Fri, 28 May 2021

12:00 - 13:00

Invariants for persistent homology and their stability

Nina Otter
(UCLA)
Abstract

One of the most successful methods in topological data analysis (TDA) is persistent homology, which associates a one-parameter family of spaces to a data set, and gives a summary — an invariant called "barcode" — of how topological features, such as the number of components, holes, or voids evolve across the parameter space. In many applications one might wish to associate a multiparameter family of spaces to a data set. There is no generalisation of the barcode to the multiparameter case, and finding algebraic invariants that are suitable for applications is one of the biggest challenges in TDA.

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the study of distances for multiparameter persistence modules is more challenging, as they rely on a choice of suitable invariant.

In this talk I will first give a brief introduction to multiparameter persistent homology. I will then present a general framework to study stability questions in multiparameter persistence: I will discuss which properties we would like invariants to satisfy, present different ways to associate distances to such invariants, and finally illustrate how this framework can be used to derive new stability results. No prior knowledge on the subject is assumed.

The talk is based on joint work with Barbara Giunti, John Nolan and Lukas Waas. 

Fri, 26 Feb 2021

12:00 - 13:00

The magnitude of point-cloud data (cancelled)

Nina Otter
(UCLA)
Abstract

Magnitude is an isometric invariant of metric spaces that was introduced by Tom Leinster in 2010, and is currently the object of intense research, since it has been shown to encode many invariants of a metric space such as volume, dimension, and capacity.

Magnitude homology is a homology theory for metric spaces that has been introduced by Hepworth-Willerton and Leinster-Shulman, and categorifies magnitude in a similar way as the singular homology of a topological space categorifies its Euler characteristic.

In this talk I will first introduce magnitude and magnitude homology. I will then give an overview of existing results and current research in this area, explain how magnitude homology is related to persistent homology, and finally discuss new stability results for magnitude and how it can be used to study point cloud data.

This talk is based on  joint work in progress with Miguel O’Malley and Sara Kalisnik, as well as the preprint https://arxiv.org/abs/1807.01540.

Wed, 24 Feb 2021

16:00 - 17:30
Virtual

The decomposability conjecture

Andrew Marks
(UCLA)
Abstract

We characterize which Borel functions are decomposable into
a countable union of functions which are piecewise continuous on
$\Pi^0_n$ domains, assuming projective determinacy. One ingredient of
our proof is a new characterization of what Borel sets are $\Sigma^0_n$
complete. Another important ingredient is a theorem of Harrington that
there is no projective sequence of length $\omega_1$ of distinct Borel
sets of bounded rank, assuming projective determinacy. This is joint
work with Adam Day.

Tue, 19 Jan 2021
16:00
Virtual

Hypergraph regularity and higher arity VC-dimension

Artem Chernikov
(UCLA)
Further Information

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

We generalize the fact that all graphs omitting a fixed finite bipartite graph can be uniformly approximated by rectangles (Alon-Fischer-Newman, Lovász-Szegedy), showing that hypergraphs omitting a fixed finite $(k+1)$-partite $(k+1)$-uniform hypergraph can be approximated by $k$-ary cylinder sets. In particular, in the decomposition given by hypergraph regularity one only needs the first $k$ levels: such hypergraphs can be approximated using sets of vertices, sets of pairs, and so on up to sets of $k$-tuples, and on most of the resulting $k$-ary cylinder sets, the density is either close to 0 or close to 1. Moreover, existence of such approximations uniformly under all measures on the vertices is a characterization. Our proof uses a combination of analytic, combinatorial and model-theoretic methods, and involves a certain higher arity generalization of the epsilon-net theorem from VC-theory.  Joint work with Henry Towsner.

Fri, 04 Dec 2020
18:45
Virtual

Symmetries and Strings of adjoint QCD in two dimensions

Konstantinos Roumpedakis
(UCLA)
Abstract

In this talk, we will review the notion of non-invertible symmetries and we will study adjoint QCD in two dimensions. It turns out that this theory has a plethora of such symmetries which require deconfinement in the massless case. When a mass or certain quartic interactions are tunrned on, these symmetries are broken and the theory confines. In addition, we will use these symmetries to calculate the string tension for small mass and make some comments about naturalness along the RG flow.

Fri, 19 Jun 2020

15:00 - 16:00
Virtual

Of monks, lawyers and airports: a unified framework for equivalences in social networks

Nina Otter
(UCLA)
Abstract

One of the main concerns in social network science is the study of positions and roles. By "position" social scientists usually mean a collection of actors who have similar ties to other actors, while a "role" is a specific pattern of ties among actors or positions. Since the 1970s a lot of research has been done to develop these concepts in a rigorous way. An open question in the field is whether it is possible to perform role and positional analysis simultaneously. In joint work in progress with Mason Porter we explore this question by proposing a framework that relies on the principle of functoriality in category theory. In this talk I will introduce role and positional analysis, present some well-studied examples from social network science, and what new insights this framework might give us.

Tue, 04 Feb 2020
14:15
L4

Tensor-triangular fields

Paul Balmer
(UCLA)
Abstract

I'll give a general introduction to tensor-triangular geometry, the algebraic study of tensor-triangulated categories as they appear in topology, geometry and representation theory. Then I'll discuss an elementary idea, that of a "field" in this theory, and explain what we currently know about them.

Wed, 06 Apr 2016

17:00 - 18:00
L1

Andrea Bertozzi - The Mathematics of Crime

Andrea Bertozzi
(UCLA)
Abstract
In the USA, law enforcement agencies have discovered that partnering with a team of mathematicians and social scientists from UCLA can help them determine where crime is likely to occur and so enable them to stop it before it happens.
 
In this lecture Andrea Bertozzi will tell the story behind her role on the UCLA team that developed a 'predictive policing' computer programme that zeros-in on areas that have the highest probability of crime. She will also discuss how mathematics play an increasing role in studying crime, especially gang crime. 

 

To book please email @email

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