in data, the practice is evolving and expanding, and now moreover draws increasingly upon data science, computer science, computational algebra, computational topology, computational geometry, and statistics.

While ideas from category theory, sheaf theory and representation theory of quivers have driven the theoretical development in the past decade, in the last years ideas from commutative algebra and algebraic geometry have started have started to be used to tackle some theoretical problems in TDA. The aim of the minisymposium is to seize this momentum and to bring together experts in algebraic geometry and researchers in topological data analysis to explore new avenues of research and foster research collaborations.

 

(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)

 

Algebraic geometry in topological data analysis: an overview

Nina Otter
UCLA, United States of America

In this talk I will give an overview on how techniques and ideas from algebraic geometry have been used so far in topological data analysis, and discuss possible new avenues of research.
 

Applications of Groebner bases

Natalia Iyudu
University of Edinburgh, United Kingdom

I will discuss the Groebner bases theory and its application to calculation of Hilbert series, and related invariants of quadratic algebras and operads. Examples where this technique can be used and on these bases, for example, Koszulity proved, will be given. It can be applied to persistent homology problems, to problems in neuroscience, etc.

The Groebner technique itself originated in computer science, more precisely, in computer algebra, but as we will see, after proper algebraic formulation it can serve for proving some structural and homological results about algebras presented by generators and relations. This in turn can serve for the solution of applied problems, for example, by the study of neural codes as pseudo monomial ideals.

 

Decomposition of 2-parameter persistence modules

Steve Oudot
Inria Saclay, France

Decomposition theorems are one of the pillars of persistence theory. They yield complete discrete invariants for persistence modules, which can be used as descriptors for data in downstream applications. While the case of 1-parameter persistence modules is by now well-understood, the multi-parameter case remains mostly unexplored and appears to be much more complicated. In this talk I will focus on the 2-parameter case, and provide an overview of the current state of the art together with some perspectives.

 

Classification of filtered chain complexes

Barbara Giunti1, Wojciech Chacholski2, Claudia Landi3
1Università di Pavia, Italy, 2KTH, Stockholm, 3Università di Modena e Reggio Emilia

Persistent homology has proven to be a useful tool to extract information from data sets. Homology, however, is a drastic simplification and in certain situations might remove too much information. This prompts us to study filtered chain complexes. We prove a structure theorem for filtered chain complexes and list all possible indecomposables. We call these indecomposables interval spheres and classify them into three types. Two types correspond respectively to finite and infinite interval modules, while the third type is unseen by homology. The structure theorem states that any filtered chain complex can be written as the unique sum of interval spheres, up to isomorphism and permutation. The proof is based on a hierarchy of full subcategories of the category of filtered chain complexes. Such hierarchy suggests an algorithm for decomposing filtered chain complexes, which also retrieves the usual persistent barcodes. This approach offers a way to retrieve more geometrical aspects of data: while homology cannot tell the difference between a point and a disk, our decomposition provides a tool to count the contractible parts of the data, thus, we can obtain not only topological but also geometrical information.