UCLA

TBA

The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.

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. 

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.

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.

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.

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.

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.